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This open access book explores the amazing similarity between paths taken by people and many other things in life, and its impact on the way we live, teach and learn. Offering insights into the new scientific field of paths as part of the science of networks, it entertainingly describes the universal nature of paths in large networked structures. It also shows the amazing similarity in the ways humans and other – even nonliving – things navigate in a complex environment, to allow readers to easily grasp how paths emerge in many walks of life, and how they are navigated. Paths is based on the authors recent research in the area of paths on networks, which points to the possible birth of the new science of “paths” as a natural consequence ‘and extension) of the science of “networks.” The approach is essentially story-based, supported by scientific findings, interdisciplinary approaches, and at times, even philosophical points of view. It also includes short illustrative anecdotes showing the amazing similarities between real-world paths and discusses their applications in science and everyday life. Paths will appeal to network scientists and to anyone interested in popular science. By helping readers to step away from the “networked” view of many recent popular scientific books and start to think of longer paths instead of individual links, it sheds light on these problems from a genuinely new perspective. --------------------------------------------------------------------------------- The path is the goal. The essence behind this short sentence is known to many people around the world, expressed through the interpretations of some of the greatest thinkers like Lao-Tze and Gandhi. It means that it is the journey that counts, not the destination. When speaking about such subjective and intangible things, philosophy and religion are some of the only approaches that are addressed. In this book, the authors address this conventional wisdom from the perspective of natural science. They explore a sequence of steps that leads the reader closer to the nature of paths and accompany him on the search for “the path to paths”.
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SHORTEST
YOURS
SHORTEST
YOURS
Paths
András Gulyás
Zalán Heszberger
József Biró
Why is life fi lled with so many
detours?
Paths
András Gulyás • Zalán Heszberger • József Biró
Paths
Why is life filled with so many detours?
András Gulyás
Budapest University of Technology
and Economics
Budapest, Hungary
Zalán Heszberger
Budapest University of Technology
and Economics
Budapest, Hungary
József Biró
Budapest University of Technology
and Economics
Budapest, Hungary
ISBN 978-3-030-47544-4 ISBN 978-3-030-47545-1 (eBook)
https://doi.org/10.1007/978-3-030-47545-1
This book is an open access publication.
© The Editor(s) (if applicable) and The Author(s) 2021
Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 Inter-
national License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation,
distribution and reproduction in any medium or format, as long as you give appropriate credit to the
original author(s) and the source, provide a link to the Creative Commons license and indicate if changes
were made.
The images or other third party material in this book are included in the book’s Creative Commons
licence, unless indicated otherwise in a credit line to the material. If material is not included in the book’s
Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the
permitted use, you will need to obtain permission directly from the copyright holder.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors, and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or
the editors give a warranty, expressed or implied, with respect to the material contained herein or for any
errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional
claims in published maps and institutional affiliations.
Cover illustration: Cover Image based on the original drawing by Lajosné F. Kiss and printed with his
permission.
This book is published under the imprint Birkhäuser, www.birkhauser-science.com, by the registered
company Springer Nature Switzerland AG.
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
The path is the goal...—Mahatma Gandhi
To our loving wives and children:
Nusi, Bandika, Gabi
Andris, Dóri, Tündi
Lili and Zsófi
Foreword: Paths We Live By
During the last generation and with the advent of the interconnectedness of people,
institutions, and ideas through computer webs, network-based metatheories of all
aspects of sciences started to flourish from metabolic mechanisms, to authors co-
citations. Now, network science of different levels of abstraction flourishes in
mathematics, physical, and biological models, as well as in sociological theories.
There are many shining stars of Hungarianscience on this path, from László Lovász,
László Barabási Albert, János Kertész to Péter Csermely and György Buzsáki.
This little book of another Hungarian trio—András Gulyás, Zalán Heszberger, and
József Bíró—takes another look at these developments.Their perspective is not the
network itself but the routes taken by neural firing patterns, handshakes, or word
activations to arrive from one node in a network to another.
Paths have been the central idea of many social sciences for more than a hundred
years. One of the most fundamental methods of the comparative psychology of
animal cognition has been and continues to be the maze learning introduced in
1901 to psychology. For a long time, we treated it as a way to study the universal
mechanism of learning. Today, we realize that it is the key to understand how
mammals are able to internalize a map of different possible paths in their world
full of orientational cues and object valences. Paths are used by animals to arrange
the knowledge about their activities like where to go and what to do.
These research paths led to the search for neural paths in the brain by assuming
specialized brain structures responsible for the long time assumed cognitive maps.
This book presents the paths connecting the words in the lexicons and in the mind,
the paths leading from corona to death during the pandemic. Several researchers,
including György Buzsáki at New York University, hope that these later cognitive,
meaning-based paths are tied to the same brain networks as the forest paths of the
animals.
This book also presents the third important path system in humans, the one that
takes one through the common past: instrumental and emotional contacts to another
person.
ix
x Foreword: Paths We Live By
This readable and easily accessible little book fills the reader with hopes and
promises towards the future of network research where paths shall be found to relate
the personal, conceptual, and neural networks.
May 3, 2020, at Budakeszi, Hungary. In the middle of the coronavirus lockdown
Psychologist and Linguist Member of the Csaba Pléh
Hungarian Academy of Sciences and Academia Europaea
Budakeszi, Hungary
Foreword: The Longest Journey
“Tell me, Master, is there a single word that one can follow throughout one’s
life?”—turned once one of his disciples to Confucius. The master replied, “Isn’t
mutuality such a word?” The disciple bowed silently and left contentedly.
The great and solemn word of mutuality is also known in our Western culture;
most often, it denotes some bilateral relationship. This relationship is mostly
considered valid by the partners for the duration of a specific ongoing action. With
its announced application, they demonstrate that they take each other’s aspects into
account to the maximum. Mutuality is mainly used in connections between you and
me,orme and the others. (According to many, this lean bilateralism is rooted deep
in the idea of monotheism.) After the action (discourse or act) between the two
parties takes place, the concept becomes invalid and practically ceases until the next
situation.
In the interpretation of Confucius and his followers, mutuality is a much more
meaningful word. In Chinese antiquity, this term referred to an entire network of
mutual relationships, a combination of paths and detours, decisions, and choices,
along with the consequences and repercussions that follow them. In contrast to
the Western-style action-like, casual-use interpretation,Confucius and his followers
never treated mutuality as a restricted bilateral relationship. For them, mutuality was
a deep principle underlying the Universe, a Weltanschauung.
This ancient Asian approach has another defining feature: The correlative
approach with modal logical judgment. Such thinking focuses on the correlated
relationship of adjacent things and allows for multiple valid judgments (statements)
at the same time. This is in high contrast to the Western binary logic, where only
one of the two statements can be correct, rejecting the other as incorrect. Any third
statement (if raised at all) is ruled out. This is the principle of the excluded third
by Aristotle. In any decision-making situation, the ancient Asian logic, however,
always recommends the application of the intermediate third’s law. This law means
that three or even more statements can be valid simultaneously. Such prudence of
minding all chances comes from the way of seeing things to be interdependent (or
in correlation to each other).
xi
xii Foreword: The Longest Journey
Well, this is precisely the principle that has been missing from European
thinking since time immemorial! It takes into account the intermediate third and
fourth statements: the permissive idea of the concurrent validity of more than
one statement. In our decision-making situations, since Aristotle,we consider a
statement to be valid (correct) or not with no further option.
I wonder why? Because we need the most effective solution in all cases.
Decisions that do not provide the most effectiveand quickest solution are considered
to be detours or misguided paths in the eyes of the progress-hungry, impatient
Western hero.
This way of thinking took us to where we are. We have progressed, progressed
undoubtedly, but maybe too fast, so fast that we have probably run over the finish
line already.And there is no way back; it is impossible to correct. Our only option is
to slow down, that is, if we have a drop of wit, at least we do not rush into a not at all
promising future robbed by increased efficiency, effectiveness, growth, and almost
completely deprived of our physical and mental living conditions.
We do it smarter if we slow down our progress, if we choose a detour to our
goals. This slowdown gives us a chance to keep a common sense of the concept
of mutuality in the broadest sense, an opportunity to be attentive, smart, and even
to spare ourselves and each other (not just action-like). During the delicate, careful
trying, and enjoyable tasting of the paths and detours, we may even get to know
human nature better. So far, we did not have time to figure out whether such a thing
exists at all. “Human science would finally be needed,sighed Ortega y Gasset
bitterly after the second great suicide attempt of the genus humanum in the twentieth
century.
And if we understand the poet’s word better, let us listen to Constantine Cavafy,
who advises that “When you set out on your journey to Ithaca, pray that the road
is long, full of adventure, full of knowledge.The longest journey gives you the
greatest gift of the city.
This book is the wisest guide to paths: an easy-to-understand book of intact,
unharmed presence, orientation, and amicable arrangement in the risky world of a
million choices. It provides exactly the prudent and relational-centered approach
that Western thinking needs the most today.
Orientalist, Writer, Hungarian Media Person László Sári (a.k.a. Su-la-ce)
Budapest, Hungary
April 10, 2020
Acknowledgements
We would like to say special thanks to our parents and grandparents for showing
us their paths and helping to smooth ours. We must express our very profound
gratitude for many of the useful ideas and for the fruitful discussions to Zsófia
Varga, Attila Csoma, Antal Heszberger, Gabriella F. Kiss, Attila K˝
orösi, István
Pelle, Dávid Szabó, and Gábor Rétvári. We are grateful for the careful perusal and
kind comments of István Papp, Csaba H˝
os, Claudia Molnár, Attila Mertzell, László
Gulyás, Mariann Slíz, Alessandra Griffa, Andrea Avena-Königsberger, Levente
Csikor, Márton Novák,Dávid Klajbár, Valentina Halasi, Máté Csigi, Erzsébet Gy ˝
ori,
Tamás Csikány, István Bartolits, Alija Pasic, Alexandra Balogh, Rudolf Horváth,
and Mária Marczinkó.
A special thanks go to Anne Comment, our ever-patient Publishing Manager
in coordinating the book project, and to Kathleen Moriarty, our Copy Editor for
her careful reading and invaluable help in revising the final manuscript. Special
gratitude to Lajosné F. Kiss for creating the fantastic illustrations.
xiii
Disclaimer
This book does not intend to communicate any scientific consensus about paths.
In fact, there is no consensus about paths. Ideas presented here, although mostly
founded on real-world data, primarily reflect the authors’ subjective (sometimes
speculative) image about the world. This work is intended to entertain, inspire, and
persuade the reader to think critically about the nature of paths as taken by people
as well as many other entities in life.
xv
Contents
1 Introduction: Long and Winding Roads .................................. 1
2 Everybody Loves Roundabouts ............................................ 5
3 The Forest of Alternative Choices ......................................... 11
4 Straight to the Point: A Short Chapter About the Shortest
Paths .......................................................................... 17
5 Finding Your Way Through the Maze .................................... 21
6 On the Trail of Nature: Collecting Scientific Evidence .................. 29
7 The Universal Nature of Paths ............................................ 45
8 Amazing Scientific Discoveries: Aspirin, Cattle, Business
Communication and Others ................................................ 67
9 Paths to the Way We Live, Teach and Learn ............................. 73
10 The Path is the Goal! ........................................................ 79
Coda ................................................................................ 83
Bibliography ....................................................................... 85
xvii
Content Path
xix
List of Figures
Fig. 1.1 The cover of Italian popular tales by Thomas Frederic
Crane [Published by the Riverside Press, Cambridge,
Massachusetts, 1885] ................................................. 2
Fig. 1.2 Ma Yuan, “Walking on a Mountain Path in Spring” [With
the permission of the National Palace Museum of Taiwan.] ........ 3
Fig. 2.1 Howard Prince’s (played by Woody Allen) portrait
hand-drawn by Lajosné F. Kiss [With the permission of
Lajosné F. Kiss] ....................................................... 6
Fig. 2.2 The working of a proxy server ....................................... 7
Fig. 2.3 A mind map about global warming by Jane
Genovese [With the permission of Jane Genovese.].
http://learningfundamentals.com.au/.................................. 9
Fig. 3.1 Euler’s Fig. 1 for the seven bridges of Königsberg
problem from ‘Solutio problematis ad geometriam situs
pertinentis,’ Eneström 53 [source: MAA Euler Archive;
http://eulerarchive.maa.org/docs/originals/E053.pdf] ............... 12
Fig. 3.2 Euler’s idea of abstracting away the network underlying the
Seven Bridges of Königsberg puzzle ................................. 12
Fig. 3.3 Logical map of the ARPANET (the ancestor of the Internet)
from 1977 [source: The Computer History Museum;
https://computerhistory.org/] .......................................... 14
Fig. 4.1 Shortest paths on a simple network .................................. 18
Fig. 4.2 The variability of shortest paths ...................................... 18
Fig. 5.1 Military hierarchy ..................................................... 22
Fig. 5.2 Military hierarchy. Shortest path vs. the regular path ............... 23
Fig. 5.3 Illustrative structure of the relevant parts of the German
Army in 1943 .......................................................... 24
xxi
xxii List of Figures
Fig. 5.4 An embryonic model of the Internet, where Castle Rock
connects to nearby Salem’s Lot directly and Dunwich in
England through Main County Trans-Atlantic (MCT) .............. 26
Fig. 5.5 A tiny model of the Internet initiated by the people at Castle
Rock to communicate with the outside world by connecting
nearby town Salem’s Lot and Dunwich in England by using
the transit services of Main County Trans-Atlantic (MCT)
and Canadian Federal Co. (CF) as a backup route .................. 27
Fig. 6.1 The flight network of the US ......................................... 31
Fig. 6.2 The word morph network is a network of three-letter English
words, in which two words are connected by a link if they
differ only in a single letter. For example, “FIT” is linked to
“FAT” as they differ only in the middle letter, but “FIT” and
“CAT” are not neighbors in this network since more than
one letter differs in them .............................................. 33
Fig. 6.3 A word morph game example with source and target words
“YOB” and “WAY”. A shortest solution is displayed in red,
while a solution given by a specific player is shown in green ...... 34
Fig. 6.4 A simple network of computers ...................................... 35
Fig. 6.5 Constructing a network based on its paths, Phase 1 ................. 38
Fig. 6.6 Phase 2 ................................................................. 38
Fig. 6.7 Phase 3 ................................................................. 38
Fig. 6.8 The Vitruvian Man depicting normal human body
proportions is often used to symbolize The Human Genom
Project as Leonardo da Vinci created it in 1490, exactly a
half a millennium before the project began in 1990. [Public
Domain; Leonardo da Vinci via Wikimedia Commons] ............ 40
Fig. 6.9 The human neural network in the brain reconstructed via
DSI, from Patric Hagmann et al. “Mapping the structural
core of human cerebral cortex”. In: PLoS biology 6.7
(2008), e159 ........................................................... 41
Fig. 6.10 Inferring path from the human brain using the shortest path
assumption ............................................................ 42
Fig. 6.11 Shortest path over the active subnetwork at a given time
instant .................................................................. 42
Fig. 6.12 Empirical paths in the human brain .................................. 43
Fig. 7.1 Six degrees of separation. The poster of the play created by
James McMullan. [With the permission of James McMullan] ...... 47
Fig. 7.2 The illustration of path stretch. The green path is the
shortest, while the red and blue paths has a stretch of 1 and
2 respectively .......................................................... 48
List of Figures xxiii
Fig. 7.3 A simplified sketch on the measured stretch of the paths
relative to the shortest one found in our real-life systems.
While most of the empirical paths exhibit zero stretch
(confirming the shortest path assumption), a large fraction
(20–40%) of the paths is “inflated” even up to 3–4 steps.
