PHYSICAL REVIEW E 85, 066122 (2012)
Complex networks in the Euclidean space of communicability distances
Ernesto Estrada
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XQ, United Kingdom
(Received 26 February 2012; published 19 June 2012)
We study the properties of complex networks embedded in a Euclidean space of communicability distances.
The communicability distance between two nodes is defined as the difference between the weighted sum of
walks self-returning to the nodes and the weighted sum of walks going from one node to the other. We give some
indications that the communicability distance identifies the least crowded routes in networks where simultaneous
submission of packages is taking place. We define an index Q based on communicability and shortest path
distances, which allows reinterpreting the “small-world” phenomenon as the region of minimum Q in the
Watts-Strogatz model. It also allows the classification and analysis of networks with different efficiency of spatial
uses. Consequently, the communicability distance displays unique features for the analysis of complex networks
in different scenarios.
DOI: 10.1103/PhysRevE.85.066122
PACS number(s): 89.75.Hc, 89.75.Fb, 02.40.Dr
I. INTRODUCTION
The analysis of complex networks has demanded an
increasing number of theoretical tools needed for extracting
useful information about their nontrivial structures [1–3].
Among them, metric properties play a fundamental role in the
study of network structure and dynamics. The best known of
these metrics is the so-called shortest path distance [4], which
is the basis of most of the metric analysis of networks [1–3].
However, other metrics like the resistance distance [5], also
known as the hitting time [6], is also useful for understanding
certain properties of networks [6]. More recently, Chebotarev
[7] has generalized the shortest path and resistance distance to
a new kind of metric. Beyond these classical “shortest-path”
and “resistance” distances, it has been shown [8–12] that many
of the topological properties that produce the unique features
of complex networks emerge naturally by embedding them
into hyperbolic spaces.
A different type of measurement also accounting for the
possible routes connecting two nodes in a network is the
communicability function, which provides information about
how well-communicated a pair of nodes in a connected
network is, Refs. [13–16]. The communicability between the
nodes p and q in a network is defined as [13]
∞
k
def A
Gpq =
= (eA )pq ,
(1)
k!
k=0
pq
where A is the adjacency matrix of the network. Here we are
considering only undirected connected networks, such that we
can express the communicability function as [13]
Gpq =
n
ϕj (p)ϕj (q)eλj ,
(2)
j =1
where φj (p) and φj (q) are the pth and q th elements of the j th
orthonormal eigenvector of the adjacency matrix associated
with the eigenvalues λj .
From (1) it is clear that the communicability function counts
all possible walks from one node to another giving more
weights to the shortest ones. A nice physical interpretation
is possible if we consider the network as a quantum harmonic
1539-3755/2012/85(6)/066122(11)
oscillators system submerged into a thermal bath with inverse
temperature β ≡ 1/kT , where k is the Boltzmann constant. In
this case the communicability
Gpq = (eβA )pq =
n
ϕj (p)ϕj (q)eβλj
(3)
j =1
represents the thermal Green’s function of the system [16].
Therefore, it indicates how a thermal oscillation is propagated
from one node to another in the network. The complete
derivation of this result is found in Ref. [16] and a resume
is given in Appendix A 1 of this work for the sake of
self-containment of the current work.
The way in which a thermal excitation at a given node
propagates throughout the network before coming back to the
same node and being annihilated is given by [17]
Gpp =
n
[ϕj (p)]2 eβλj ,
(4)
j =1
which is known as the subgraph centrality or selfcommunicability of the corresponding node.
In this work we are interested in analyzing how much of
an excitation is absorbed by the pairs of nodes and how much
of it is transmitted from one to the other. That is, supposing
that there is a thermal disturbance to the whole network, how
large is the difference between the absorbed and transmitted
excitation between two nodes in the network? This difference
is given by
def
ξpq (β) = Gpp (β) + Gqq (β) − 2Gpq (β).
(5)
Therefore, we study here whether the amount of perturbation absorbed by the two nodes is larger [ξpq (β) > 0] or smaller
[ξpq (β) < 0] than that transmitted among the two nodes. We
also prove that (5) is a Euclidean distance for the pair of nodes,
which provides important insights about the network structure.
II. COMMUNICABILITY DISTANCE
Because we are considering an undirected network it
is obvious that Gpq (β) = Gqp (β). By using the spectral
representation of the communicability function we can express
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ERNESTO ESTRADA
PHYSICAL REVIEW E 85, 066122 (2012)
(5) as follows:
ξpq (β) =
n
where
2 βλj
[ϕj (p)] e
+
j =1
−2
n
X=
2 βλj
[ϕj (q)] e
ϕj (p)ϕj (q)eβλj
j =1
=
n
eβλj {[ϕj (p)]2 + [ϕj (p)]2 − 2ϕj (p)ϕj (q)}
III. COMMUNICABILITY DISTANCE AND
NETWORK STRUCTURE
j =1
=
n
eβλj [ϕj (p) − ϕj (q)]2 ,
(6)
j =1
which means that ξpq (β) 0, implying that Gpp (β) +
Gqq (β) 2Gpq (β). Obviously, ξpq (β) = 0 if p = q. The case
ϕj (p) − ϕj (q) = 0 for all j implies that there are two identical
rows in U, which also implies that det(U) = 0, where U is the
matrix of orthonormal eigenvectors of the adjacency matrix.
