[Show abstract][Hide abstract] ABSTRACT: Data transfer is one of the main functions of the Internet. The Internet
consists of a large number of interconnected subnetworks or domains, known as
Autonomous Systems. Due to privacy and other reasons the information about what
route to use to reach devices within other Autonomous Systems is not readily
available to any given Autonomous System. The Border Gateway Protocol is
responsible for discovering and distributing this reachability information to
all Autonomous Systems. Since the topology of the Internet is highly dynamic,
all Autonomous Systems constantly exchange and update this reachability
information in small chunks, known as routing control packets or Border Gateway
Protocol updates. Motivated by scalability and predictability issues with the
dynamics of these updates in the quickly growing Internet, we conduct a
systematic time series analysis of Border Gateway Protocol update rates. We
find that Border Gateway Protocol update time series are extremely volatile,
exhibit long-term correlations and memory effects, similar to seismic time
series, or temperature and stock market price fluctuations. The presented
statistical characterization of Border Gateway Protocol update dynamics could
serve as a ground truth for validation of existing and developing better models
of Internet interdomain routing.
[Show abstract][Hide abstract] ABSTRACT: Common sense suggests that networks are not random mazes of purposeless connections, but that these connections are organized so that networks can perform their functions well. One function common to many networks is targeted transport or navigation. Here, using game theory, we show that minimalistic networks designed to maximize the navigation efficiency at minimal cost share basic structural properties with real networks. These idealistic networks are Nash equilibria of a network construction game whose purpose is to find an optimal trade-off between the network cost and navigability. We show that these skeletons are present in the Internet, metabolic, English word, US airport, Hungarian road networks, and in a structural network of the human brain. The knowledge of these skeletons allows one to identify the minimal number of edges, by altering which one can efficiently improve or paralyse navigation in the network.
[Show abstract][Hide abstract] ABSTRACT: Represented as graphs, real networks are intricate combinations of order and
disorder. Fixing some of the structural properties of network models to their
values observed in real networks, many other properties appear as statistical
consequences of these fixed observables, plus randomness in other respects.
Here we employ the $dk$-series, a complete set of basic characteristics of the
network structure, to study the statistical dependencies between different
network properties. We consider six real networks---the Internet, US airport
network, human protein interactions, technosocial web of trust, English word
network, and an fMRI map of the human brain---and find that many important
local and global structural properties of these networks are closely reproduced
by $dk$-random graphs whose degree distributions, degree correlations, and
clustering are as in the corresponding real network. We discuss important
conceptual, methodological, and practical implications of this evaluation of
network randomness.
[Show abstract][Hide abstract] ABSTRACT: Prediction and control of network dynamics are grand-challenge problems in
network science. The lack of understanding of fundamental laws driving the
dynamics of networks is among the reasons why many practical problems of great
significance remain unsolved for decades. Here we study the dynamics of
networks evolving according to preferential attachment, known to approximate
well the large-scale growth dynamics of a variety of real networks. We show
that this dynamics is Hamiltonian, thus casting the study of complex networks
dynamics to the powerful canonical formalism, in which the time evolution of a
dynamical system is described by Hamilton's equations. We derive the explicit
form of the Hamiltonian that governs network growth in preferential attachment.
This Hamiltonian turns out to be nearly identical to graph energy in the
configuration model, which shows that the ensemble of random graphs generated
by preferential attachment is nearly identical to the ensemble of random graphs
with scale-free degree distributions. In other words, preferential attachment
generates nothing but random graphs with power-law degree distribution. The
extension of the developed canonical formalism for network analysis to richer
geometric network models with non-degenerate groups of symmetries may
eventually lead to a system of equations describing network dynamics at small
scales.
[Show abstract][Hide abstract] ABSTRACT: Networks representing many complex systems in nature and society share some
common structural properties like heterogeneous degree distributions and strong
clustering. Recent research on network geometry has shown that those real
networks can be adequately modeled as random geometric graphs in hyperbolic
spaces. In this paper, we present a computer program to generate such graphs.