The plot appropriately represents the distribution of path
stretch that is found to be stunningly similarity in all four
previously presented networks ....................................... 48
Fig. 7.4 Military hierarchy with 3 lieutenants ................................. 50
Fig. 7.5 Illustration of paths with regard to the internal logic of
the network. A path is regular if it does not contain a
large-small-large pattern forming a “valley” anywhere in
its centrality sequence (green and orange paths). Red paths
show examples of non-regular paths. An upstream path
contains at least one step upwards in the hierarchy of the
network (orange paths), while in downstream paths, the
centrality decreases all the way (green paths) ....................... 51
Fig. 7.6 Military hierarchy: downstream and upstream paths ................ 52
Fig. 7.7 Confirmation of the prefer downstream rule. The plot
shows the percentage of regular paths containing no more
than a given number of upstream steps before entering
the downstream phase. The empirical paths tend to avoid
stepping upwards in the hierarchy, which is reflected by the
much lower number of upstream steps, in comparison with
the randomly selected regular paths of the same length ............. 53
Fig. 7.8 Organizational hierarchy in the story with the magnet with a
path containing a “valley” through a cross-hierarchy edge
from Anthony to Mark ................................................ 56
Fig. 7.9 A possible interpretation of the dispute wheel,a
theoretical object illustrating the unpredictable behavior
of communicating actors or nodes making decisions
independently of each other not possessing the Gao-Rexford
conditions. In the figure, Alice, Bob, and Carol,
the employees of an imaginary small organization,
communicate with each other, with the intention of passing
possibly unpleasant news to their boss. Each of them is
reluctant to confront the boss with the bad news, so they all
try to persuade each other to relay the message to the boss,
but none of them actually does so. The wheel exemplifies
that the message never arrives at its destination, even so, the
nodes in the network are well connected ............................. 60
xxiv List of Figures
Fig. 7.10 Our previously developed tiny model of the Internet initiated
by the people of Castle Rock to communicate with the
outside world by connecting to nearby town Salem’s Lot
and Dunwich in England using the transit services of Main
County Trans-Atlantic (MCT) and Canadian Federal Co.
(CF) as a backup route ................................................ 60
Fig. 7.11 Paths in nature lie between pure randomness and pure
rationality .............................................................. 63
Fig. 7.12 A curved pathway in the Japanese garden of the Budapest
Zoo overlayed with an artificial pathway constructed by
joining two segments each being a third of a circle. Walking
along the synthetic path makes the distance between the two
endpoints around 20% longer. The photo is the property of
the authors ............................................................. 64
Fig. 7.13 The Yin Yang, a Tai Chi symbol with the indication of the
middle path by a red line .............................................. 65
Fig. 8.1 A small part of human metabolism by Evans Love. [With
the permission of Evans Love] ....................................... 68
Fig. 8.2 Handling system for dipping cattle with curved races. As
appeared in the publication [11]. [With the permission of
Temple Grandin] ...................................................... 70
Fig. 8.3 Visualization of a part of Twitter by Elijah Meeks. [With the
permission of Elijah Meeks] .......................................... 71
Fig. 9.1 Role of shortest and regular paths .................................... 74
Fig. 10.1 The official map of Central Park in New York City. [With
the permission of the Central Park Conservancy] ................... 80
List of Tables
Table 6.1 Possible setting of routing tables for the network in Fig. 6.4 ....... 36
Table 6.2 Setting of routing tables leading to a loop for the network in
Fig. 6.4 ................................................................ 37
Table 7.1 Basic properties of our networks and paths ......................... 46
xxv
Chapter 1
Introduction: Long and Winding Roads
Once upon a time, there was a cock and a mouse. One day the mouse said to the
cock, “Friend cock, shall we go and eat some nuts on yonder tree?” “As you like.
So they both went under the tree, and the mouse climbed up at once and began to
eat. The poor cock began to fly, and flew and flew, but could not come where the
mouse was. When it saw that there was no hope of getting there, it said, “Friend
mouse, do you know what I want you to do? Throw me a nut.” The mouse went and
threw one and hit the cock on the head. The poor cock, with its head broken and all
covered with blood, went away to an old woman. “Old aunt, give me some rags to
cure my head.” “If you will give me two hairs, I will give you the rags.” The cock
went away to a dog. “Dog, give me some hairs. The hairs I will give the old woman.
The old woman will give me rags to cure my head.” “If you will give me a little
bread,” said the dog, “I will give you the hairs.” The cock went away to a baker.
“Baker, give me bread. I will give the bread to the dog. The dog will give hairs. The
hairs I will carry to the old woman. The old woman will give me rags to cure my
head. [5]...” (Fig. 1.1).
We could go on with the story, but to quickly reassure the reader we state that
the poor cock finally managed to cure his head after going through several other
interesting adventures in the forest. Telling such cumulative tales to children is
always great fun. They quickly pick up the rhythm of the story and listen to you with
curious eyes throughout. But what makes those cumulativetales, like the Italian one
above, so fascinating that children always listen and watch intently? Well, of course
they are worried about the little cockerel and wonder if he can cure his head. But if
that is all, then the tale could end after the nut hit the cock on the head by saying
that “The poor cock, with its head broken and all covered with blood, went away to
an old woman who gave him rags, and the cock cured his head.” Not so brilliant.
If we put it this way, the story would lose its meaning–its essence. But, what is at
the heart of the tale that makes it exciting? We could say, a long chain of events
that has to happen before the cock can finally heal his head. An intricate path of
events which can take unexpected turns and may go on forever. A path which we
go down with the little cockerel and almost forget why he desperately needs all of
© The Author(s) 2021
A. Gulyás et al., Pa t h s ,https://doi.org/10.1007/978-3-030-47545-1_1
1
2 1 Introduction: Long and Winding Roads
Fig. 1.1 The cover of Italian
popular tales by Thomas
Frederic Crane [Published by
the Riverside Press,
Cambridge, Massachusetts,
1885]
those things. When listening to the tale, we are so preoccupied with following his
path, that the goal almost vanishes from our horizon.1The whole adventure slowly
becomes to exist in its own right, perhaps more important than the goal itself, and
gains its own, independent meaning. Does it mean that we avoid getting to the goal?
Well, not exactly. Wandering around pointlessly would become tiresome over time.
But we seem to have a strange desire to meander a bit before finishing the story.
Is it maybe to warm up or to attune ourselves to the story? Or is it simply a quest
for some pleasure? Or do we just need time to prepare for an important message?
Regardless of the reasons, the path eventually becomes the essence of the story, and
the goal loses its meaning entirely!
If you have ever watched the classic Columbo crime series with Peter Falk,
you will surely understand this idea. Each episode of Columbo starts by showing
a murder exactly as it happened. So, from the very beginning, we know who the
victim is, who the murderer is and how the murder has been committed. The ending
of the story is not a question: Columbo will arrest the murderer. So, we don’t watch
this series for the excitement of whether the murderer will be caught or not. Then
why do we watch it? Well, for the specific way Columbo solves the crime with
all the tiny, seemingly insignificant details that are slowly pieced together to create
an unwavering proof. In short, we watch it for Columbo’s particular path towards
solving the case. And, of course, for one more thing: Columbo’s rigorously funny
character.
1This thought is beautifully captured by the painting of Ma Yuan, where the figure in the painting
walks on a mountain path, quickly vanishes (see Fig. 1.2).
1 Introduction: Long and Winding Roads 3
Fig. 1.2 Ma Yuan, “Walking on a Mountain Path in Spring” [With the permission of the National
Palace Museum of Taiwan.]
How universal are those seemingly useless turns in paths that humans make
in their everyday life, and why do they exist? How winding should they be? Can
we collect and analyze data about them to discover their properties? And how can
we use such information to predict the behavior of diverse, real-life systems that
clearly implement paths? In this book, we try to address those questions. We seek
a functional sequence of steps that take us closer to the nature of real-world paths.
Searching for a path to paths? This sounds crazy enough. Let’s to jump in and start
our long and winding journey. . .
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as long as you give appropriate
credit to the original author(s) and the source, provide a link to the Creative Commons licence and
indicate if changes were made.
The images or other third party material in this chapter are included in the chapter’s Creative
Commons licence, unless indicated otherwise in a credit line to the material. If material is not
included in the chapter’s Creative Commons licence and your intended use is not permitted by
statutory regulation or exceeds the permitted use, you will need to obtain permission directly from
the copyright holder.
Chapter 2
Everybody Loves Roundabouts
On a typical day, before starting an activity, we set an explicit or implicit goal. The
path that leads towards that goal is often selected without carefully designing it.
It just comes naturally to select the right route to work or the appropriate series
of actions to prepare breakfast. It is no wonder that an optimal execution is rarely
considered. There are, however, cases when our target is in some distant future,
giving us an opportunity to mull over it and to discover an energy-saving solution.It
comes as a surprise that we rarely take advantageof it. To us, the adopted path seems
to contain superfluous steps requiring extra effort. What is more, sometimes people
cannot stand making things more complicated. Is that simple human negligence or
is there more to be discovered? In the following, let us make an attempt to unfold
the mystery through a series of real-world examples. We first start with a story from
the postwar America, and then we study human activities on the Internet, and finally
we discuss a superb idea for project presentations.
2.1 Hiding Behind Proxies
1947–1960 was not an easy period for Hollywood artists, writers, and directors.
After the beginning of the cold war, the political witch-hunt in search of communists
culminated in the primitive act of blacklisting more than 300 artists as they were
accused of having communist ties or sympathies. Orson Welles, Arthur Miller,
Charlie Chaplin are just a few names who had lost their jobs and reputations because
they were blacklisted. To continue their careers, many of the blacklisted wrote under
the names of friends who posed as the actual writers. These friends were called
“fronts”. The motion picture titled “The Front” (1976), directed by Martin Ritt, was
based on these regrettable events. Howards Prince (Woody Allen) (Fig. 2.1), the
restaurant cashier and illegal bookie, is asked by his friend Alfred Miller (Michael
Murphy), the blacklisted screenwriter, to sign his name to Miller’s television scripts.
Howard is a good friend and desperate for money and success, so he agrees. Miller’s
© The Author(s) 2021
A. Gulyás et al., Pa t h s ,https://doi.org/10.1007/978-3-030-47545-1_2
5
6 2 Everybody Loves Roundabouts
Fig. 2.1 Howard Prince’s
(played by Woody Allen)
portrait hand-drawn by
Lajosné F. Kiss [With the
permission of Lajosné F.
Kiss]
scripts make both of them wealthy and Howard becomes a famous screenwriter.
Howard proves to be a good “front” for Miller and attracts other “clients”. He
proceeds to publish their scripts under his own name. The business is booming and
Howard becomes one of the most prominent screenwriters in Hollywood.
What a surreal way of becoming successful and famous! If we start to think
a bit deeper about Howard’s success, it becomes even more surreal. Why do the
writers turn to him with their problems? Restaurant cashiers are not the typical
supporters of writers in Hollywood. Directors, artists, businessmen are more likely
to have the resources writers need. They have the money, social contacts, influence,
and reputation. Howard does not have any of those. Despite that, many writers
turn to Howard for help. What can Howard provide then, which is so valuable to
Hollywood’s famous writers? Instead of money, contacts or influence, he could
have offered his harmless personality. He was a nobody and that is exactly what
the writers needed. Through this representative, the desperate writers reached their
goals: publish their scripts and continue their careers. Acting as a front for others,
Howard simply provided a path; path to the goals, which otherwise would have been
unreachable.
“Fronts”, like Howard Prince, are widely used on the Internet too. They are called
proxies, and in the most basic setting, they can be used to act as “straw men” for
users when accessing an Internet service. In the example Fig.2.2, Bob is running a
current time service, meaning that he tells the current time for a few cents if someone
asks. Alice uses a proxy to ask Bob the time. Why does she do it? First of all, to
save some money. Alice has a friend named Carol who also uses Bob’s service very
frequently.They agree to use the same proxy to ask Bobthe time. If Alice and Carol
are curious about the time almost simultaneously, then the proxy can ask Bob once
and tell both of the girls. The more people there are eager to know the time, the
more money they can save by using the proxy. Secondly, as the proxy acts on behalf
of Alice, Bob will never know that Alice uses his service; only knows the proxy.
2.1 Hiding Behind Proxies 7
Fig. 2.2 The working of a
proxy server
In 2003 Vivek Pai and his colleagues at Princeton University decided to set up
many open proxies [22] all over the Internet for research purposes. Open means that
the proxies can be used by anybody to indirectly access Internet services, like Alice
and Carol did. Unfortunately open servers on the Internet are a hackers delight,
so their intentional release was not a good idea. The researchers assumed that
“an unpublicized, experimental research network of proxies would not be of much
interest to anyone”. They were wrong. They underestimated how long it would take
for others to discover their system, and the scope of activities for which people
sought open proxies. Vivek and his colleagues experienced extraordinary attention
in their system immediately after its launch. They quickly detected a very large
volume of traffic (emails, chats, downloads, casts) going through their open proxies.
The question is, can a non-advertised open proxy server farm draw the attention
of anybody? We can agree with the researchers, that it is hard to imagine that such
a seemingly valueless thing would interest more than a few moping networking
fellows. What is it in this system that is so attractive to a surprisingly large amount
of people? What does this system provide that is useful to many people with such
diverse purposes?
Among the unforeseen activities of the open proxy experiment, the researchers
observed that proxies in Washington and California received a very high amount
of connections with both sources and destinations located along the eastern rim of
Asia. The multi-megabyte downloads appeared to be for movies, though the reason
that these clients chose round-trip access across the Pacific Ocean was not clear. A
direct connection would presumably be much faster. A reasonable explanation for
this behaviour is that these clients were banned from some websites and required fast
proxies to access them without disclosing their identity. Given the high international
Internet costs in Asia, proxies in the Western United States were probably easier to
find. Regardless of the real motivation of these users, they all picked characteristic
paths through the Internet to reach their services, their goals. In fact in this story,
the paths play a more important role than the users and services they connect. The
path, visible on an Internet map, in itself means something. It has its own reason
8 2 Everybody Loves Roundabouts
for existence and tells us how people try to solve their problems, how they think
and how they manage their lives. Besides financial or legal causes, there can be
other more elusive human motivations to create perplexing systems of paths, like
presenting complex ideas.
2.2 Mind Maps: The Revolution of Presenting Ideas
Showing a sequence of slides is the most widespread way of presenting ideas to
an audience. In a slide-based presentation, the speaker goes through the slides,
supporting the talk, in a linear fashion. Besides this mainstream slide-oriented
approach, a new wave of storytelling tools have appeared in the market, centered
around so-called mind maps. The users of such tools (e.g., Prezi, Mindmeister) can
collect various materials (texts, images, videos, slides) related to a specific topic and
organize them like a “mind map”. A mind map is a drawing that visually organizes
information. It is generally hierarchical and shows relationships among pieces of the
whole(seeFig.2.3). It is often created around a single concept, drawn in the center
of the map, to which associated representations of ideas such as images, texts, videos
and slides are added. Major ideas are connected directly to the central concept, and
other ideas branch out from those.
The mind map oriented presentation approach quickly became popular among
presenters and continues to attract millions of users. Mind maps are indeed beautiful
and eye-catching, but it is hard to think that the main motivation of millions of users
is to draw and present aesthetically appealing, didactic mind maps. Is it the pure
concept of mind maps that enabled small startups to compete with giants of the IT
industry like Microsoft, Googleand Apple in the area of presentation softwares? Or
is there more to it than meets the eye?
When you present using a mind map tool, you can define a presentation path
through your mind map and only focus on parts of the whole map that are
related to your specific talk. Your particular message depends on the audience. For
example, explaining Newton’s second law to elementary school students requires
a fundamentally different path than presenting the same in a university lecture,
although they can share common parts as well (e.g., illustrative figures, experiment
descriptions). The identification of paths as the main tools of storytelling is one of
the core innovations of mind map based approaches. Arguably, this feature is what
draws the attention of millions of users. A story can be told in many ways, and
each version holds a specific footprint, a particular path lying in the background.
Are there good or bad paths in storytelling? What is the difference between them?
