The Physics of Communicability in Complex Networks

Abstract

A fundamental problem in the study of complex networks is to provide
quantitative measures of correlation and information flow between different
parts of a system. To this end, several notions of communicability have been
introduced and applied to a wide variety of real-world networks in recent
years. Several such communicability functions are reviewed in this paper. It is
emphasized that communication and correlation in networks can take place
through many more routes than the shortest paths, a fact that may not have been
sufficiently appreciated in previously proposed correlation measures. In
contrast to these, the communicability measures reviewed in this paper are
defined by taking into account all possible routes between two nodes, assigning
smaller weights to longer ones. This point of view naturally leads to the
definition of communicability in terms of matrix functions, such as the
exponential, resolvent, and hyperbolic functions, in which the matrix argument
is either the adjacency matrix or the graph Laplacian associated with the
network. Considerable insight on communicability can be gained by modeling a
network as a system of oscillators and deriving physical interpretations, both
classical and quantum-mechanical, of various communicability functions.
Applications of communicability measures to the analysis of complex systems are
illustrated on a variety of biological, physical and social networks. The last
part of the paper is devoted to a review of the notion of locality in complex
networks and to computational aspects that by exploiting sparsity can greatly
reduce the computational efforts for the calculation of communicability
functions for large networks.

Full text

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Technical Report
TR-2011-012
The Physics of Communicability in Complex Networks
by
Ernesto Estrada, Naomichi Hatano, Michele Benzi
Mathematics and Computer Science
EMORY UNIVERSITY

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The Physics of Communicability in Complex Networks
Ernesto Estrada1
Department of Mathematics and Statistics, Department of Physics, SUPA, and Institute of
Complex Systems, University of Strathclyde, Glasgow G1 1XQ, UK.
Naomichi Hatano
Institute of Industrial Science, University of Tokyo, Komaba, Meguro, Tokyo 153-8505,
Japan.
Michele Benzi
Department of Mathematics and Computer Sciences, Emory University, Atlanta, Georgia
30322, USA


1 Corresponding author. E-mail: ernesto.estrada@strath.ac.uk

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ABSTRACT

A fundamental problem in the study of complex networks is to provide quantitative measures
of correlation and information flow between different parts of a system. To this end, several
notions of communicability have been introduced and applied to a wide variety of real-world
networks in recent years. Several such communicability functions are reviewed in this paper.
It is emphasized that communication and correlation in networks can take place through
many more routes than the shortest paths, a fact that may not have been sufficiently
appreciated in previously proposed correlation measures. In contrast to these, the
communicability measures reviewed in this paper are defined by taking into account all
possible routes between two nodes, assigning smaller weights to longer ones. This point of
view naturally leads to the definition of communicability in terms of matrix functions, such
as the exponential, resolvent, and hyperbolic functions, in which the matrix argument is either
the adjacency matrix or the graph Laplacian associated with the network.
Considerable insight on communicability can be gained by modeling a network as a system
of oscillators and deriving physical interpretations, both classical and quantum-mechanical,
of various communicability functions. Applications of communicability measures to the
analysis of complex systems are illustrated on a variety of biological, physical and social
networks. The last part of the paper is devoted to a review of the notion of locality in
complex networks and to computational aspects that by exploiting sparsity can greatly reduce
the computational efforts for the calculation of communicability functions for large networks.

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CONTENTS
I. INTRODUCTION ............................................................................................................................... 4
A. Overview: interaction and correlation ............................................................................................ 4
B. Correlation function ....................................................................................................................... 8
C. Plan of the article .......................................................................................................................... 10
II. COMMUNICABILITY IN NETWORKS ....................................................................................... 10
A. Combinatorial definition .............................................................................................................. 10
B. Some combinatorial formulae ...................................................................................................... 15
III. PHYSICAL ANALOGIES ............................................................................................................. 17
A. Oscillator Networks ..................................................................................................................... 17
B. Network Hamiltonians ................................................................................................................. 18
C. Network of Quantum Oscillators ................................................................................................. 20
D. Network of Classical Oscillators .................................................................................................. 26
IV. COMPARING COMMUNICABILITY FUNCTIONS .................................................................. 29
A. Study of a social conflict ........................................................................................................... 30
B. Study of biomolecular networks ............................................................................................... 35
V. COMMUNICABILITY AND THE ANALYSIS OF NETWORKS ............................................... 39
A. Microscopic analysis of networks ................................................................................................ 40
B. Mesoscopic analysis of networks ................................................................................................. 42
C. Macroscopic analysis of networks ............................................................................................... 47
D. Multiscale analysis of networks ................................................................................................... 52
E. Communicability at negative absolute temperature ..................................................................... 55
VI. COMMUNICABILITY AND LOCALIZATION IN COMPLEX NETWORKS ......................... 59
A. Generalities ............................................................................................................................... 59
B. Exponential decay in communicability ........................................................................................ 62
VII. COMPUTABILITY OF COMMUNICABILITY FUNCTIONS .................................................. 66
A. Analytical results .......................................................................................................................... 66
B. Numerical experiments ................................................................................................................ 73
VIII. CONCLUSIONS AND PERSPECTIVES ................................................................................... 74
REFERENCES ..................................................................................................................................... 77