Besides real-world-like networks, the program can generate random graphs from
other well-known graph ensembles, such as the soft configuration model, random
geometric graphs on a circle, or Erd\H{o}s-R\'enyi random graphs. The
simulations show a good match between the expected values of different network
structural properties and the corresponding empirical values measured in
generated graphs, confirming the accurate behavior of the program.
[Show abstract][Hide abstract] ABSTRACT: We introduce and explore a new method for inferring hidden geometric
coordinates of nodes in complex networks based on the number of common
neighbors between the nodes. We compare this approach to the one in [1], which
is based on the connections (and disconnections) between the nodes, i.e., on
the links that the nodes have (or do not have). We find that for high degree
nodes the common-neighbors approach yields a more accurate inference than the
link-based method, unless heuristic periodic adjustments (or "correction
steps") are used in the latter. The common-neighbors approach is
computationally intensive, requiring $O(t^4)$ running time to map a network of
$t$ nodes, versus $O(t^3)$ in the link-based method. But we also develop a
hybrid method with $O(t^3)$ running time, which combines the common-neighbors
and link-based approaches, and explore a heuristic that reduces its running
time further to $O(t^2)$, without significant reduction in the mapping
accuracy. We apply this method to the Autonomous Systems (AS) Internet, and
reveal how soft communities of ASes evolve over time in the similarity space.
We further demonstrate the method's predictive power by forecasting future
links between ASes. Taken altogether, our results advance our understanding of
how to efficiently and accurately map real networks to their latent geometric
spaces, which is an important necessary step towards understanding the laws
that govern the dynamics of nodes in these spaces, and the fine-grained
dynamics of network connections.
Physical Review E 02/2015; 92(2). DOI:10.1103/PhysRevE.92.022807 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Exponential random graph models have attracted significant research attention
over the past decades. These models are maximum-entropy ensembles under the
constraints that the expected values of a set of graph observables are equal to
given values. Here we extend these maximum-entropy ensembles to random
simplicial complexes, which are more adequate and versatile constructions to
model complex systems in many applications. We show that many random simplicial
complex models considered in the literature can be casted as maximum-entropy
ensembles under certain constraints. We introduce and analyze the most general
random simplicial complex ensemble $\mathbf{\Delta}$ with statistically
independent simplices. Our analysis is simplified by the observation that any
distribution $\mathbb{P}(O)$ on any collection of objects $\mathcal{O}=\{O\}$,
including graphs and simplicial complexes, is maximum-entropy under the
constraint that the expected value of $-\ln \mathbb{P}(O)$ is equal to the
entropy of the distribution. With the help of this observation, we prove that
ensemble $\mathbf{\Delta}$ is maximum-entropy under two types of constraints
that fix the expected numbers of simplices and their boundaries.
[Show abstract][Hide abstract] ABSTRACT: All real networks are different, but many have some structural properties in
common. There seems to be no consensus on what the most common properties are,
but scale-free degree distributions, strong clustering, and community structure
are frequently mentioned without question. Surprisingly, there exists no simple
generative mechanism explaining all the three properties at once in growing
networks. Here we show how latent network geometry coupled with preferential
attachment of nodes to this geometry fills this gap. We call this mechanism
geometric preferential attachment (GPA), and validate it against the Internet.
GPA gives rise to soft communities that provide a different perspective on the
community structure in networks. The connections between GPA and cosmological
models, including inflation, are also discussed.
[Show abstract][Hide abstract] ABSTRACT: The common sense suggests that networks are not random mazes of purposeless
connections, but that these connections are organised so that networks can
perform their functions. One common function that many networks perform is
targeted transport or navigation. Here with the help of game theory we show
that minimalistic networks designed to maximise the navigation efficiency at
minimal cost share basic structural properties of real networks. These
idealistic networks are Nash equilibria of a network construction game whose
purpose is to find an optimal trade-off between the network cost and
navigability. They are navigation skeletons that we show are present in the
Internet, {\it E. coli} metabolic network, English word network, US airport
network, and the Hungarian road network. The knowledge of these skeletons
allows one to identify the minimal number of edges by altering which one can
dramatically improve or paralyse the navigation in the network.