That is the million dollar question to answer. One thing is certain: getting straight to
the point is rarely didactic; making detours for instructive examples can be key to a
successful presentation. Everybody loves roundabouts!
2.2 Mind Maps: The Revolution of Presenting Ideas 9
Fig. 2.3 A mind map about global warming by Jane Genovese [With the permission of Jane Genovese.]. http://learningfundamentals.com.au/
10 2 Everybody Loves Roundabouts
2.3 Short but Winding Roads
Despite their apparent independence, our nursery tale about the little cockerel,
Howard Prince, the open proxy system and mind map based presentation tools
share something which makes them so compelling that they attract much attention.
It turns out that, although in different forms of appearance, they all provide different
forms of paths. Paths to entertain us, paths to reach our goals, paths to solve our
problems and paths to deliver our messages. These examples indicate that paths play
an important role in diverse areas of life. It seems that paths are somewhat universal.
They are abstractionswhich can emerge in various kinds of guises. Is it possible that
the paths coming from differentareas share some common properties? Is it possible
that these paths are the product of some general laws that can be identified? What
are the possible steps that a path might include? These are difficult questions to
tackle. First, we can make an interesting observation. Common sense suggests that
we should favor “short” paths. We don’t like lengthy talks, we don’t want to forget
our goals when seeking a path, and we don’t have infinite time and energy to solve
our problems. Does it automatically mean that we should use the “shortest” possible
path? Our earlier examples hint that the shortest path may not always be the best
choice either. Shortening the nursery tale will diminish its entertainment value, the
Asian users of the open proxy system do not use a shorter direct path and a short
talk concentrating exclusively on the essentials of a topic may be boring or hard
to interpret. They should not be too long, but some windings may be necessary to
reach their goals. To analyze our paths in the following chapters, we need formal
concepts to grasp their most essential properties.
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adaptation, distribution and reproduction in any medium or format, as long as you give appropriate
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Chapter 3
The Forest of Alternative Choices
Watch the path of your feet And all your ways will be
established.
—Proverbs 4:26
In general, a path can be thought of as a sequence, timely ordered sequence of con-
secutive events or choices which can lead us far from the starting point.
Following a path means that we choose a specific sequence of steps from a pool
of possibilities or alternative choices. So what does such a pool look like? Well,
sometimes it is small, concrete and well-defined, while other times it is seemingly
infinite and may be obscure and intricate. To illustrate, let’s consider a very famous
and classic event pool from European century.
In the eighteenth century, the city of Königsberg, Prussia was wealthy enough
to have seven bridges across the river Pregel. The seven bridges connected four
parts of lands separated by the branches of the river. The situation is shown in
Fig. 3.1 where letters A, B, C, D denote the lands and the corresponding handwriting
(ending with the B. and Br. abbreviations) mark the locations of the bridges. This
scenario inspired the fantasy of the leisured inhabitants of Königsberg who made
a virtual playground from the bridges and lands. Their favorite game was to think
about a possible walk around the bridges and lands in which they cross over each
bridge once and only once. Nobody could come up with such a fancy walk and
nobody managed to prove that such a walk was impossible, until Leonhard Euler,
the famous mathematician, took a look at the problem. Euler quickly noticed that
from the perspectiveof the problem, most of the details of the map shown in Fig.3.1
could be omitted and a much simpler figure could be drawn, focusing more on the
problem (see Fig. 3.2).
This new representation contains only “nodes” marked with letters A, B, C, D
in circles which represent the lands and “edges” drawn with curved lines between
the nodes representing the bridges. A walk now can be described as a sequence of
nodes and edges. For example the sequence A E1 CE3 DE4 A
represents a walk starting from land A which proceeds to land C via bridge E1, then
to land D via bridge E3, and finally back to land A via bridge E4. Using only the
nodes and edges, all sorts of walks can be created. In fact, all the possible walks
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11
12 3 The Forest of Alternative Choices
Fig. 3.1 Euler’s Fig. 1 for the seven bridges of Königsberg problem from ‘Solutio problematis ad
geometriam situs pertinentis,’ Eneström 53 [source: MAA Euler Archive; http://eulerarchive.maa.
org/docs/originals/E053.pdf]
Fig. 3.2 Euler’s idea of abstracting away the network underlying the Seven Bridges of Königsberg
puzzle
that one can imagine throughout the bridges and lands is captured by this simple
representation. The collection of nodes and edges called a network N(n, e) turned
out to be so powerful in modeling real-world problems that a whole new branch of
mathematics, called graph theory,1was defined based on them.
1In the first ever graph theoretic argumentation Euler showed that to find a walk crossing each
bridge once and only once requires that the underlying network can contain only two nodes with
an odd number of edges. In Fig. 3.2 one can see that all nodes have an odd number of edges (A has
five, while B, C and D has three), which makes the problem insolvable in this network.
3 The Forest of Alternative Choices 13
We can observe that the walks around the lands and bridges are nothing more
than the ordered sequences of consecutive events (bridge crossings) in Königsberg.
These walks are very similar to our paths and the network N(n, e) seems to
effectively contain all the possible paths that can be taken, i.e., the paths2that can
be differentiated by the sequence of bridge crossings in Königsberg. So, a network
seems to be a good representation of the pools from which paths can take shape.
The network in the case of the Königsberg bridges is very small and well-defined
(contains four nodes and seven edges); however, the number of possible paths that
people can take in this network is theoretically infinite as the length of the paths
is not limited. In practice, the pool of all possible paths is much smaller as people
become tired or bored after a few hours of walking. Even if we remove E2 and E3
and we are allowed to cross a maximum of ten bridges during the walk, there are still
2330 possible paths to choose from. Would people generally have a preference when
choosing their afternoon walk? Will they choose randomly from all the possibilities?
Or is there a hidden order affecting their choices? Those questions get even more
complicated when, as in many real-life situations, a few more orders of magnitude
of choices are at hand. For the sake of extending our scope for other connected
systems, let us take a slightly more abstract network from the social sciences.
The small world experiments conducted by Stanley Milgram, a famous social
psychologist, in the 1960s targeted to understand the network of human contacts in
society. The main goal was to study the connectedness of people formed by their
acquaintances. In the experiments, several random people were asked independently
to send a letter (postcard) to a randomly chosen common target. Anyone, who did
not personally know the target, was asked to send the letter to a friend who possibly
would. Then the selected friends were subsequently asked to act the same way
until the message arrived. In such a way, the persons were the nodes, and the
friendships made up the edges of the network, while the traveler was the letter.
Although many of the messages never arrived, those that did found a surprisingly
short way through the chain of acquaintances. In many cases, even two or three
middlemen were just enough for the letter to arrive (the average path length fell
close to 6), in spite of the fact that the endpoints of the chain were carefully picked
to be sufficiently far away from each other either geographically or socially. The
so-called small world phenomenon is quite arresting in itself, the way the path is
formed on the social graph is also fascinating. The arbiter behind the wanderer in
this case is not a single entity but several independent ones, thus the established
route is a collective phenomenon. Is there any similarity between the paths taken by
individuals or the members of a community having a partially divergent perception
on their environment? Do they achieve better at finding the shortest paths? Or do
they require some superfluous sidesteps as well?
We will be able to answer such questions soon, but for now, let’s be satisfied
with finding networks as good representatives of all the imaginable paths belonging
2Note that the word path as used in this book corresponds to walks in the terminology of graph
theory.
14 3 The Forest of Alternative Choices
to a specific situation, because we will use them throughout this book. So, we have
networks overwhich one can take paths by traversing nodes and edgesin a particular
sequence. But who or what will take the paths? Well, sometimes they are people as
in the Bridges of Königsberg problem, sometimes a letter controlled by a small
community as in the Milgram experiments. But in a broader scope, there can be
many things that can take paths. Gossip, fashion styles, memes and all sorts of
information seem to travel over social networks. If we look inside the human brain,
we can identify the neurons as nodes and their axons as edges. What travels through
this network? All kinds of information encoded into the specific firing patterns of
neural cells. Similarly to networks (which can represent all kinds of paths), we need
to find a name for the something which will travel through the network. From now
on, we will call these travelers “packets”. Networks and packets will be all we need
to discuss paths in the broadest scope. Now let’s consider a much more intricate
network, over which the traveling “packets” will be indeed: packets.
The Internet is the greatest network man has ever built. Starting from a small
research network funded by the US government, it became a huge interconnection
network of thousands of computers all over the world. In its early phase, the Internet
was similar in size to the network lying behind the bridges of Königsberg problem.
It had so few nodes and edges that one could draw its map on a single piece of paper
(see Fig. 3.3). After opening the network to the rest of the world, making it possible
for almost anybody to connect, an interesting game began which still goes on today.
Fig. 3.3 Logical map of the ARPANET (the ancestor of the Internet) from 1977 [source: The
Computer History Museum; https://computerhistory.org/]
3 The Forest of Alternative Choices 15
Nodes started to join the network in an uncoordinated fashion, which meant that
nodes and edges could have appeared almost anywhere in the network. As a result
of this process, the Internet evolved into a large, complex network, the topology
of which changes heavily day-by-day. Even drawing an approximate contemporary
map was a great challenge for networking researchers and its visualization needed
newly developed algorithms. Despite the researchers’ best efforts, such maps were
able to grasp only a limited subset of the edges present on the Internet. Above this
large and evolving network, our emails, chat texts, web pages and videos travel day-
by-day. All these data, converted to small information packets, are delivered through
paths determined by the Internet’s so-called “routing” system. This routing system
has no central authority which could compute the paths for every single packet.
Quite the contrary, the Internet’s routing system is heavily decentralized, meaning
paths are determined through the complex interactions of thousands of nodes. What
kinds of paths come out of such a process? We know that latency is crucial if it
comes to Internet services. Nothing is more irritating than a website that is slow to
respond, a lagging video conference or a frozen video game. So it is natural that we
expect the provisioning of low latency paths. But does it mean that we will have the
shortest path between our computer and the desired service? Recalling the example
of the Asian users of the open proxy system can make us suspicious. As usual, the
truth will lie between the two extremes. But what exactly are these shortest paths?
Now it is time to get acquainted with them.
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as long as you give appropriate
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The images or other third party material in this chapter are included in the chapter’s Creative
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the copyright holder.
Chapter 4
Straight to the Point: A Short Chapter
About the Shortest Paths
A passenger walks in the ˝
Orség (region of Hungary) and asks a
man mowing at the fringe of the forest in Szalaf˝o: – How far is
˝
Oriszentpéter from here? – Five kilometers in a straight line, but
I can get a shorter ride through the woods.
—György Moldova, Tökös-mákos rétes, Magvet˝o 1982
There is something compelling about shortest paths. They are so simple and rea-
sonable. They seem to be the most efficient paths for traveling between nodes in a
network. They may take the lowest amountof distance, time or energy.For grasping
the idea of shortest paths, let’s consider the network in Fig. 4.1. In this network, the
shortest path between nodes D and H is the path (DCEGH) marked
with red arrows. Its length is the number of edges crossed which is 4 and this is
the only shortest path between D and H. Green arrows mark the shortest paths from
node C to node F. There are two shortest paths (C BAF) and (C E
GF) and both have a length of 3. Shortest paths are also pretty straightforward
to compute by a few lines of code e.g., by using Edsger W. Dijkstra’s [7] method.
The compelling concept of shortest paths makes them first-class citizens in
many areas of life. Everybody tries to take the shortest path from the store to
the car or from home to the workplace, to save time and energy. Engineers of
computer networks kindly favor the shortest paths because of their low latency and
low resource usage (they load the smallest possible amount of routers and links).
Shortest paths are also kindly used to predict information flow in social, biological
and transportation networks. Researchers also use them to categorize networks and
predict their behaviour under unusual circumstances (e.g., testing the behavior of
the Internet during a massive natural disaster).
Although shortest paths are definitely desirable, there are also some problems
with them. First, to find the shortest paths, one needs to explicitly know the whole
network. Any program computing shortest paths requires the whole network as an
input to run. To illustrate how much the shortest paths may change, imagine that
we forgot a single edge in Fig.4.1, namely the C G edge, which is drawn with a
dashed line in Fig. 4.2. In this modified network, the red path is not the shortest path
between D and H anymore, since the path D CGH is shorter. The shortest
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A. Gulyás et al., Pa t h s ,https://doi.org/10.1007/978-3-030-47545-1_4
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18 4 Straight to the Point: A Short Chapter About the Shortest Paths
Fig. 4.1 Shortest paths on a simple network
Fig. 4.2 The variability of shortest paths
4 Straight to the Point: A Short Chapter About the Shortest Paths 19
path between C and F is neither of the green paths since C GF is shorter
than both of them. In addition, in this new situation there is only one shortest path
between C and F.
Well, providing the exact structure of the network is not a problem in the case of
small and quasi-static networks (e.g., the Bridges of Königsberg),but it is clearly not
an option for large, complex and dynamic networks like the Internet. Secondly, there
is something strange, something artificial in shortest paths. It seems that shortest
paths sometimes fall too short and do not coincide with the underlying natural logic
of the network (just think about the little cock or the users of the open proxy system
or mind map presentations). Wait a minute! Can a network have a natural internal
logic? Let’s take a closer look at how such a logic might look!
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as long as you give appropriate
credit to the original author(s) and the source, provide a link to the Creative Commons licence and
indicate if changes were made.
The images or other third party material in this chapter are included in the chapter’s Creative
Commons licence, unless indicated otherwise in a credit line to the material. If material is not
included in the chapter’s Creative Commons licence and your intended use is not permitted by
statutory regulation or exceeds the permitted use, you will need to obtain permission directly from
the copyright holder.
Chapter 5
Finding Your Way Through the Maze
Orr was crazy and could be grounded. All he had to do was ask;
and as soon as he did, he would no longer be crazy and would
have to fly more missions. Orr would be crazy to fly more
missions and sane if he didn’t, but if he was sane, he would have
to fly them. If he flew them, he was crazy and didn’t have to; but
if he didn’t want to, he was sane and had to.
—Joseph Heller, Catch-22, Simon & Schuster 1961
Have you ever wondered how you would be able to navigate yourself through the
labyrinthine street network of a town without any central knowledge base like a
map or a GPS device? One thing is sure, to wander around would result in an
inadmissibly long journey, even in a smaller settlement. How about the letter in the
social acquaintance network of Milgram’s small world experiments? Is it a feasible
scenario that the letter just accidentally finds its way towards an addressee without
any central guidance for the messengers passing it randomly to each other? With
only a maximum of ten bridges across Königsberg’s four islands resulted in 2330
different paths; what would happen in a network containing 300 million nodes with
a hundred times more edges between them? The turmoil would be inconceivable!
The very existence of short paths between the nodes of a network is one thing, to
find them is completely another thing. It is reasonable to assume that there must be
some landmarks or traffic signs in even the smaller networks if we are to find an
adequately short way through it. There must be some internal logic that helps us
to navigate from node to node towards our predefined destination without spending
too much time roaming in the maze.
The internal logic of a network is something that is, on one hand, strongly
connected to its outlook or construction. However there is sometimes something
that is even more important than that: it is the rules of how to use paths among the
nodes. In real networks, it is not uncommon that, although a path exists between
nodes, it cannot be used due to some rules. For example think about a traffic sign
that indicates one cannot enter a road unless invited by a resident. Just like the sign
at the house of Winnie the Pooh’s friend Piglet: “TRESPASSERS W.” (Or was it
really the name of his grandfather?) Or what about a carpool lane where a path can
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21
22 5 Finding Your Way Through the Maze
only be used by cars shared by multiple travelers? Those also very much belong
to the internal logic of a network: it is about how a network may be used.Inthe
following let us take a look at some more complex examples, one from history and
one from technology: the military organization network, and the Internet.