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I. INTRODUCTION
A. Overview: interaction and correlation
In social system studies it is frequently found that agents belonging to the same
group tend to behave similarly (Manski, 2000). Social scientists use the term „interaction‟ to
explain this empirical regularity and use terms such as “social norms”, “peer influences”,
“neighborhood effects”, “conformity”, “imitation”, “contagion”, “epidemics”, “bandwagons”
or “herd behavior” to refer to them (Merton, 1957; Granovetter, 1979; Manski, 2000). In
physics, one is often warned (but not as often heard) that interaction and correlation are two
different concepts. Consider a solid, for example. Each atom in a solid interacts with
neighboring atoms mostly, and perhaps with next- and second-next neighboring ones at most.
However, if we hit one end of a solid bar, the effect of the action propagates to the other end,
which is a manifestation of the fact that an atom on the one end of solid is correlated with an
atom on the other end. Correlation is indeed the driving force of most phase transitions of
many-body systems. When a substance undergoes successive phase transitions from gas to
liquid on to solid, the interaction range of each atom does not change much but the
correlation range grows singularly and becomes macroscopic eventually when the body is
solidified. It is not difficult then to realize that the term „interaction‟ widely used in social
sciences actually refers mostly to the „social correlation’ that is produced by the networked
characteristic of social systems (for a review on the statistical physics of social dynamics see:
Castellano et al., 2009). One illustrative example of macroscopic (global) correlation is the
cell-phone adoption during the 1990s; it was a “contagion” effect that induced many to buy
phones simply because their friends and colleagues were buying them, which then propagated
in a correlated way across Europe (Michard and Bouchard, 2005) and the world.
The atomic/social metaphor has been widely used in both social and physical
sciences. Some of the pioneers of statistical physics, like Maxwell and Boltzmann, were

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inspired by the works of social scientists Buckle and Quetelet; see (Ball, 2004). More
recently, the metaphor of a „social atom‟ and the tools of statistical mechanics were used to
explain the structure and dynamics of social and economic systems (Buchanan, 2007), giving
rise to the fields of socio- and econophysics (Mantegna and Stanley, 1999; Chakrabarti et al.,
2006). This analogy functions very well when the properties studied depend mainly on the
networked structural properties of the system analyzed. A good example is a linear chain in
which the ith node is connected to the (i 1)th one. It has been proven that a purely one-
dimensional system never becomes a solid (Peierls, 1936; Ruelle, 1969). It is roughly
explained as follows. In one dimension, correlation could grow only along the chain. A
disturbance at only one point of the chain, destroying correlation locally, results in the global
destruction of the correlation between the atoms on both ends. However, social networks are
much more complex than a linear chain and the analogy with a three-dimensional atomic
system is more illuminating. In the three-dimensional system, we know very well that the
global correlation can overcome the local disturbance, such as the removal of one atom, to be
a solid at low temperatures. This is because there are many more paths along which the
correlation can grow than in one dimension. In other words, there is a topologically more
complex networked structure in these systems and the complexity of the structure promotes
the growth of correlation throughout the systems.
It is clear that not only atomic and social systems display network-like structures.
Complex networks are also ubiquitous in many biological, ecological, technological,
informational, and infrastructural systems (Albert and Barabási, 2002; Barabási and Oltvai,
2003; Caldarelli, 2007; da Fontoura Costa et al., 2011; Dorogovtsev and Mendes, 2003;
Newman, 2003; 2010; Strogatz, 2001; Watts, 2003). A complex network is a representation of
a complex system in which the nodes
niVvi
,1,


of a graph

,
G V E

represent the
entities of the systems and the interactions between pairs of entities are accounted for by the