[Show abstract][Hide abstract] ABSTRACT: Networks often represent systems that do not have a long history of studies
in traditional fields of physics, albeit there are some notable exceptions such
as energy landscapes and quantum gravity. Here we consider networks that
naturally arise in cosmology. Nodes in these networks are stationary observers
uniformly distributed in an expanding open FLRW universe with any scale factor,
and two observers are connected if one can causally influence the other. We
show that these networks are growing Lorentz-invariant graphs with power-law
distributions of node degrees. New links in these networks not only connect new
nodes to existing ones, but also appear at a certain rate between existing
nodes, as they do in many complex networks.
New Journal of Physics 10/2013; 16(9). DOI:10.1088/1367-2630/16/9/093031 · 3.56 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In statistical physics any given system can be either at an equilibrium or away from it. Networks are not an exception. Most network models can be classified as either equilibrium or growing. Here we show that under certain conditions there exists an equilibrium formulation for any growing network model, and vice versa. The equivalence between the equilibrium and nonequilibrium formulations is exact not only asymptotically, but even for any finite system size. The required conditions are satisfied in random geometric graphs in general and causal sets in particular, and to a large extent in some real networks.
Physical Review E 08/2013; 88(2):022808. DOI:10.1103/PhysRevE.88.022808 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: As the Internet AS-level topology grows over time, some of its structural
properties remain unchanged. Such time- invariant properties are generally
interesting, because they tend to reflect some fundamental processes or
constraints behind Internet growth. As has been shown before, the
time-invariant structural properties of the Internet include some most basic
ones, such as the degree distribution or clustering. Here we add to this
time-invariant list a non-trivial property - k-dense decomposition. This
property is derived from a recursive form of edge multiplicity, defined as the
number of triangles that share a given edge. We show that after proper
normalization, the k- dense decomposition of the Internet has remained stable
over the last decade, even though the Internet size has approximately doubled,
and so has the k-density of its k-densest core. This core consists mostly of
content providers peering at Internet eXchange Points, and it only loosely
overlaps with the high-degree or high-rank AS core, consisting mostly of tier-1
transit providers. We thus show that high degrees and high k-densities reflect
two different Internet-specific properties of ASes (transit versus content
providers). As a consequence, even though degrees and k-densities of nodes are
correlated, the relative fluctuations are strong, and related to that, random
graphs with the same degree distribution or even degree correlations as in the
Internet, do not reproduce its k-dense decomposition. Therefore an interesting
open question is what Internet topology models or generators can fully explain
or at least reproduce the k-dense properties of the Internet.
[Show abstract][Hide abstract] ABSTRACT: Parallel Discrete Event Simulation (PDES) is based on the partitioning of the simulation model into distinct Logical Processes (LPs), each one modeling a portion of the entire system, which are allowed to execute simulation events concurrently. This ...
[Show abstract][Hide abstract] ABSTRACT: Prediction and control of the dynamics of complex networks is a central problem in network science. Structural and dynamical similarities of different real networks suggest that some universal laws might accurately describe the dynamics of these networks, albeit the nature and common origin of such laws remain elusive. Here we show that the causal network representing the large-scale structure of spacetime in our accelerating universe is a power-law graph with strong clustering, similar to many complex networks such as the Internet, social, or biological networks. We prove that this structural similarity is a consequence of the asymptotic equivalence between the large-scale growth dynamics of complex networks and causal networks. This equivalence suggests that unexpectedly similar laws govern the dynamics of complex networks and spacetime in the universe, with implications to network science and cosmology.