Military organizations have a strong internal logic: a hierarchy. As we will see,
this strict hierarchy has a fundamental influence on the internal communication
paths. The network representation of an imaginary military organization is shown
in Fig. 5.1. On the lowest level of the hierarchy, there are the privates (Pvt Gump,
Ryan and X). They are usually under the command of a sergeant (Sgt Drill and
Horvath). Above sergeants, we find lieutenants (Lt Dan and Dewindt), commanded
by the captain (Captain Miller). The typical order of command in the military is
that soldiers at lower levels of the hierarchy report to one level above, while higher
level soldiers give commands to one level below. For example, the path of some
imaginary information from Pvt Gump to Pvt Ryan could be: (1) Pvt Gump reports
to Sgt Drill, (2) Sgt Drill includes this information in his report to Lt Dan, (3) Lt
Dan also includes the info in his report to Captain Miller, (4) the captain makes
a decision and gives a command to Lt Dewindt, (5) Lt Dewindt commands Sgt
Horvath accordingly, (6) Sgt Horvath then gives the corresponding command to Pvt
Ryan. Such a path may describe a situation where Pvt Gump observes something
important in the battlefield which should be reported to higher levels, from which
the reacting commands seep down to the lower levels.
How does this regular path relate to shortest paths? In our imaginary organiza-
tion, this regular path is also the shortest path, as we cannot find a path between Pvt
Gump and Pvt Ryan with fewer steps. In fact, the organization is so simple that we
only have one reasonable path between Pvt Gump and Pvt Ryan. All other paths
will contain loops, meaning that there is at least one soldier that appears twice on
the path. Let’s make our organization a little more complex and realistic.
Consider that Pvt X is doing a special service for the military and spends half of
his day under the command of Sgt Drill and the other half under Sgt Horvath (i.e.,
he is part of a liaison squad enablingcommunication between the units commanded
Fig. 5.1 Military hierarchy Captain Miller
Lt Dan
Sgt Drill
Pvt Gump Pvt X
Lt Dewindt
Sgt Horvath
Pvt Ryan
5 Finding Your Way Through the Maze 23
Fig. 5.2 Military hierarchy.
Shortest path vs. the regular
path
Captain Miller
Lt Dan
Sgt Drill
Pvt Gump Pvt X
Lt Dewindt
Sgt Horvath
Pvt Ryan
Shortest path
Regular path
by the two sergeants). The network representation (see Fig. 5.2) of this modified
organization differs in only one edge going between Pvt X and Sgt Horvath. This
small modification, however, uncovers an interesting phenomenon. In the modified
network, the shortest path between Pvt Gump and Pvt Ryan is no longer through
lieutenants and captains. The regular path is unchanged, but the shortest path is Pvt
Gump Sgt Drill Pvt X Sgt Ho rvath Pvt Ryan. The corresponding story
could be that Pvt Gump reports to Sgt Drill, who orders Pvt X to report something
to Sgt Horvath, who gives the command to Pvt Ryan. This absolutely can be done
and fits within the norms of the army, but it is rather unusual. The shortest path
seems odd and breaches the everyday logic of the military network. We can say that
the shortest path is theoretically usable, but it seems practically non-traversable.
Moreover, in this case, the regular path coinciding with the internal logic of the
network is longer than the shortest possible path.
Besides the clear conflict between the shortest path and the regular path,
implementing the two paths will have different effects on the organization.By using
the regular path, high-level decision makers are notified about the event happening
at lower levels and can make use of this knowledge in later decisions. The usage of
the shortest path, while it enables a faster reaction, prevents the information from
escalating and leaves the army in a different state. Can such illogical but short paths
be used within the army under any circumstances? Are there unusual situations in
which the everyday practices can be overridden? Well, it depends on the level of
unusuality. Let’s illustrate this with a short story about Hungary’s participation in
the Second World War.
The participation of the Hungarian 2nd Army in the Second World War on the
side of Nazi Germany was undoubtedly surrounded by a great amount of unusuality.
There are many books[24] covering the stories of the battles facing Russian soldiers
on the eastern theatre of the war near the Don River. During its 12 months of activity
in 1942–1943 on the Russian front in the framework of the Operation Barbarossa,
the 2nd Hungarian Army’s losses were enormous. Of an initial force of about
200,000 Hungarian soldiers 125,000 were killed in action, wounded or captured.
These losses were the result of the power of the Russian army, the extreme cold
24 5 Finding Your Way Through the Maze
Hitler
Wehrmacht high command
Weichs, General
Commander of Army Group B
Jány, Colonel General
commander of the Hungarian nd Army
Witzleben, Major General
commander of the German liaison staff
assigned to the Hungarian nd Army
Non-regular path
Regular path
Fig. 5.3 Illustrative structure of the relevant parts of the German Army in 1943
and the commanding structure of the German Army Group B under which the
Hungarian 2nd Army appeared as a sub-unit. The commander of the Hungarian
2nd Army was Gusztáv Jány (see Fig.5.3). A German liaison staff commanded by
General Hermann von Witzleben was assigned to work under the Hungarian army,
coordinating the movements of the German and Hungarian armies. Thus his troops
were somewhat subordinates of the two armies at the same time. The German Army
Group B was commanded by Maximilian von Weichs, who received orders from the
Wehrmacht high command and eventually from Adolph Hitler.
After a few months of Hungarians and Russians peering at each other on the
banks of the Don, the Russians began their counter-attack on the front line of the
Hungarian 2nd Army on 12 January 1943. During this attack, most of the Hungarian
units were quickly encircled and either annihilated or forced to open terrain where
they succumbed to the extreme cold. Facing the situation and the casualties Jány
tried to obtain a command to withdraw his troops, using the standard chain of
commands in the army. He sent messages to his superiors with the immediate
request for retreat. After days of bloody massacre, the answer from the German
high command remained the same: “In accordance with the Führer’s decision,
the positions ...must be held to the last man under all conditions”. As a parallel
action, on 15 January, Weichs asked Witzleben to meet Jány and to unofficially
persuade him to order the immediate retreat. The reason for this unofficial action
was that Weichs himself did not want to give a withdrawal order contradicting
Hitler’s instructions. On 17 January Jány eventually ordered his troops to commence
5 Finding Your Way Through the Maze 25
retreating. It was only on 22 January when the headquarters of Army Group B
decided, with Hitler’s permission, to withdraw the Hungarian 2nd Army from the
front line.
So what happened? Jány tried to use the regular path (chain of command) to
report and react according to the decision of high-level decision makers. However,
his situation was really unusual. He lost thousands of soldiers day-by-day and the
regular path was too slow to properly react to the situation. While Weichs and other
German commanders on site agreed with Jány about the immediate retreat, they
didn’t want to conflict with higher decisions. Instead, Weichs unofficially notified
Jány through his German subordinate, by communicating that “high orders should
always be interpretedin accordance with the situation”. This act convinced Jány, that
he really had his last men standing, so it was time to withdraw his troops. As a result,
he retreated 5 days before the official permission. Those few days saved thousands
of his people. In this case, the non-regular path was undoubtedly odd but worked and
saved lives. How frequently do such events, that require non-regular paths, happen
in the military? Well, horse sense suggests that if such events were prevalent, then
the military would not work at all. So we should expect the great majority of paths
to be regular and just a small portion of the paths to be non-regular.
The large scale Internet also possesses definite internal logic. It does in spite of
the fact that it has a strong self-organizing characteristic in its evolution. Indeed, the
Internet has grown into an intricate interconnection system through the uncoordi-
nated process of nodes freely joining to the network. Although this joining process
was truly without central coordination, it wasn’t without laws. Relations between
nodes have evolved to show a similar structure to what we saw in the military
example. To shed some light on how this can happen, let’s follow the imaginary
story of the people of the little town Castle Rock, Maine.
During strong winters in Castle Rock, people are doomed to stay in their houses.
One day lonely locals decide to connect to each other to communicate (e-mail, chat,
video chat, etc.) by using their computers. Out of that purpose, they form a civil
company that creates a local network across the whole village. Interestingly, almost
in parallel, a similar series of actions takes place at the nearby settlement of Salem’s
Lot. No wonder that soon the two towns decide to establish a communication cable
to connect to each other.
People seem to be happy for a while, but not much later it comes to the locals’
knowledge that an English town, Dunwich, also built a local area network for
similar purposes. Would it be possible to also connect Castle Rock to Dunwich?The
distance seems to be too much for the poor little town. Fortunately an entrepreneur
at the Main County Trans-Atlantic Co. (MCT) undertakes the task of building an
underwater cable through the Atlantic Ocean and connects the MCT to Castle Rock
and than to Dunwich. Thus providinglong distance communication services to both
towns for a monthly fee. Our newly born communication network, at this state, has
four nodes: Maine County Trans-Atlantic Co, Dunwich, Castle Rock and Salem’s
Lot. The little town of Castle Rock initiated the building of a large-scale computer
26 5 Finding Your Way Through the Maze
Castle Rock
Salem’s Lot
MCT
Dunwich
Fig. 5.4 An embryonic model of the Internet, where Castle Rock connects to nearby Salem’s Lot
directly and Dunwich in England through Main County Trans-Atlantic (MCT)
network, the nodes of which are networking companies. It is just like a tiny Internet
(see Fig. 5.4).
Soon enough, however, conflicts arise when people at Salem’s Lot take to
regularly using the network spending most of their spare time communicating with
the nice people from Dunwich! Notice that Salem did not spend money itself on
building the network or buying the service from MCT, even so they can reach
Dunwich using the networking resources of the nearby Castle Rock. But is it fair
to Castle Rock to load its network, possibly slowing its communication service, due
to the traffic of Salem’s Lot? Through a connection that was initially established out
of mutual agreement to exchange network traffic free of charge? Castle Rock pays
Maine Trans-Atlantic to reach Dunwich, but it is free for Salem’s Lot! The path
from Salem to Dunwich does not seem to be regular at all! It is not in the logic of
the network to use such a path. Castle Rock would surely cease the cooperation or
ask for some compensation, thus making Salem’s Lot a customer of Castle Rock.
Furthermore, a more interesting situation arises when the people of Castle Rock
begin to feel unsafe about the single existing communication path to Dunwich. A
Canadian company called Canadian Federal Communications (CF), noticing the
business opportunity, comes forward also offering data transit services across the
Atlantic for a fee. So, Castle Rock also chooses to connect to Canadian Federal as a
backup plan for when the Maine Trans-Atlantic link goes down due to some error.
Our tiny Internet has grown into a complex 5-node super-highway (see Fig.5.5).
5 Finding Your Way Through the Maze 27
Castle Rock
Salem’s Lot
MCT CF
Dunwich
Fig. 5.5 A tiny model of the Internet initiated by the people at Castle Rock to communicate with
the outside world by connecting nearby town Salem’s Lot and Dunwich in England by using the
transit services of Main County Trans-Atlantic (MCT) and Canadian Federal Co. (CF) as a backup
route
All seems to be fine at the moment and will remain so as long as the Canadians do
not decide to communicate with Maine County Trans-Atlantic, generating an even
more serious conflict. Currently MCT and CF are only connected via their common
customer’s network. Should Castle Rock allow the CF to communicate with MCT?
Absolutely not! It is not logical to provide transit services to them for free! The
resources alone would not be enough to serve the transmission requirements of such
large companies. Canadian Federal and Maine Trans-Atlantic should build their own
network or use the transit services of even larger telecom service providers. It does
not matter if the path connecting the two becomes longer as using the shortest path
through a customer is not a viable solution! It is not considered to be regular!
As we can see the labyrinthine network, which we now call the Internet, connect-
ing the telecom companies of the world also has some structure to it and works along
some rules. Similarly to the military system, it contains a communication hierarchy
formed by customer-provider relationships generating strict communication policies
governed by complex business interests.
Is it possible that other networks also have similar internal logic? Is it possible
that similar reasons can make paths longer than the shortest path? It is time to not
just philosophies about real paths, but to do some ground truth measurements and
see what phenomena are actually supported by real data.
28 5 Finding Your Way Through the Maze
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Chapter 6
On the Trail of Nature: Collecting
Scientific Evidence
Over every mountain there is a path, although it may not be
seen from the valley.
— Theodore Roethke
To get closer to understand the nature of paths we need two kinds of data about
the same networked system like the Internet, or the Bridges of Königsberg. First,
we need at least an approximate network connecting its nodes and a large number
of paths collected from real traces of packets. Using the words of the Bridges of
Königsberg problem, we need the network representation of the lands and bridges
(Fig. 3.2) and the footprints of people’s afternoon walks.
In the last two decades, the flurry of network science [2] in all fields (biology,
physics, sociology, technology) has resulted in the reconstruction of thousands of
networks lying behind real-world systems [9]. Ranging from the classic Kevin
Bacon game1over the network of Holywood movie actors, through the metabolic
and social networks to the sexual contact of people, we now have systematically
collected well-organized and publicly available data repositories about real-world
networks (e.g., SNAP [17]). So downloading and computing something interesting
over the network representation of cell metabolism in our cells is now an afternoon
of laboratory work for an undergraduate student. What about the paths? Well,
gathering paths seems to be a very different task compared to inferring simple
connections in a network. The techniques working for the identification of network
edges are generally not usable for gathering paths. Recall the Bridges of Königsberg
again as an illustration. The map of downtown Königsberg is easy to get. Just jump
into a map store and buy one, or draw an approximate map after a few days of
walking in the streets of the city. The map or the network of the city is a form of
public information. But what about the paths? Well, the paths belong to people.
The paths describe the habits of people and tell us about them. About their favorite
1https://oracleofbacon.org/.
© The Author(s) 2021
A. Gulyás et al., Pa t h s ,https://doi.org/10.1007/978-3-030-47545-1_6
29
30 6 On the Trail of Nature: Collecting Scientific Evidence
places, the location of their homes and even about their health (if they prefer long or
short walks). The nature of paths seems to be somewhat confidential. Some people
may talk about it and give their names, others may talk about it anonymously and
others may ignore you if you ask them about their paths.
Although gathering information about paths is not particularly easy, it is not
hopeless either. Now we present four very different systems for which both the
network data and the path data can be obtained to an appropriate extent. Our
collection here will be based on the recent study of Attila Csoma and his colleagues
about paths [6].
6.1 Flight Paths
When you board a plane of some airlines and get seated, you can find many
things stuffed into the rear pocket of the seat in front of you. There are life-saving
instructions, maps of the aircraft with the locations of exits, a sanitary bag, but
there are also airline’s magazines. In these magazines, among the advertisements
about the most attractive flight destinations, there are usually nice maps showing
all the flights operated by your airlines. If we could collect all magazines from the
back pockets of all airlines, then we could easily reconstruct the flight network of
the world, by considering the airports as the nodes of our network and the flights
between them as the edges, no matter which airlines operate them. Although it
would be quite time-consuming, it is absolutely doable.
Fortunately, there is a much simpler methodfor constructing the flight map of the
world. Since flight information is public, there are public online data repositories
which accumulate all the information about the flights all over the world. For
example, the OpenFlights [21] project collects such data and makes the whole
database publicly accessible. By listing, for example, all the flights of US airlines,
the flight map of the US can be drawn (see Fig.6.1).
Therefore, the reconstruction of the flight network is not rocket science, having
the online datasets at hand. What about the paths? Well, a path is the multi-flight
travel of somebody between the departing and destination airports through the
flight network. Having a path means that we know the detailed flight information
including the flight transfers for a given passenger or a set of passengers. Path
information reveals how people choose transferring options at various airports, in
cases where there is a lack of direct flights between the source and destination.
Knowing a large number of paths is equal to knowingall the transfers of passengers
for their trips, which is not something we should know without their consent and
as such, there are no online databases for them. How can we obtain paths then?