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links 
,
ij
v vE

of the graph. (In graph theory, a node is also called a vertex and a link an
edge.) Consequently, global correlation effects are also observed in a variety of complex
networks. For instance, it is well documented that the extinction of one species in an
ecological system produces a “cascade” of effects that propagate well beyond the nearest
neighbors of the extinguished species (Dunne et al., 2002; Jordán and Scheuring, 2002). In a
biological cell where protein-protein interactions form biomolecular networks, it is well
documented that „perturbing‟ one protein can trigger a cascade of effects that change or
modify the synthesis and folding of several other proteins not necessarily directly interacting
with the targeted one (Zotenko et al., 2008). For infrastructural network scenarios, the
cascade of local failures in power grids that have produced mass blackouts such as the one in
eleven USA states and two Canadian provinces on 14th August 2003 affecting 50 million
people, or those in London, U.K., Sweden-Denmark, and Italy, are palpable examples of
correlation effects in complex networks (Makarov et al., 2005). Finally, the propagation of
crisis effects in a world-wide networked economy (Count and Bouchard, 2000; Eguíluz and
Zimmermann, 2000) alerts us about the importance of the study of correlation in complex
networks as a tool of great relevance to understand the structure and functioning of many
complex systems in nature and society. It is then essential to understand what topology
indeed promotes the growth of correlation and what disturbs it in such systems.
Unfortunately, the term „correlation‟ is frequently used in other sciences under
different meanings than the one in physics. For instance, the term “correlated effects” is used
in social sciences to refer to “interactions” in which “agents in the same group tend to behave
similarly because they have similar individual characteristics or face similar institutional
environments” (Manski, 2000). In other contexts it mainly refers to the linear
interdependence of two or more variables in the statistical sense as measured by a correlation

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coefficient. Subsequently, in the context of complex networks we have proposed the use of
the term “communicability” to refer to the situations in which a perturbation on one node of
the network is „felt‟ by the rest of the nodes with different intensities. The concept of network
communicability was introduced by Estrada and Hatano (2008). The intuition behind this
concept is that in many real-world situations the communication between a pair of nodes in a
network does not take place only through the optimal shortest-path routes connecting both
nodes, but through all possible routes connecting both nodes, the number of which can be
enormous in the complex topology of the systems. The information can also go back and
forth before arriving at the end node of a given route. The network communicability
quantifies such correlation effects in the communication between nodes in complex networks.
The most important point that we would like to stress in the present paper is that the number
of routes along which the correlation can grow is crucial in the analyses of the structures of
complex networks.
There have been other proposals of indices for the communication through complex
networks. These indices are mainly used to quantify the self-communicability of a given node
in the form of a centrality measure. The most characteristic of these indices are the closeness
(Freeman, 1979) and betweenness centrality (Freeman, 1979) and some of their modifications
like the information centrality (Stephenson and Zelen, 1989), and betweeness accounting not
only for shortest paths (Freeman et al., 1991; Newman 2005; Estrada et al., 2009). In some
way the eigenvector centrality introduced by Bonacich (1972; 1989), which is the principal
eigenvector of the adjacency matrix of the network, can be considered as a self-
communicability function. In this context, we can consider the number
 
i of walks of
k
N
length k starting at node i (see further for proper definition) of a non-bipartite connected
network. If
  
i
 
j
1
1
n
kkk
j
s iNN










, for k  , the vector
 
1 ,
 
2 ,
 
T
kkk
sss n





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tends towards the eigenvector centrality (Cvetković et al., 1997). This means that the
eigenvector centrality of node i represents the ratio of the number of walks of length k that
departs from i to the total number of walks of length k in a non-bipartite connected network
when the length of these walks is sufficiently large.
In the present review, we will analyze four other kinds of communicability indices. It
is important to note that it is not a matter of deciding which index is the „correct‟ one to
indicate the communication. There is indeed no standard that we can refer to in judging the
„correctness‟ of an index. It is a matter of which index is more appropriate to a specific
problem than others. In judging the appropriateness, we will have to resort to our intuition
and experience. Typically, we would make predictions from various indices and compare
them with the result of analyzing actual datasets or sometimes even with a plausible guess. In
the present review, we therefore show various specific examples where one index is more
appropriate than others.
B. Correlation function
Correlation effects can be quantified by the correlation function. The definition of
the correlation function depends on the problem under study, but the general idea is to
measure how a tiny disturbance at one point of the system propagates to another point of the
system; hence the aliases, the propagator and the Green‟s function. For a general reference on
this topic the interested reader is referred to Landau and Lifshitz (1980). In quantum
mechanics, the system with the Hamiltonian H evolves in time according to the time-
evolution operator
 