[Show abstract][Hide abstract] ABSTRACT: The principle that 'popularity is attractive' underlies preferential attachment, which is a common explanation for the emergence of scaling in growing networks. If new connections are made preferentially to more popular nodes, then the resulting distribution of the number of connections possessed by nodes follows power laws, as observed in many real networks. Preferential attachment has been directly validated for some real networks (including the Internet), and can be a consequence of different underlying processes based on node fitness, ranking, optimization, random walks or duplication. Here we show that popularity is just one dimension of attractiveness; another dimension is similarity. We develop a framework in which new connections optimize certain trade-offs between popularity and similarity, instead of simply preferring popular nodes. The framework has a geometric interpretation in which popularity preference emerges from local optimization. As opposed to preferential attachment, our optimization framework accurately describes the large-scale evolution of technological (the Internet), social (trust relationships between people) and biological (Escherichia coli metabolic) networks, predicting the probability of new links with high precision. The framework that we have developed can thus be used for predicting new links in evolving networks, and provides a different perspective on preferential attachment as an emergent phenomenon.
[Show abstract][Hide abstract] ABSTRACT: Recent years have shown a promising progress in understanding geometric
underpinnings behind the structure, function, and dynamics of many complex
networks in nature and society. However these promises cannot be readily
fulfilled and lead to important practical applications, without a simple,
reliable, and fast network mapping method to infer the latent geometric
coordinates of nodes in a real network. Here we present HyperMap, a simple
method to map a given real network to its hyperbolic space. The method utilizes
a recent geometric theory of complex networks modeled as random geometric
graphs in hyperbolic spaces. The method replays the network's geometric growth,
estimating at each time step the hyperbolic coordinates of new nodes in a
growing network by maximizing the likelihood of the network snapshot in the
model. We apply HyperMap to the AS Internet, and find that: 1) the method
produces meaningful results, identifying soft communities of ASs belonging to
the same geographic region; 2) the method has a remarkable predictive power:
using the resulting map, we can predict missing links in the Internet with high
precision, outperforming popular existing methods; and 3) the resulting map is
highly navigable, meaning that a vast majority of greedy geometric routing
paths are successful and low-stretch. Even though the method is not without
limitations, and is open for improvement, it occupies a unique attractive
position in the space of trade-offs between simplicity, accuracy, and
computational complexity.
[Show abstract][Hide abstract] ABSTRACT: Many social, biological and technological systems can be conveniently
represented as bipartite networks, consisting of two disjoint sets of
elements along with edges connecting only elements from different sets.
Many of such systems are characterized by high values of bipartite
clustering coefficient. We also find that pairs of elements in these
bipartite systems tend to have many common neighbors. We present a
natural interpretation of these observations. We suggest that elements
of the above bipartite systems exist in underlying metric spaces, such
that the observed high clustering is a topological reflection of the
triangle inequality, the key property of metric space. We propose a
simple stochastic mechanism of formation of bipartite networks embedded
in metric spaces. We prove that this mechanism is able to reproduce the
observed topological properties of bipartite networks. We also discuss
the possibility of constructive embedding of real bipartite systems into
metric spaces. In my talk I will overview the concept of hidden metric
spaces with respect to both unipartite and bipartite networks. I will
also discuss existing methods used to infer hidden metric spaces in real
networks and possible applications for bipartite networks.
[Show abstract][Hide abstract] ABSTRACT: We introduce and study random bipartite networks with hidden variables. Nodes
in these networks are characterized by hidden variables which control the
appearance of links between node pairs. We derive analytic expressions for the
degree distribution, degree correlations, the distribution of the number of
common neighbors, and the bipartite clustering coefficient in these networks.
We also establish the relationship between degrees of nodes in original
bipartite networks and in their unipartite projections. We further demonstrate
how hidden variable formalism can be applied to analyze topological properties
of networks in certain bipartite network models, and verify our analytical
results in numerical simulations.