Well, we can take an “indirect” path to these paths. There are various flight-trip-
planner portals that offer tickets between arbitrary sources and destinations all over
the world. On such websites we can plan our whole journey and buy the tickets
online. We can safely assume that many of the passengers buy their tickets using
similar websites. So, what we can do is observe flights between randomly picked
6.1 Flight Paths 31
Fig. 6.1 The flight network of the US
airports and consider these offerings as paths that passengers could really choose
for their trips. Gathering thousands of such flight offerings can give us a fairly
usable approximation of paths used by real people, without raising confidentiality
issues. Note that a particular trip can be the result of intricate business interactions
among many different airlines and the passenger itself, so similarly to the Internet,
the airport network also seems to be working without central coordination. What
expectations may we have about the paths coming out of such a network? For
starters, consider the position of Hungary’s flagship airlines (which has stopped
flying recently) in the flight system of the whole world.
MALÉV Hungarian Airlines was the principal airline of Hungary from 1946 to
2012. It had its head office in Budapest, with its main operations at Budapest Liszt
Ferenc International Airport. In its best years, Malév operated direct flights between
Budapest and New York, and Budapest and Moscow. In this respect, Moscow and
New York could be connected by a path of length two through Budapest. What can
we say about this two-step path offered by Malév? Well since one could travel from
Moscow to Budapest and from Budapest to New York with Malév, we must say
that this path is usable by passengers. However, Moscow and New York are huge
metropolises with airports serving around 30 and 60 million people respectively,
while Budapest airport is used by around 10 million people in a year. So, connecting
these cities through the relatively small airport of Budapest looks a bit odd and
we may suspect that the great majority of people would use other airports (e.g.,
Heathrow, Charles de Gaulle, Schiphol or Frankfurt) for changing flights between
Moscow and New York. Although the background and the way of operation of the
32 6 On the Trail of Nature: Collecting Scientific Evidence
flight network are very different, we can suspect a similar underlying hierarchy of
the airports as we have seen in the case of the military or the Internet.
6.2 Paths from a Word Maze
Word games are fun and entertain people regardless of their age. The Last and
First game, for example, is frequently played between children and parents or
grandparents. The essence of the game is to say a word which begins with the final
letter of the previous word. For example, the word chain camel lion napkin
nest tiger raven can be the result of an afternoon game between granny
and grandchild. Wait a minute! Doesn’t this look like a path? A path of words? Sure
it is! But instead of leading to somewhere, the aim of this chain is to go on as long
as granny is awake and the grandchild is not bored. In this respect, the game does
not have a destination (at least in terms of words). But can we twist the game a little
bit so that the word paths will lead somewhere? Word ladder games are designed
just for this purpose.
In a word ladder game, players navigate between fixed length source and destina-
tion words step-by-step by changing only a single letter at a time. For example, the
word path fit-fat-cat is a good solution of a game with source word “fit” and target
word “cat”. This path is now very similar to our flight paths, in the sense that they
have a definite source and destination and “transfers” can be made between words.
Is there a public repository accumulating solutions of word ladder games played
by people? Well, luckily there is[15]. Recently, Attila Csoma and his colleagues
have developed a word ladder game for smartphones in a framework of a scientific
project, and collect the word paths of people. After the users install the game, they
are asked to transform a randomly picked three-letter English source word into an
also random three-letter target word through meaningful intermediate three-letter
English words by changing only a single letter at a time. The word paths entered
by the users are collected anonymously. Fortunately, word path game solutions do
not seem to be as confidential as flight information, as hundreds of users shared
thousands of word paths (despite the clear deficiencies of a game developed by
university researchers). These paths can be considered as the footprints of humans
navigation over the word morph network of the English language.
More specifically, the collected paths are footprints of the process by which
people master their navigational skills in the network lying behind the game. The
word morph network is a network of three-letter English words, in which two words
are connected by an edge if they differ in only a single letter at the same position
(see Fig. 6.2). For example, the word “FIT” is connected to the word “FAT” as they
differ only in their middle letter. “FAT” is linked to “CAT” as they differ in their
first letter, but “FIT” and “CAT” are not connected in this network since they differ
in more than one letter. The paths collected from players are paths in this network
and reflect valuable information about how people try to navigate between nodes.
6.2 Paths from a Word Maze 33
Fig. 6.2 The word morph network is a network of three-letter English words, in which two words
are connected by a link if they differ only in a single letter. For example, “FIT” is linked to “FAT”
as they differ only in the middle letter, but “FIT” and “CAT” are not neighbors in this network
since more than one letter differs in them
Figure 6.3 shows a small portion of the word morph network and illustrates two
solutions for a puzzle between source and target words “YOB” and “WAY”.
What can we expect from these word paths? How will they look? Will we find
“odd” paths and “regular” paths similarly to the chain of commands in the military
and flight paths? As a sanity check, we present a common finding the players
reached after playing some games. They realized that words are not equal in this
game and some words can be used for various functions. The most basic puzzles,
like the “FIT” “CAT” one, can be solved by simply getting closer and closer
to the destination in terms of matching letters. In “FIT” there is one matching letter
with “CAT”, in “FAT” there are two matching letters, while in the destination “CAT”
all letters are matched. How about the “TIP” “ALE” puzzle? This is much more
complicated since the consonants and vowels are at completely opposite positions.
In this case, the above strategy simply doesn’t work. Now the players have to find
words with back-to-back consonants or vowels, where such letters can be swapped.
For example, the “TIP” “TIT” “AIT“ALTALE” is a solution, where
the intermediate word “TIT” is just there to turn to “AIT” at which vowels are back-
to-back at the front, which then can be changed to “ALT” at which consonants are
34 6 On the Trail of Nature: Collecting Scientific Evidence
Source word (yob)
Target word (way)
Shortest path words (bob-boy-bay)
User path words (rob-rot-rat-wat)
baa
bad
bag
bat
bay
boa
bob
bod
bot
boy
cat
cay
cob
cot
day
eat
fob
rat
rob
rot
wat
way
yob
Fig. 6.3 A word morph game example with source and target words “YOB” and “WAY”. A
shortest solution is displayed in red, while a solution given by a specific player is shown in green
back-to-back at the end and is just one step from “ALE”. People quickly memorize
such “trade” words like “AIT” or “ALT” and reuse them in further puzzles. So, it
seems that there is also some underlying logic in this simple word game. But is this
logic similar to what we can find on the Internet or in the flight network?
6.3 Internet Paths
The reconstruction of the network to which the Internet has evolved after more than
three decades, grasped the attention of many researchers worldwide. As the Internet
is built over electronic devices, its topology could be reconstructed by collecting
all the connection-related data residing in each of its constituting nodes (i.e.,
computers). However, unlike the airport network where the flights between airports
is public information, the connection information between Internet providers is not
easy to obtain. The traffic agreements between internet providers are usually kept
confidential. Therefore, it seems that we cannot even get the underlying network of
6.3 Internet Paths 35
the Internet in a straightforward way, not to mention the paths we are curious about.
It turns out, however, that some Internet hacks can help us find the paths. In this
respect, the Internet is a unique platform for researching paths.
To get a picture of how packets go through the Internet, we have to understand
some fundamentals of computer networking first. A packet traveling between
computers is nothing more than a few bits of information encoded as electronic
signals. Every packet has a source and destination address and a payload which
should be delivered to the destination. Packets usually do not change and do not
think, which is in high contrast with the people walking through the bridges of
Königsberg. Instead of the packets, the computers (i.e., the lands) “think”. What
does a computer (let’s refer to them as nodes in a network context) do when
receiving a packet? First, it looks into the destination address. If the destination
address is the current node, then it “consumes” the packet. After extracting the
payload data, the packet is destroyed. If the destination address is not the node
receiving the packet, it has to find out how to forward the packet to its destination.
Who tells the node how to find this out? People! Not ordinary people of course, but
networking people whose job it is to operate networks. There is a routing table in
every node, which is very similar to road signs. It indicates the next turn a packet
should take on the way to a specific region of the network.
Consider Fig. 6.4 as an example. There are seven computers marked with letters
(A, B, C, D, E, F , G) forming a very simple network of seven edges. Now suppose
that Dwants to send a packet to G. As a first step, it creates a packet containing the
source address D, the destination address Gand also the payload data, just like
a postal letter. Now node Dhas to send the packet to G. The situation of Dis
extremely simple as it has no choice where to send the packet, its only option is B.
Quite the contrary, at node B(as it is not the destination) there are plenty of options
to forward to. How will Bdecide? Well, a capable network operatorconfigured Bto
solve such situations. The operator creates the routing table of B, from which Bcan
read the next step of the packet destined to G:itmustbegiventoC. Similarly, Cis
instructed to send packets with destination address G, to node G. As a result finally
Greceives the packet successfully. The full operation can be made according to the
routing tables in the nodes (see Table 6.1). From these routing tables, all the paths
Fig. 6.4 A simple network of
computers
36 6 On the Trail of Nature: Collecting Scientific Evidence
Tab l e 6 . 1 Possible setting of
routing tables for the network
in Fig. 6.4
Routing tables of the nodes
Node Destination Next step
B D D
E E
A A
C, F, G C
C F F
G G
A A
B,D,E B
AB,D,E B
C, F, G C
DALL B
EALL B
FALL C
GALL C
between any pairs of nodes can be reconstructed. The problem is that the nodes
of the Internet belong to various networking corporations, which do not intend to
disclose the routing tables. So, collecting the paths in such a way is not an option.
What can we do then to get our paths?
Fortunately, there is something in computer networks which all networking
companies are scared of. So scared, that they implement several mechanisms to
detect and avoid them. These daemons are called loops. Consider that in our simple
computer network in Fig. 6.4, every node is administrated by a distinct company,
i.e., a different operating person. Consider that the operator of Anotices that its
direct connection to Cis weak, e.g., it provides a slow connection. So the operator
of Asets the routing table in Ato forward every packet destined for C, F and G
to B, avoiding the laggy direct connection to C. Independently, Balso considers its
connection to Cas pretty weak and forwards all packets heading to C, F and Gto
A(see all the modified routing tables in Table 6.2, with the modifications shown in
boldface).
Now, what happens with the packet destined for Gafter these tiny, uncoordinated
modifications in the routing tables? Well, it starts at Das seen before. Bsends it to
Aaccording to its routing table, but Asends it back to B,Bsends it back to A,A
sends it back to B...andso on forever.Thereisaninnite loopbetweenBand A.
After some time, node Band Aare only occupied by looping the packet infinitely,
which eats their resources, pointlessly generates a lot of heat, and most importantly
ruins the operation of the whole network. You may think that the routing settings
are carefully negotiated between networking operators, so such things could not
happen. Well, routing settings are usually well negotiated, but the human factor is
always there. Misconfigurations happen every day on the Internet. One of the most
famous examples was in 2008 when due to a routing table misconfiguration in a
6.3 Internet Paths 37
Tab l e 6 . 2 Setting of routing
tables leading to a loop for
the network in Fig. 6.4
Routing tables of the nodes
Node Destination Next step
B D D
E E
A A
C, F, G A
C F F
G G
A A
B,D,E B
AB,D,E B
C, F, G B
DALL B
EALL B
FALL C
GALL C
node of the Pakistan Telecom, a large portion of YouTube’s traffic was hijacked and
discarded in Pakistan.
Thus, loops are dangerous things in computer networking and one should
immediately detect and avoid them. The current solution for that is to include so-
called time-to-live (TTL) information in the packets. This is a simple number which
is decremented by every node the packet visits. If this number becomes zero, the
packet will be destroyed even if it hasn’t reached its destination. This way, bad
configurations will have limited effects as packets cannot travel for an infinite
time between the nodes. When a packet’s TTL becomes zero and it is not at its
destination, that is a good sign that something is wrong with the network. In this
case, the node which destroys the packet sends an alert to the source address found
in the packet stating that something is wrong. And this is where the approximate
tracing of packets becomes possible.
Consider the following hack. We want to know the nodes on the path towards
a destination node Dfrom source node S. First, we send out a packet from Sand
set its TTL value to 1. This packet will reach one of Ss neighbors (let’s say A),
which will destroy the packet and notify Sthat something went wrong. From this
notification, we record, that our packet has visited node A.Nowwestartagain
and send out the packet, but this time setting its TTL to 2. The packet would not be
destroyed by node Aas its TTL becomes 1 when Adecrements it. So Awill forward
the packet to somewhere, let’s say to B. Since at Bthe TTL is decremented again,
it becomes zero, so Bdestroys the packet and sends back a notification to Sthat
something went wrong. At S, we record that the packet has visited Bso the path to
Bis SAB. The process continues until a large enough TTL setting lets our
packet reach its destination. Sounds a bit complicated but this is all we have. This
method (called traceroute) gives us approximate paths and we can use this method
38 6 On the Trail of Nature: Collecting Scientific Evidence
from any node connected to the Internet, even from your laptop. Fortunately, there
are public datasets which contain such Internet paths collected from thousands of
different locations. These datasets (see for example the website of the Center for
Applied Internet Data Analysis[4]) can give us millions of paths from which an
approximate map of the Internet can be recovered.
How can we construct the topology of the network from paths? Suppose that we
have three paths: Pat h 1 :ABC, Path 2:ABDE, Path 3:E
CFA. By analyzing Path 1 , we see that there are nodes A, B and C and
there are edges between A and B, and between B and C. From this, we can draw a
network shown in Fig.6.5.
Now we analyze Pa t h 2 . We realize, that there are also nodes D and E in the
network, and we locate two edges: B DandDE(seeFig.6.6). Finally, the
observation of Path 3 adds a new node F and three edges: E C, C F, and
FA. So, after processing the three paths, we get the network in Fig. 6.7.After
processing more and more paths, we will have more and more appropriate pictures
of the whole network.
Fig. 6.5 Constructing a network based on its paths, Phase 1
Fig. 6.6 Phase 2
Fig. 6.7 Phase 3
6.4 Paths from the Human Brain 39
6.4 Paths from the Human Brain
The human brain is one of the most complex networks one could imagine.
Understanding even parts of its functionality is extremely challenging and is still
one of the biggest mysteries of human life. Here we are interested in the paths inside
the brain: the paths over which information can travel between different parts of the
brain. Getting realistic paths from inside the human brain is extremely hard, if not
impossible. As a consequence, almost all current studies concerning path-related
analysis simply assume that signaling uses shortest paths, meaning that we suppose
brain signals follow the shortest possible path in the brain. Similarly to these studies,
we have to accept that we cannot get paths out from the brain in a direct manner. In
the case of the Internet and the flight network, the confidentiality of the path related
data was the main obstacle of getting direct paths. In the case of the brain, we just
simply don’t have the appropriate technology (yet) which could identify the paths
for us. What can we do then? Is there a similar hack for the brain that we used over
the Internet? What kind of data is currently available about the flow of information
inside the brain? We will go through these questions in the following paragraphs.
The Human Genome Project (Fig. 6.8) was one of the biggest endeavors of
mankind and was surrounded by the most remarkable scientific collaboration across
many nations. Its target was to determine the sequence of nucleotide base pairs that
make up the human DNA. Upon its completion, at a press conference at the White
House on the 26th June 2000, Bill Clinton evaluated the resulting map of the human
genome as: “Without a doubt, this is the most important, most wondrous map ever
produced by humankind.”
A similar endeavor started in 2011 when the Human Connectome Project was
awarded by the National Institutes of Health. This project is targeted to construct the
“map of the brain”, i.e., to discover the structural and functional neural connections
within the human brain. The structural map means that we locate specific brain areas
(these will give us the nodes) and the physical connections (which will give us the
edges) between them. How can one do this without slicing up somebody’s brain?