exp
i
U tHt









. (1)
(For some readers, the form

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  
G E

1
EH


, (2)
may be more familiar. Indeed, Eq. (2) is simply the one-sided Fourier transform of Eq. (1)
(Sakurai, 1985).) Then, the typical definition of the propagator may be given as
 
†
0
, vacvac
r
C r taU t a

, (3)
where vac denotes the vacuum state and
†
0a denotes the particle creation operator at the
origin 0, whereas
ra denotes the particle annihilation operator at the distance r from the
origin. Equation (3) describes how the impact of creating a particle at the point 0 propagates
over the distance r and affects the point r after the time t . Obviously, the correlation is
strong if there are many paths on which the effect can propagate.
In equilibrium statistical physics, the thermal disturbance is important. Instead of the
real-time dynamics in Eq. (3), we then often consider the thermal correlation function, or the
thermal Green‟s function, which may be given in the form
 
†
0
,vacvac
r
C raa
 
, (4)
where
 
1exp
ZH
 


, (5)
is the density operator of the Gibbs equilibrium distribution, where

exp
Z trH




 is the

partition function. We will take full advantage of the thermal Green‟s function (4) throughout
the paper. It describes the propagation of disturbance through the system in a thermal bath at
the inverse temperature
1/kT
 
, where k here is the Boltzmann constant. The thermal
Green‟s function (4) is indeed the analytic continuation of the Green‟s function (3) onto the
imaginary time axis /
it

.

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C. Plan of the article
We will define the communicability in Sec. II, presenting several definitions. Section
III elaborates the analogy between the communicability of complex networks and the
correlation of physical systems. We will show that classical and quantum statistical
calculations with the adjacency-based and Laplacian-based models result in four different
versions of the communicability. Then in Sec. IV we compare the four versions in two
specific examples. Section V presents a variety of usages of the communicability in analyses
of various complex networks, namely analyses at the microscopic level, the mesoscopic level,
the macroscopic level, and the multi-scale level. We also present an interesting application of
the communicability with a negative temperature to the analysis of bipartite networks. We
discuss the locality of the communicability in Sec. VI, showing instances of exponential
decay and slow decay of the communicability, or the correlation function. We finally review
recent advances in computing the communicability of large networks. It is a heavy task to
compute the communicability as an exponentiated operator. Taking advantage of the sparsity
of the adjacency matrix can greatly improve the computational efficiency. The final section is
devoted to conclusions.
II. COMMUNICABILITY IN NETWORKS
A. Combinatorial definition
In this section we introduce the concept of communicability in networks by using a
graph-theoretic (combinatorial) approach. In general, we will refer to simple graphs

,
GV E

in which there are no self-loops or multiple-links. When directionality or weights
of the links are considered it is explicitly stated; otherwise, we will refer to undirected and
unweighted graphs. The concept of network communicability briefly described in the
Introduction of this work immediately invokes the concept of walks in networks. A walk of

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length k is a sequence of (not necessarily distinct) nodes
011
, , ,,
kk
v vvv

such that for each
1,2,
ik

there is a link from
1
iv to
iv (Cvetković et al., 1997). Using the concept of walk
we define the communicability between two nodes as follows. The communicability between
the nodes p and q in a network is the weighted sum of all walks starting at node p and
ending at node q, in which the weighting scheme gives more weight to the shortest walks
than to the longer ones.
Mathematically, the communicability function can be expressed as follows (Estrada and
Hatano, 2008):

0
k
pq k
pq
k
GcA



, (6)
where Ais the adjacency matrix, which has unity in the 
, p q -entry if the nodes p and q
are linked to each other and has zero otherwise. In Eq. (6), we have used the fact that the

, p q -entry of the k th power of the adjacency matrix, 
k
pq
A
, gives the number of walks of
length k starting at the node p and ending at the node q (Harary and Schwenck, 1979). The
terms

0
k
ppk
pp
k
GcA



represent the self-communicability of a node and they provide a
centrality measure known as the node subgraph centrality (Estrada and Rodríguez-Velazquez,
2005a).