Well, this is what the “non-invasive” brain mapping methods are used for. With a
quite complicated method called DSI (Diffusion Spectrum Imaging), the diffusion
of water molecules can be observed inside the brain. To get a picture of how DSI
works, think about constructing a road network by observing only the movement
of cars at various observation points throughout the area you want to map. You
cannot see the roads themselves, but you can see the cars at these observation points
and you can write the direction and intensity of their movements. By collecting all
this information from the observation points, after some non-trivial computerized
post-processing, we can create an approximate map of roads and cities in the given
area. Interestingly, the process is very similar to the operation of WAZE, a popular
navigation software (now owned by Google), where the positions of WAZE users are
collected in an anonymized database. In this case, however, the exact map is drawn
by volunteer editors,using the draft map deduced from the database. In DSI, the cars
40 6 On the Trail of Nature: Collecting Scientific Evidence
Fig. 6.8 The Vitruvian Man depicting normal human body proportions is often used to symbolize
The Human Genom Project as Leonardo da Vinci created it in 1490, exactly a half a millennium
before the project began in 1990. [Public Domain; Leonardo da Vinci via Wikimedia Commons]
are water molecules, which are observed at various points in the brain by using MRI
(Magnetic Resonance Imaging) devices. A picture about the human connectome,
i.e., an approximate picture of one’s map of neural connections in the brain, obtained
via DSI, can be seen in Fig. 6.9.
Thanks to DSI we can have one’s connectome, i.e., we have the network over
which our paths form. What can we say about the paths? It seems that at this
time we can say nothing about them in a direct manner. But there is something we
can do to at least estimate brain paths better than simple shortest paths? fMRI[20]
(functional Magnetic Resonance Imaging) is a method with which one can reason
about brain activity. With fMRI, the blood oxygenation of various regions in the
brain can be measured. Since blood flow and oxygenation are correlated with brain
activity (active brain regions use more energy and require a higher level of oxygen
in the blood), the changes in blood oxygenation reveal the neural activity. Back to
our city-roads-cars analogy, fMRI is quite similar to the task of reasoning about the
operation of a city, by observing the density of cars in its various districts.
6.4 Paths from the Human Brain 41
Fig. 6.9 The human neural network in the brain reconstructed via DSI, from Patric Hagmann et
al. “Mapping the structural core of human cerebral cortex”. In: PLoS biology 6.7 (2008), e159
How can we approximate paths in the brain? Well, DSI delivers an approximate
“network” of the brain, meaning that it gives us the nodes and the physical
connections (the bridges in the Königsberganalogy) between them. The fMRI gives
a different “network” in which brain regions are not physically, but functionally
or logically connected, meaning that they frequently act together, so they seem to
implement similar functionality. Can we make use of some trick and infer something
path-like from these data? Here is what we can do. By combining structural (DSI)
and functional (fMRI) data, we estimate paths through which neural signals might
propagate using the following hack. First, we have to identify the sources (i.e., the
starting node) and destinations (the nodes where the path ends) of our paths. From
the fMRI signals, we can identify brain regions, which frequently exhibit neural
activity at the same time. Simultaneous activity hints that these brain regions are
working on the same task and are likely to exchange information in the form of
neural signals. We identify these simultaneously active brain regions as the source-
destination pairs of our paths. Now we have to figure out the path between these
sources and destinations. In cases where there is a lack of information, we could
determine the shortest path between the endpoints of our paths using, for example,
Dijkstra’s algorithm over the structural connectivity network obtained from DSI.
Figure 6.10 shows an illustrative brain network of 15 nodes. Over this network, we
would like to approximate the possible signaling path between regions 1 and 15.
The shortest path approximation will give the 1 512 15 path for this. In
fact, most studies in the related literature use this simple approximation.
42 6 On the Trail of Nature: Collecting Scientific Evidence
Fig. 6.10 Inferring path from
the human brain using the
shortest path assumption
Fig. 6.11 Shortest path over
the active subnetwork at a
given time instant
Due to the extreme complexity of the brain, as of now, we do not have direct
information about the pathsinside, but we can do slightly better than simple shortest
paths. We can use the fMRI to identify regions with neural activity and from the DSI
network, we can exclude the inactive regions during signal transmission between the
endpoints of our paths. We can do this because inactive regions are not likely to pass
on any information. By excludinginactive regions,we will get the active subnetwork
for every information exchange we are curious about. Figure 6.11 shows the same
network we can see in Fig.6.10, but the red regions (2,7,9,12,14) are inactive, and
thus are excluded from the path approximation. Therefore, we will find the shortest
path between 1 and 15, but we cannot step onto the red regions. The shortest path in
this new scenario is 1 5811 15, which is longer than the shortest path
in the original DSI network.
While we cannot validate with empirical data whether these paths (see Fig. 6.12)
are actually used for the flow of neural signals, we can at least consider these paths
as lower bounds on the length of the real brain paths.
6.4 Paths from the Human Brain 43
Fig. 6.12 Empirical paths in
the human brain
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Chapter 7
The Universal Nature of Paths
The way is like the bending of a bow. To achieve its ends the top
must bend down and the bottom rise up.
— Tao Te Ching LXXVII
Although it was not always particularly straightforward, by the end of the last
chapter we came up with methodsenabling the exact measurement orthe estimation
of empirical paths in various real-life networks. From now on, we will refer to the
paths coming from measurements in real networks as empirical paths, to clearly
distinguish them from other paths (e.g., shortest paths) with which we will compare
them later. Before taking a look at the properties of the empirical paths, let’s take
some time to overview some numbers about our networks and paths (see Table 7.1).
The first row of Table 7.1 shows the numberof nodes in each network. We can see
that those networks are not just from completely different corners of life, but their
sizes also vary significantly. In the case of the Internet, it contains more than 50
thousand nodes, while there are only 1015 three-letter English words constituting
the word morph network. The second row presents the number of edges in each
network. Read with the node sizes, we can conclude that these networks are much
bigger than the network behind the seven bridges of Königsberg problem (which
had only four nodes and seven edges, see Fig.3.2).
The third-row reports on the so-called diameter of the networks, which is the
longest among the distances of any two nodes. Remember that the distance is
the shortest path among two nodes. To understand the concept, we can take, for
example, the diameter of the Universe as the distance of the two galaxies that are the
farthest away from each other. In that case, the distance is measured with the shortest
possible straight line through free space. Measuring the distance in a network of
course is done by counting the number of links from node to node. The diameter
of the Königsberg network in Fig. 3.2 is two, as no two nodes are farther away
from each other by walking the shortest path. Interestingly our networks, although
larger than the Königsberg network by orders of magnitude, have an extremely low
diameter. This property, that the diameter can be very small despite the network
being very large is also known as the small world property [25], which most of the
real networks readily exhibit. To intuitively grasp the small world property, think
© The Author(s) 2021
A. Gulyás et al., Pa t h s ,https://doi.org/10.1007/978-3-030-47545-1_7
45
46 7 The Universal Nature of Paths
Tab l e 7 . 1 Basic properties of our networks and paths
Network Airport Internet Brain Word morph
Number of nodes 3433 52,194 1015 1015
Number of edges 20,347 117,251 12,596 8320
Diameter 13 11 6 9
Average shortest path 3.98 3.93 2.997 3.52
Number of emp. paths 13,722 2,422,001 394,072 2700
Average empirical path 4.67 4.21 4.16 3.82
about the friendship network (e.g., Facebook) of people around the world. Although
there are billions of people in this network, any two persons can be connected by
using a friendship path of around six people. A friendship path starts with some
guy and goes on to one of his friends, then to one of his friend’s friends, then to
one of his friend’s friend’s friends, and so on. The small world phenomenon is
frequently illustrated by the popular term “Six degrees of separation” [13,14]used
in John Guare’s play (Fig. 7.1), in which Ouisa Kitteridge says: “I read somewhere
that everybody on this planet is separated by only six other people. Six degrees of
separation between us and everyoneelse on this planet. The President of the United
States, a gondolier in Venice, just fill in the names. I find it extremely comforting
that we’re so close.
The final metric belonging to the networks presented in the fourth row of
Table 7.1, is the average distance between their nodes. This means, that we compute
the lengths of shortest paths (e.g., by using Dijkstra’s algorithm) between all
possible pairs of nodes and then we take the average of all these lengths. This will
give a smaller number than the diameter (which is the maximum among the shortest
paths) and is remarkably similar for all our networks. For the Königsberg network
in Fig. 3.2, the average distance is 1.16667.
Regarding the empirical paths, we have two rows in Table 7.1.Thefifthrow
presents the number of paths we have been able to collect by our measurement
hacks in various networks (ranges from several thousand to millions, in the case of
the Internet). The final (sixth) row shows the average length of our empirical paths
given by the traceroutes over the Internet, ticket bookings over the airport networks,
path estimations in the brain, and puzzle solutions in the word-morph game. We can
see that the average empirical path is longer than the average of the shortest paths
(distances), which insinuates that nature does not always use the shortest possible
path over its networks, not even in networks where the shortest path could easily be
found. Although the difference is not extremely large, it is not negligible, especially
compared to the length of the paths (3–4). So it seems that real empirical paths are
10–30% longer on average than the shortest paths.
Now recall our introductory examples! The tale of the little cock, the users of
the open proxy system and the mind map presentations. Our impression about these
examples was that paths used in real life may be somewhat longer for some reason
than the shortest possible path. This impression about the presence of detours is now
7.1 Rule 1: Pick a Short (But Not Necessarily Shortest) Path 47
Fig. 7.1 Six degrees of separation. The poster of the play created by James McMullan. [With the
permission of James McMullan]
confirmed by real measurements in four networks having very diverse backgrounds.
In short, not just people, but many other things seem to favour detours. But is it just
the average length of the paths that exhibit similarities or is there more in common?
Let’s continue with a bit deeperexamination of the length of the empiricalpaths and
the possible path selection rules used by nature.
48 7 The Universal Nature of Paths
7.1 Rule 1: Pick a Short (But Not Necessarily Shortest) Path
Let’s define a metric which can show to what extent the empirical paths are longer
than the shortest possible path. We will call the difference between the length of the
empirical paths and the shortest path as the “stretch” of the empirical path. In the
example of Fig. 7.2, the shortest path between A and C is the green path (of length
2). The red path is of length three, thus it will have a stretch of 32=1. Similarly,
the blue path has a stretch of 2.
Now let’s see what percentage of the empirical paths exhibit a stretch of zero (i.e.,
the empirical path is the shortest path), stretch of 1, 2, 3, etc. Figure 7.3 depicts a
ABC
D E
FGH
Fig. 7.2 The illustration of path stretch. The green path is the shortest, while the red and blue
paths has a stretch of 1 and 2 respectively
01234
0
25
50
75
100
Stretch
Percentage of empirical paths
Fig. 7.3 A simplified sketch on the measured stretch of the paths relative to the shortest one found
in our real-life systems. While most of the empirical paths exhibit zero stretch (confirming the
shortest path assumption), a large fraction (20–40%) of the paths is “inflated” even up to 3–4 steps.
The plot appropriately represents the distribution of path stretch that is found to be stunningly
similarity in all four previously presented networks
7.2 Rule 2: Use Regular Paths 49
simplified sketch of the summarized findings in our four real life networks showing
the percentage of empirical paths as the function of stretch. Remarkably, all of our
networks show very similar behavior in that regard. As the stretch increases, the
percentage of empirical paths having that particular stretch decays pretty similarly.
This means that it is not just the average stretch which is similar in real networks
and paths, but in each network, we seem to find paths of a particular stretch with
a similarly decaying chance. The overall behavior is also interesting. While around
60–80% of the empirical paths have zero stretch, the remaining paths exhibit stretch
which can exceed up to 3–4 steps, or even more in some networks. From this result,
two things follow. First, the plot confirms the efficiency of nature in the sense that
most of its paths are shortest indeed. In this respect, nature definitely “prefers short
paths”. However, a non-negligible portion (20–40%) of stretched paths suggests that
there may be other considerations when paths are picked in real networks. What kind
of path selection rules produce similar results regarding the stretch of the paths?
What are the guidelines when picking a path? For understanding this, we have to
recall our main ideas about the internal logic of networks.
7.2 Rule 2: Use Regular Paths
We have seen before that there can be some kind of internal logic in networks, in the
form of various hierarchies, which can affect the structure of paths. In case of the
army example, this is quite obvious as the army is a fully hierarchical organization.
In the case of the Internet or the air transportation system, a similar hierarchy can be
reasoned; however, their presence is not so obvious. In the case of the human brain or
the word-morph game even, reasoning about hierarchies behind the network seems
non-trivial at this time.
How can we check if the stretch of the paths has something to do with these
underlying hierarchies? How can we prove that the reason of an empirical path being
slightly longer than the shortest path is to match the internal logic of the network?
How can we define the hierarchy that can be used for all of our networks in the
first place? A possible resolution to this problem is to use the so-called closeness
centrality number of the nodes as the measure of hierarchy level. The closeness
centrality, or centrality in short, of a node can be obtained by taking the number of
nodes in the network except the node itself and dividing it by the sum of the lengths
of the shortest paths from the node to every other node. Notice that the number is
higher for nodes located more centrally in the network.
For our military example (with an extra lieutenant added) in Fig.7.4, for Captain
Miller we have 1 +1+1+2+2+3+3+3=16 as the sum of the lengths
of the shortest paths to the others, and 8 as the number of nodes in the network
except Captain Miller. Thus his centrality is 8/16 =0.5. Computing the centrality
of the other soldiers as well (see Fig. 7.4), we get a clear reflection of the military
hierarchy. The nodes in the inner part of the network with higher centrality can be
50 7 The Universal Nature of Paths
Fig. 7.4 Military hierarchy
with 3 lieutenants Captain Miller
Lt Dan
Sgt Drill
Pvt Gump Pvt X
Lt Dewindt
Sgt Horvath
Pvt Ryan
Lt x
considered as the core and the nodes with lower centrality as the periphery of the
network.
By assigning a number to every node of a network reflecting the position in
the hierarchy, we get their role in the internal logic of the network. Now, the
question arises as if our empirical paths in the different networks have anything
to do with those numbers. After analyzing our paths, we find that most (around
90%) of the empirical paths do not contain a large-small-large pattern forming a
“valley” anywhere in their centrality sequence. For example, the path from Sgt. Drill
towards Lt Horvath trough Lt Dan, Captain Miller and Lt Dewindt in Fig. 7.4 has
a centrality sequence of 0.4,0.47,0,5,0.42,0.33, which contains no large-small-
large patter in it (no “valley”). However, would there be a link between Pvt X and
Sgt Horvath, the path from Sgt Drill towards Sgt Horvath through Private X would
have a centrality sequence of 0.4,0.3,0.33, containing a “valley”.1The fact that
the probability of finding such valleys in the empirical paths is very low suggests
that in real networks higher level nodes do not prefer the exchange of information
through their subordinates even if there are short paths through them. On most of
the empirical paths, the centrality increases monotonically at first (upstream), or in
other words goes “deeper into the center of the network”, then starts to decrease
(downstream), going “out of the network”, until it reaches the destination. Or in
other cases, the path goes upstream or downstream all the way. So the empirical
paths coming from our measurements seem to follow the underlying hierarchy of
the network. In other words, almost all empirical paths follow the internal logic of
1The watchful reader may argue that adding a new link toPvt X would change also hiscentrality in
the network (in our case, increasing his centrality above even that of his direct superiors), however,
this odd artifact would diminish fast as new soldiers were enlisted in the army. For the sake of
keeping our network example perspicuous, we omitted this correction here.
7.3 Rule 3: Prefer Downstream 51
N
E
T
W
O
R
K
P
E
R
I
P
H
E
R
Y
Increasing centrality
NETWORK CORE
X
Y
R
e
g
u
l
a
r
u
p
s
t
r
e
a
m
p
a
t
h
N
o
n
-
r
e
g
u
l
a
r
p
a
t
h
Regular downstream path
Fig. 7.5 Illustration of paths with regard to the internal logic of the network. A path is regular
if it does not contain a large-small-large pattern forming a “valley” anywhere in its centrality
sequence (green and orange paths). Red paths show examples of non-regular paths. An upstream
path contains at least one step upwards in the hierarchy of the network (orange paths), while in
downstream paths, the centrality decreases all the way (green paths)
the networks; they are “regular” by following the chain of commands. Figure 7.5
graphically illustrates such paths, where regular paths are colored green or orange.
Now we can recall our example of the Hungarian 2nd army. Then we settled on
the horse-sense conclusion that the great majority of paths were regular and we
expected only a small subset of the paths to be non-regular. Well, measurements in
this section now quantify the “great majority”as 90% and confirm our expectations.
7.3 Rule 3: Prefer Downstream
All right, so empirical paths follow the internal logic of the network even though it
produces slightly longer paths. Can we say anything else? Well, there can be subtle
differences between regular paths of similar length. For example, a path can contain
upstream then downstream steps or only downstream steps. Recall that an upstream
52 7 The Universal Nature of Paths
Fig. 7.6 Military hierarchy:
downstream and upstream
paths
Captain Miller
Lt Dan
Sgt Drill
Pvt Gump Pvt X
Lt Dewindt
Sgt Horvath
Pvt Ryan
Downstream path
Upstream path
step goes upwards in the hierarchy, while a downstream step goes towards the
periphery of the network (see Fig. 7.5). Is there a preference among those? Should
military sergeants turn towards their commander, or can they give orders directly to
the units under their command? If we ask the question in such a form, the answer
seems pretty clear: sergeants surely can issue orders directly to their units. Thus, a
military sergeant would prefer the path going downwards in the military hierarchy,
although there can be other regular paths to its units, e.g., through a lieutenant (see
Fig. 7.6).
What is the situation in other networks? To answer that, let us plot the percentage
of regular paths containing no more than a given number of upstream steps before
going downwards in the hierarchy. In Fig. 7.7 we can compare the results for the
empirical paths to some randomly chosen ones from all the possible regular paths
of the same length. We can observe that the empirical paths contain fewer upstream
steps, which means that those paths try to avoid stepping upwards in the hierarchy.
We can see that around 50% of the empirical paths contain no more than one
upstream step, while the random path’s percentage is below 10%. This finding adds
“prefer downstream” as a third identifiable rule that nature seems to consider when
picking a path. So it is not only our military sergeant who should use the downstream
path to issue a command, but this rule seems to be universal and present in other
real-life systems. This finding may sound somewhat contradictory to regularity,
which says that paths should first go upstream, i.e., towards the core, and then
downstream, towards the periphery ofthe network. However, this is just an apparent
contradiction. The prefer downstream rule only says to pick the downstream path if
available. For example, the bottom part of Fig. 7.5 shows two paths between nodes
Xand Y, one beginning with an upstream step followed by several downstream
ones, and one containing only downstream steps, marked as orange and green
paths respectively. In this case, the sergeant can choose between upstream and
downstream regular paths. The prefer downstream rule means that in such cases,
the downstream path is favorable, avoiding stepping upwards in the hierarchy.
7.4 Checkpoint 53
012345
50
100
Empirical paths
Random paths
Number of upwards steps
Percentage of paths
Fig. 7.7 Confirmation of the prefer downstream rule. The plot shows the percentage of regular
paths containing no more than a given number of upstream steps before entering the downstream
phase. The empirical paths tend to avoid stepping upwards in the hierarchy, which is reflected by
the much lower number of upstream steps, in comparison with the randomly selected regular paths
of the same length
7.4 Checkpoint
Let’s stop here for a second, take a deep breath and summarize our findings about
the structure of paths. First, we have seen that empirical paths can be slightly longer
than shortest paths. Although in some particular cases, the stretch can be more than
four steps, the paths are only around 10–30% longer on average. This inevitably
means that nature prefers the usage of short paths. Secondly, we have seen that real
networks seem to have an internal logicor an internal hierarchy,which the empirical
paths follow in the majority of cases. This simply means that empirical paths go first
upwards in the hierarchy and then downwards and paths cannot contain down-up
jumps. Finally, paths avoid stepping upwards in the hierarchy if possible, meaning
that if there are both downstream and upstream paths available, then the downstream
one should be picked.
Our findings hint at the operation of the “prefer short paths”, “prefer regular
paths” and “prefer downstream” path selection rules. Are these rules equally
important, or is one more important than the other? Are there any clear relative
priorities among the identified rules? In what follows we argue that there is a
reasonable prioritization among those components, which allows us to set up a
toy path selection rule set imitating nature’s path picking process. According to
Fig. 7.3 the prefer shortest path rule can only have lower priority than the “prefer
regular path” and the “prefer downstream” otherwise we would not experience
54 7 The Universal Nature of Paths
stretch at all. Since “prefer downstream” implies the “prefer regular path” rule, the
only reasonable choice is to: prefer regular paths at first, then prefer downstream
if there is a downstream path and from the remaining paths, prefer the short paths.
Remarkably, the length of the paths is just the third thing on this checklist.
7.5 Imitating Nature’s Path Picking Procedure
Now let’s check how close our argument about path selection above comes to real
empirical paths. In order to do this, we define our toy path selection rule set more
precisely and compare the selected paths to shortest and empirical ones. We define
our toy path selection rule set to:
Rule 1 Use regular paths only
Rule 2 Pick the downstream paths if available
Rule 3 From the paths remaining after Rule 1 and Rule 2, pick the shortest ones
Rule 4 If there are still multiple paths remaining break ties randomly
It turns out that the above simple path selection rule set gives very realistic path
stretch, close to the stretch computed for the real empirical paths. However, since we
explicitly prohibit the use of non-regular paths, all of them are regular, unlike real
empirical paths. We have done a hell of a job and made our paths always regular.
So, what our simple path selection method cannot reproduce is that empirical
paths sometimes violate the “prefer regular path” and the “prefer downstream”
rules, although in only a minority of cases. However, the slight randomization of
the centrality values of the nodes fixes this. Why would someone randomize the
centrality values of the nodes? Well, we have seen that in the case of large and
dynamic networks (like the Internet or a social network) we cannot even reconstruct
an up-to-date map. Therefore, it doesn’t sound reasonable to suppose that any real
person or entity representing a node in the network would know the exact structure,
i.e., the correct hierarchy of suchnetworks. So, the randomization can be interpreted
as simulating the case in which nodes have an approximate picture of the network
and therefore their position in the hierarchy is known only with some random error.
In that case, the nodes can only have an approximate picture of the internal logic or
the hierarchy of the network. This small modification recovers both the stretch and
the level of regularity exhibited by the empirical paths.
Our findings are suitable for gaining a vivid estimation of the traffic situation in a
large city during busy hours. Monday mornings are always a great stress for the road
network and the public transportation system, as everybody goes to work roughly
at the same time. On the one hand, due to stretch, empirical paths impose larger
average load on the network nodes, on the other, the load is even more concentrated
in the inner parts of the network hierarchy (i.e., on nodes located more centrally
in the network). The above presented path selection rule set seems to explain the
load footprint of real paths better, than resting simply on the assumption that people
always using shortest paths. In the context of the public transportation example, the
7.6 Two Short Illuminating Stories About Paths 55
toy policy allows us to better estimate the mass of people at various stations and the
possible length of lines at the ticket offices compared to the approximations based
on pure shortest paths.
7.6 Two Short Illuminating Stories About Paths
In short, our examinations so far hint at the fact that empirical paths follow the
underlying hierarchy (or logic) of the network. They avoid stepping upwards in
the hierarchy and they are short, although not always the shortest. How can we
make sure that our data cannot be explained by some other path selection rules
completely different from what we have found? In short, we can’t and this will give
us a wealth of possibilities for future research. But we can summarize here two
interesting stories investigating paths from a completely different angle, yet come
to a remarkably similar conclusion.
An Attracting Story About a Magnet
Anthony was a diligent man. He worked at the subsidiary of a large international
relocation company. His bosses took notice of his professional calling early in his
carrier, so he went steadily up the ladder becoming a regional leader in his younger
years. He was proud to be a fairly autonomous person solving emerging problems
on his own.
One day, a peculiar problem arose in connection with the relocation of a whole
medical laboratorywith some expensive medical equipment.The core of the problem
revolved around a specific part of an MRI machine used to monitor the physiological
processes inside the human body. Or more specifically, one of its components: a
high power magnet that quickly had to be moved overseas, leaving the only possible
choice but to carry it by plane. But air cargo companies were reluctant to ship the
magnet without a certification issued by an expert stating that transportation by air
was safe. Anthony decided to resolve the problem himself; he knew that none of
his subordinates had any experience in such matters. However, he knew that Mark,
who he did not know personally and was the leader of the Asian branch office, had
already been involved in relocating such medical imaging appliances. So he asked
one of his friends, Charles, a truck driver at the Asian sub-office to request for help
from his boss. In the following week, Charles tried to meet with Mark, but his efforts
were in vain; his boss was too busy to make time for him. Eventually, Anthony gave
up on Charles and tried sending direct e-mails to Mark, which were also to no avail;
it generated no reaction. Ultimately, one last option remained for Anthony: to call
the Central Office and ask for some official advice from his bosses. The answer to
his questions arrived the very same day. He got the contact details of a university
department with experimental physicists who have widely recognized competence in
assessing the effects of high power magnets on the navigation system of air flights.
With their help, Anthony succeeded in arranging the relocation of the laboratory,
although was not happy at all about being forced to resort to his superiors.
56 7 The Universal Nature of Paths
The above story perfectly reflects how an employee typically navigates around
his organization. The cheapest, and most often, the fastest way, is to look for a
subordinate to solve a problem. One should knowabout their subordinate employees
and their capabilities. Also, it is the employee’s responsibility to constantly listen
to his superiors’ orders. They are likely to be the best option when it comes to
problem-solving. Turning to superiors, however, takes time, and we also implicitly
communicate that we cannot cope by ourselves. You are generally expected to
minimize the amount of your boss’s time youwaste. Finding the help of a co-worker
may even be a better option, at least when you have an opportunity to ask a favor
of them. Indeed, impeding a superior is the most expensive alternative. Charles,
the truck driver, could not even manage to see his boss. Had Anthony known any
of Mark’s bosses directly, he could have avoided disturbing his own superiors at
the Central Office. Finally, we also note that the path from Anthony towards Mark
through Charles was not a regular way of connecting the two, as Charles did not
have any authority over Mark; he created a non-regular “valley” between the two
superiors (Fig. 7.8).
A group of scientists, Peter Sheridan Dodds, Duncan J. Watts, and Charles F.
Sabel at Columbia University conducted research in a closely related area some 15
years ago. They studied the information flow in organizational networks [8], for
example, the communication of people inside business firms. Their focus was on
the robustness of paths of information exchange between entities like departments
or individual persons forming the nodes of a network under stress caused by envi-
Fig. 7.8 Organizational
hierarchy in the story with the
magnet with a path
containing a “valley” through
a cross-hierarchy edge from
AnthonytoMark
Anthony Mark
Charles
Cross-hierarchy edge
7.6 Two Short Illuminating Stories About Paths 57
ronmental changes. They were particularly curious about the congestion situations
when the network disintegrates according to heavy load on some nodes at strategic
points. Just think about a business firm performing poorly due to overloading
employees in strategic positions, e.g., overstressed managers. Organizations are
supposed to have a strict hierarchical structure according to the relationship of sub-
ordination, but they also assume random bonds between individuals, representing
informal acquaintances between colleagues. These relationships form additional,
so-called cross-hierarchical, edges. When studying this phenomenon, they needed
a realistic path selection model, which approximated the load of the nodes in
the network to an appropriate extent. After the in-depth study of the literature of
organizations, Dodds, Watts, and Sabel settled on a simple Path selection model
with a three-step decision mechanism at each node traversed by the communication:
Step 1 If an employee of the organization looks for somebody that serves under
them somewhere on the working team, then she asks a direct subordinate that
connects her to the target person. It may also happen that the direct subordinate
is the target person, which ends the path right away.
Step 2 If the employee knows a co-worker through a cross-hierarchy edge, e.g.,
has an informal relationship with somebody who is the superior of the target
person or is the target person, then they choose that connection as their second
best option.
Step 3 Finally if none of the above holds, then the employee asks their direct
superior.
Now, let’s examine the structure of paths coming out of the Path selection model
above. The three-step decision process suggests that if in the organizational network
the target person (or target node) is not a subordinate nor an acquaintance of the
person holding the message, then they pass the message up in the hierarchy to a
direct superior. If the message reaches a node of which the target is a subordinate,
the message should be passed downwards in the hierarchy. First upwards, then
downwards. Sounds pretty familiar, doesn’t it? We have required our paths to go first
upwards and then downwards in the hierarchy avoiding containing a large-small-
large value pattern in the centrality sequence, thus non-regular paths cannot occur.
The Path selection model also suggests that if a node is a superior to the target then
it should pass the message downwards in the hierarchy, so downstream is preferred
in the hierarchy skipping upward steps whenever possible. If Anthony would have
applied the Path selection model in our story, then he would have solved the MRI
relocation problem much quicker, as in Anthony’s case the model suggests turning
immediately to his superiors.
Let’s carry on with our second story about a chaotic elevator system in a multi-
story office center.
A Single Story of a Multi-Story Office Center
Our second story is about Kate, a business consultant, who 1 day gets a very
interesting job. Her task is to design the layout of offices and working pathways
of people in a newly built 50-level office block. The main source of the problem
58 7 The Universal Nature of Paths
comes from the fact that a typical work-flow in the company is complicated. The
path that should be taken by an employee touches several floors on a daily basis.
Additionally, considering the number of offices, there are too few lifts built into the
building to serve the requirements. Even though they are large enough to carry a
few dozen people simultaneously. It is not the size that counts but the time it takes to
get to the caller. If the employees spend half the working hours waiting at the silver
doors, the overall efficiency drops to an unacceptable level.
After several weeks of speculation, Kate finally reaches a conclusion. The best
solution to cut down delays caused by time-consuming inter-story trips is to design
an efficient control algorithm for the lifts. It should be taken into account that the
control circuits of the elevators do not have any memory at all. The only information
that may be counted on in the movement decision is the direction it was headed right
before stopping at a floor: upward or downward. One of Kate’s design principles
is that unused lifts should always rest near the busiest floors. Furthermore, and
most importantly, short waiting times can only be achieved if an elevator, after a
stop, always continues to travel in the direction of the nearest floor where the call
button is pushed (independent of its previous direction). It doesn’t matter much if
more people are collected into the same cabin or the elevator takes a few additional
detours towards other floors on the way.
After careful planning, Kate has the specially designed lift control mechanism
implemented and lets the daily work begin in the office block. Several weeks later
she makes a visit to the office to take pride in a job well done. What she observes
extremely surprises her. People just gave up taking the elevators and everybody runs
up and down the stairs. By questioning a randomly picked employee caught in the
stairwell, she learns that the newly designed lift control system generally did prove
to be fast enough. Even the amount of energy consumed by the elevators dropped,
as the sum of all the paths taken were minimized. However, from time to time people
who traveled too far, for example from the top of the building to the first floor, almost
never seemed to reach the destination. On the way down there was always a new
calling from an upper floor, which was closer than the first floor. On one hand, the
inter-story trips were faster on average. On the other, however, people just could not
plan the duration of the trip in advance. The system simply became unpredictable.
Many times it took only a minute to arrive, but every now and then it lasted more
than half a day.
Finally, Kate drew the conclusion that her algorithm failed to fulfill its task
and she had to look for some additional advice and redesign the elevator control
algorithm.
What can cause such chaos in the office? Kate has a single design parameter
in mind, namely the total efficiency of the whole system. She doesn’t take into
account however, other aspects of the problem. If workers with strict deadlines find
a system unpredictable, they rarely venture a trip with an uncertain duration. A
similar observation can be taken in many walks of life, especially in the area of
engineering: an unpredictable operation of a machinery or an artificial system is
rarely beneficial or desired.