Centrality measures were originally introduced in social sciences (Freeman, 1979;
Wasserman and Faust, 1994) and are now widely used in the whole field of complex network
analysis (Newman, 2010). The coefficients
kc need to fulfill the following requirements: (i)
making the series (1) convergent, (ii) giving less weight to longer walks, and (iii) giving real
positive values for the communicability. Then, assuming a factorial penalization we obtain
the following communicability function:

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
 
0
!
k
pq
EA
pq
A
pq
k
A
Ge
k




, (7)
where
A
e is a matrix function that can be defined using the following Taylor series (Higham,
2008):
23
2! 3!!
k
A
AAA
eIA
k
 
. (8)
Note that the inclusion of the identity matrix in the expansion (8) does not affect neither the
subgraph centrality nor the communicability between pairs of nodes since in the first case it
only adds a constant to every value of the centrality measures and in the second case the off-
diagonal entries are unchanged. Using the spectral decomposition of the adjacency matrix,
the communicability function can be expressed as:
 
p
 
q e
,
,,
1
j A
n
EA
pqj Aj A
j
G




, (9)
where
1,2,,
AAn A

are the eigenvalues of the adjacency matrix in a non-increasing
order and
 
, j Ap

is the p th entry of the j th eigenvector which is associated with the
eigenvalue
, j A

of the adjacency matrix.
It is straightforward to realize that the shortest paths connecting any pair of nodes
always make the largest contribution to the communicability function. That is, if
  l
rs
P
is the
number of shortest paths between the nodes r and s having length l and
 
k
rs
W
is the number
of walks of length kl
 connecting the same nodes, the communicability function is given
by

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 
l
rs
 
k
rs
!!
EA
rs
k l

PW
G
lk

, (10)
which indicates that
EA
pq
G
accounts for all channels of communication between two nodes,
giving more weight to the shortest path connecting them. Therefore, the name of
„communicability‟ has been proposed to designate this function.
We can generalize the concept of communicability in three different ways. First, the
analogy of the communicability with the thermal Green‟s function in statistical physics
motivates us to introduce the temperature T , or its inverse  as a weighting parameter:


0
!
kk
pq
EA
pq
A
pq
k
A
Ge
k






, (11)
where

23
2!3!!
k
A
AAA
eIA
k



. (12)
The „physical meaning‟ of this parameter  will be evident in the next sections of this paper.
An interesting way of utilizing the parameter  is to consider the negative
temperature. The communicability function of a network can be separated into the
contributions coming from walks of even and odd lengths. For instance, for the case of
EA
pq
G

we can write
 
p
 
q

 
p
 
q


,,,,,,
11
coshsinh
evenodd
nn
EA
pqj A j Aj Aj Aj Aj A
jj
EA
pq
EA
pq
G
GG





(13)
We can separate these two contributions as

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 


1
even
2
EA
pq
EA
pq
EA
pq
GGG




,

 


1
odd
2
EA
pq
EA
pq
EA
pq
GGG




.

For a network having link weights
ij
w


, the communicability function is obtained
by using the weighted adjacency matrix

ij
n n

Ww

as


0
!
k
pq
EA
pq
W
pq
k
W
Ge
k




. (14)
In this case it has been proposed to normalize the weighted adjacency matrix in order to avoid
the excessive influence of links with higher weights in the network (Crofts et al., 2009). The
normalization used so far transforms the weighted adjacency matrix as:
1/2

1/2

WKWK

,
where K is a diagonal matrix of weighted degrees.
The third useful generalization of the communicability function is obtained by
considering other penalization coefficients ck in the expression (6), which can give rise to
different matrix functions. For instance, let
1
1/

and let us take
k
kc


in Eq. (6)
(Estrada and Higham, 2010). Then, we obtain the following communicability function:


1
00
RA
pq
kkk
k
pq pq
kk
G c WWIW






, (15)
where we have replaced the adjacency matrix in (6) by its weighted version. The index
RA
pq
G

was introduced as early as 1953 by Katz (1953) as a centrality measure for the nodes in social
networks.