7.6 Two Short Illuminating Stories About Paths 59
In 2001, Lixin Gao and Jennifer Rexford at Princeton University studied the pre-
dictability of the Internet routing system [10], which, among computer networking
fellows, is the technical term for the path selection of packets on the network. We
have already seen that paths, which packets take over the Internet, are determined
by routing tables configured in each node by its operating personnel.Since different
nodes may belong to different operating staff working at different networking
companies, they will place their own, independent communication tricks on those
routing tables. For example, to reach a given destination, an Internet company
may prefer to avoid some insecure regions of the Internet. If the companies follow
utterly different rules irrespective of each other, the resulting system may become
chaotic. Some packets may circulate in the network for an indeterminate amount
of time, just like the people in the office elevator in the story above. Due to such
chaotic behaviour, complete regions of the Internet may become unreachable even
if they are connected properly. How can we make the system stable without forcing
individuals to synchronize their actions all the time? Are there any general, but not
too restrictive, rules that the companies on the Internet (or the employees in the
story) should follow, resulting in a tractable system?
Well, Gao and Rexford came to the conclusion that if the Internet companies(the
nodes) agree on some simple and reasonable rules when generating their routing
tables, then the network will behave nicely even if its structure changes in time. The
rules have become famous among networking theoreticians and practitioners under
the name of the “Gao-Rexford conditions”. They demonstrated the problem on a
simple object called the dispute wheel [12], a pathological case of chaotic behaviour
widely known among Internet routing theoreticians (see Fig.7.9).
The solution started by observing that Internet companies act on different levels
of a hierarchy called the “service chain”. Do you remember our small Internet
in Chap. 5? Canadian Federal Co. and Maine Trans-Atlantic Co. were at the top
level, and Castle Rock, Salem’s Lot and Dunwich were below it. The Gao-Rexford
conditions refer to the acts that should be performed when a packet travels from one
level of the service chain to another.
TheRuleofHierarchy: If a packet comes from an upper level of the service
chain, it should not be sent up again. Or more specifically, the routing tables
should be configured in such a way that packets should never need to turn back
upwards while heading down (see Fig. 7.10).
The Rule of Preferring the Downward Direction: When it is equally appropri-
ate to send a packet up to a higher level of the hierarchy or down to a lower level,
the downward direction should be chosen.
And that is all. If those two simple rules are followed, the Internet is nice and
safe. These two rules are based on a simple observation, that there is an internal
logic, an order on the Internet. A built-in hierarchy of Internet companies. This
hierarchy enables the definition of “up” and “down” and the Gao-Rexford conditions
simply use these directions to prevent packets infinitely circulating in the network.
But there is more. If we meditate on the first rule a bit more, we find that, besides
ensuring system predictability, it has an additional advantage for the Internet as
60 7 The Universal Nature of Paths
Boss
Alice
Bob Carol
Fig. 7.9 A possible interpretation of the dispute wheel, a theoretical object illustrating the
unpredictable behavior of communicating actors or nodes making decisions independently of
each other not possessing the Gao-Rexford conditions. In the figure, Alice, Bob, and Carol, the
employees of an imaginary small organization, communicate with each other, with the intention of
passing possibly unpleasant news to their boss. Each of them is reluctant to confront the boss with
the bad news, so they all try to persuade each other to relay the message to the boss, but none of
them actually does so. The wheel exemplifies that the message never arrives at its destination, even
so, the nodes in the network are well connected
Castle Rock
Salem’s Lot
MCT CF
Dunwich
Fig. 7.10 Our previously developed tiny model of the Internet initiated by the people of Castle
Rock to communicate with the outside world by connecting to nearby town Salem’s Lot and
Dunwich in England using the transit services of Main County Trans-Atlantic (MCT) and Canadian
Federal Co. (CF) as a backup route
7.6 Two Short Illuminating Stories About Paths 61
well. We require that packets heading towards companies at lower layers of the
hierarchy cannot be sent up again. That means it cannot happen that two higher
layer companies communicate through a lower layer. Canadian Federal Co. and
Main County Trans-Atlantic, the companies handling high traffic volumes, may
never use the, possibly quite lightweight, networking infrastructure of Castle Rock.
That sounds quite reasonable that majors should not overload the system of the
lesser. They should use a direct connection or even larger transit companies to make
the connection.
Applying the Rule of Preferring the Downward Direction also has some addi-
tional advantage that we can uncover with a little analysis. In the hierarchy of the
Internet, lower layer companies pay the higher ones to be connected to the big
network, pretty much like your home Internet subscription. You pay your provider
to forward your data to the Internet, but your provider won’t pay you for receiving
data. Thus, the connection is free for the one being higher in the hierarchy. Which
one is the better choice? Using a free connection down to an inferior party or going
upstream for a price instead? Let’s prefer the downstream!
Okay, now can we apply these magic rules forthe elevator problem as well? Let’s
continue and see how Kate finally solves the problem by studying some advanced
computer networking principles. After a more intensive investigation of the business
procedures within the company, Kate realizes that the employees’ inter-story paths
show some orderliness. Most of the time they visit one or more of their superiors
then return. So, a better office arrangement would be to move the employees to
different levels of the building according to their position in the chain of command.
More senior officials should be put on the higher floors, with the general manager
at the top. In such a way, the routes of the employees become less random, the
movements can be harmonized a bit more. In short, we build a hierarchy in the
office according to the journeys of the employees.
Now, it is not too difficult to see the parallels in the two different areas of
engineering. By rephrasing the elevator problem in new terms, we are ready to
relieve the difficulties of the office block: the elevator makes a decision every time
it stops at a floor during operation. Those decisions depend on the demands (i.e.,
destination floors) of the actual travelers and the current direction of the elevator,
independent of previous decisions due to the lack of memory. If you recall how
Internet routing works (see Chap. 6.3), you find the situation very similar. Now we
can match the control decision of the elevator with the forwarding policy of the
Internet companies in sending the packets to lower or higher hierarchy levels. Both
decisions are local in time, meaning that past decisions along the route cannot be
taken into account. It doesn’t matter how complicated the lift control mechanism is:
if at a minimum, it keeps itself to the Gao-Rexford principles, the resulting global
behavior remains predictable.
The Rule of Hierarchy tells us that we should never change the recent direction
of the elevator even if one of the current travelers decides so. Unless, of course,
there are no further demands to keep going in that direction. In such a way, none of
the travelers need to take unnecessary detours, that is taking up-down-up or down-
up-down “valleys” during their path as the Rule of Hierarchy forbids the down-up
62 7 The Universal Nature of Paths
pattern. Of course, an employee may lose their temper while waiting for the elevator
and push the call button for both directions announcing an ambiguous request. In
such a case, she is doomed to be taken on a possibly longer journey, infuriating her
even more.
Now consider what happens if the control mechanism loses the information of
the current direction, for example, due to a power failure or a circuit damage. This
is when the Rule of Preferring the Downward Direction comes into life. If there
was not a default action included in the control mechanism for such cases, the lift
might still roam up and down for eternity between neighboring floors changing its
mechanical mind each time. The choice for the downward direction for an elevator
as a default action seems to be more advantageous yielding an escape route in the
course of emergency e.g., fire or earthquake, should the careless employee decide
upon using the elevator in such a case.
Had Kate been aware of the above two rules when designing the elevator control
system in the office center for the first time, the employees would have been much
less fatigued when finally getting home at the end of the day. How fortuitous that
elevator companies in the real world generally stick by it!
Now we can see that Gao and Rexford came to remarkably similar path selection
rules to ours, albeit their target was to ensure the predictable operation of the Internet
via theory. Although the definition of the hierarchy is not the same as ours, the Rule
of Hierarchy sounds familiar to our finding that empirical paths rarely contain a
large-small-large pattern anywhere in their centrality sequence. Although at this
time it is not clear if the Gao-Rexford conditions can be mapped exactly to our
centrality-based rules, it is worth wondering whether our observations about the
stretch of paths enable the predictability of operation in other networks similar to
the Internet. Additionally, also the Rule of Preferring the Downward direction has a
ghostly resemblance to our observations on paths in real-world systems.
These short stories have identified similar logic of path selection in two distinct
fields of networks. We found something very similar through measurements and
not just only on the Internet but also in three networks coming from very different
corners of life. Can all these still be a mere coincidence? In these stories, the authors
use very different methods and definitions for constructing the underlying hierarchy
and they formulate the rules of path selection quite differently. Our arguments
about the structure of paths generalize these sporadic results and show that there
can be an amazing universality in path selection across many different networks.
Path selection rules built upon centrality creates a common ground for speaking
about underlying hierarchies. Thus reasoning about networks and paths coming
from possibly arbitrary fields of life is possible. Now we have a rough picture about
the structure of real-world paths.
The reader at this point can ask the very reasonable question: So what? We
have identified some characteristic similarities of path selection in various systems
coming from a wide spectrum of life. What can we do with all this? For what
purpose can we use it? Well, before delving into this, we suggest sitting back for
a second and just simply being amazed by the possible universality of the nature
of paths. One should never underestimate the benefits of just sitting back, relaxing
7.6 Two Short Illuminating Stories About Paths 63
CHOICE
Randomness Rationality
Fig. 7.11 Paths in nature lie between pure randomness and pure rationality
and being amazed by something. So, paths in real life are clearly not the results of
pure randomness. They are not like mucking about in the city. Quite the contrary,
they are short and right on target. On the other hand, paths are not governed by
pure rationality either. They are not always the most efficient; those would be the
“soulless”, monotone and dry shortest paths. They are somewhat stretched. Real
paths exist somewhere in the middle of these two extremes.2Now, what is the
similarity between pure randomness and pure rationality? There is no possibility
of choice in either of them. When our paths are governed by randomness, we are
“just floatin’ around accidental-like on a breeze”,3while at the other extreme, pure
rationality dictates taking the path with the highest supposed benefits or lowest cost,
depriving us of our free will. So, it seems our paths lie in between. Although our
results suggest they are closer to rationality. Recall, that our empirical paths were
only around 10–30% longer than the shortest paths, so real paths are nearly as
effective, or energy-efficient as shortest paths. But while being efficient and target
oriented, real systems can still make a choice. The 10–30% enables us to form
messages about the way we live our lives, our way of solving problems. Through
these little extra steps, we are allowed to make a difference and form our own stories.
10–30%. This is our playground.
Even ancient Chinese wisdom has something interesting in store for us about
choosing a path of appropriate length. The more than 6000-year-old art of Feng
Shui is about harmonizing people with their direct or indirect environment. It is
actually one of the many Taos of the big Tao, just like the art of tea ceremonies or
paysage. According to Feng Shui, everything in the world, life or lifeless, is a form
of liquid energy, the Chi, that also flows constantly around any objects, connecting
them together. It streams inside and outside of things, nothing can block its way,
not even the thickest concrete wall. Objects, however, can modify its direction or
speed, which can alter the quality of energy. By rearranging the objects that surround
us, creating appropriate paths, we can avoid negative energy and increase positive
energy, which can have a beneficial effect on our mood, emotions, or even our
2See Fig. 7.11.
3I don’t know if we each have a destiny, or if we’re all just floatin’ around accidental-like on a
breeze But I, I think maybe it’s both.
— Forrest Gump.
64 7 The Universal Nature of Paths
health. The course that the ever-flowing Chi takes has a crucial influence on the
energy.So, what is it, that Feng Shui tells us about paths?
At this part of the book, you should not be surprised that Feng Shui is absolutely
against shortcuts. Straight pathways or trails should not be used to connect objects,
for example, in your garden. According to the Tao, it is because the flow of Chi
becomes speedy, streaming too fast out of your garden. But trail crossings or sharply
curving paths just like sharp corners or other saliences have a negative effect,
sending forth “poisonous arrows” (negative energy or Sha) blocking the flow of
Chi. Straight short paths are not correct, but lengthy curving driveways are equally
wrong. Brightly winding watercourses are of less positive effect, but cutting down
the curves also emits Sha. The ultimate advice of Feng Shui is to form slightly
curved pathways in your garden (Fig.7.12). Similarly to the curved middle path in
the Yin Yang symbol (Fig. 7.13), if it finds you in an even more philosophical mood.
Fig. 7.12 A curved pathway in the Japanese garden of the Budapest Zoo overlayed with an
artificial pathway constructed by joining two segments each being a third of a circle. Walking
along the synthetic path makes the distance between the two endpoints around 20% longer. The
photo is the property of the authors
7.6 Two Short Illuminating Stories About Paths 65
Fig. 7.13 The Yin Yang, a
Tai Chi symbol with the
indication of the middle path
by a red line
p
a
t
h
T
h
e
m
i
d
d
l
e
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Chapter 8
Amazing Scientific Discoveries: Aspirin,
Cattle, Business Communication
and Others
Besides the philosophical arguments, one could think of a series of possibilities
where all these observationsabout paths could be used. Have you everwondered, for
example, what happens in your body after you swallow an Aspirin? As incredible
as it sounds, this question was answered only 74 years after the development of
the medicine. It was 1897 when the young German chemist Dr. Felix Hoffmann
managed to stabilize the agent of Aspirin. After patenting in Germany in 1899 and
in the US in 1900, Aspirin started its great triumph and became the most popular
painkiller worldwide. Even Neil Armstrong took an Aspirin pill in his medical-kit
when going to the Moon on the Apollo 11. In the early 1970s, more and more
researchers asked the question: How and where does Aspirin work in the body?
Pharmacologist Sir John Vane was the first to demonstrate the classical effect profile
of Aspirin, for which he received the Nobel Prize in 1982.
What is this foggy mystery about the effect of drugs? Well, the effects and side-
effects of drugs are mainly characterized by the path of molecules which the drug
interacts until it has its targeted effect. Usually, the drugs are first converted into
so-called metabolites which then interact with the metabolic network of the cells.1
The metabolic network is comprised of the maze of chemical reactions in our cells
over which various materials are converted into each other. A specific path or set
of paths in this network can basically correspond to a chain of chemical reactions
happening after somebody swallows a pill. For example, after taking an Aspirin,
it is readily hydrolyzed to salicylic acid, which in turn undergoes conjugation
reactions generating the major metabolites salicyluric acid and glucuronides. And
this is just the major path of Aspirin’s metabolism, there are other minor paths
governing the whole metabolic process and thus the effects and side effects of
Aspirin. Interestingly, simple shortest paths in the metabolic network do not always
reflect the biochemical facts. Such paths may introduce biologically infeasible
1See, Fi g. 8.1.
© The Author(s) 2021
A. Gulyás et al., Pa t h s ,https://doi.org/10.1007/978-3-030-47545-1_8
67
68 8 Amazing Scientific Discoveries: Aspirin, Cattle, Business Communication. . .
Fig. 8.1 A small part of human metabolism by Evans Love. [With the permission of Evans Love]
8 Amazing Scientific Discoveries: Aspirin, Cattle, Business Communication. . . 69
shortcuts [18]. Thus, a deeper understanding of the structure of paths can be used to
estimate the side effects of drugs more carefully even before anyone takes them.
The sum of everyday paths taken by people in larger cities to reach their
workplaces or homes constitutes the load which public transportation and road
systems have to carry day-by-day. The appropriate knowledge about these paths can
support the design and operation of such systems and approximate their behavior
in unforeseen situations, such as scheduled network changes, roadworks, natural
disasters or walkouts. In the era of in-pocket GPS powered route planners, the
assumption that people use the shortest paths for traveling between their sources
and destinations seems more than reasonable, it seemssomewhat obvious. Recently,
Shanjiang Zhu and David Levinson at the University of Minnesota decided to
check the validity of this assumption [27]. They evaluated the paths followed by
residents of the Minneapolis-St. Paul metropolitan area and they had an interesting
observation. For some reason, people don’t always use the shortest possible path
for their journeys. They found that, if the destination is near (around 1.5 km),
80% of people follow the shortest paths, the remaining 20% prefer a longer ride.
Interestingly, if the destination lies farther, the larger portion of people tend to take
a longer ride compared to the shortest possible path. For example, if the destination
is around 16 km away, then only around 17–18% of people follow the shortest path
and the majority of them will choose a longer path. Thus, the authors’ claim is that
the available path selection methods based on the shortest path assumption cannot
reveal the majority of paths that individuals take and they promote future effort