However, this is not possible because U is an orthogonal
matrix, i.e., det(U) = ±1. Consequently, ξpq (β) = 0 if and
only if p = q.
This first result indicates that no matter what the structure
of the network is, the amount of excitation absorbed by a
pairs of nodes is always larger than the amount of excitation
transmitted among them. We will see now that this
is a
consequence of a more general result by proving that ξpq (β)
is a Euclidean distance among the corresponding two nodes
in a network. In order to prove it we start by considering the
spectral representation of the adjacency matrix A = UT U ,
where is the diagonal matrix of the eigenvalues of A. Let us
define the vector
ϕ p = [ϕ1 (p) ϕ2 (p) · · · ϕn (p)]T ,
(7)
which is obtained by transposing the pth row of the matrix
U of eigenvectors of the adjacency matrix. We then can write
ξpq (β) as
ξpq = (ϕ p − ϕ q )T eβ (ϕ p − ϕ q ),
(8)
which can be regrouped as
ξpq = [eβ/2 (ϕ p − ϕ q )]T eβ/2 (ϕ p − ϕ q )
= (eβ/2 ϕ p − eβ/2 ϕ q )T (eβ/2 ϕ p − eβ/2 ϕ q )
= (xp (β) − xq (β))T (xp (β) − xq (β))
= xp (β) − xq (β)2 ,
where xp (β) = eβ/2 ϕ p . Consequently,
ξpq (β) = xp 2 + xq 2 − 2xp · xq ,
(9)
(10)
is a Euclidean distance between the nodes p and q of the
graph [18,19]. Let us call this distance the communicability
distance between two nodes in a network.
The sum of all communicability distances in a network is
an invariant that characterizes how much of the excitations
occurring at the nodes is absorbed with respect to that
transmitted in the network. This invariant can be defined as
def
ϒ(G) = 12 1T X1,
(12)
is
the
communicability
distance
matrix,
s=
[G11 (β) G22 (β) · · · Gnn (β)]T is a column vector of
self-communicability for every node in the graph, 1 is an
√
all-ones column vector, and ◦ is the entrywise square root.
j =1
n
◦
s1T + 1sT − 2eβA ,
(11)
In order to gain insights about the structural meaning of
the communicability distance in complex networks we start
by studying all 12 111 connected graphs having n = 3 to 8
nodes. All the calculations in this section are carried out by
using β ≡ 1. When studying this set of graphs, we have found
some interesting and revealing structural information. First, we
have seen that the complete graph is always the graph having
the smallest value of the ϒ(G) index among all graphs with n
nodes. That is, the complete graph is the most “packed” one
in terms of the communicability distance, which means that
the amount of excitation returned to any pair of nodes in the
complete graph is almost identical to that transmitted among
them. The communicability distance index of the complete
graph is given by (see Appendix A 2)
√ −β/2
2e
.
(13)
ϒ(Kn ) = n(n − 1)
2
The identification of the graphs having the maximum value
of ϒ(G) among the graphs with n nodes is a more complex
task. As a consequence, we start by studying how this index
changes when the density of the graphs increases. That is, we
study the average of the communicability distance index for
all graphs having m links, ϒ(G)m . In Fig. 1 we illustrate the
results of the plots of ϒ(G)m for all connected graphs having
n = 7 and 8 nodes. In addition, we also plot the mean of the
¯ m and that of the average Watts-Strogatz
average path length l
clustering coefficient C̄m [20] for the corresponding group
¯ m decreases as the number of links in
of graphs. As expected l
the graph increases. That is, acyclic graphs, i.e., trees, display
the largest values of the average path length and this index
decreases as the density of the graph increases. At the same
time C̄m increases monotonically with the increase of the
graph density. However, ϒ(G)m displays a very interesting
dependence with the number of links. As can be seen in Fig. 1,
ϒ(G)m does not display a maximum or minimum for trees.
Instead, monocyclic graphs (m = n) have larger values of
ϒ(G)m than trees, and the index increases up to a maximum
which depends on the size of the graphs, i.e., m = 9 for graphs
with 7 nodes and m = 15 for graphs with 8 nodes. After this
point the values of ϒ(G)m decreases monotonically with the
increase of the number of links.
This remarkable characteristic of the communicability
distance is confirmed by the analysis of the individual graphs
that extremize it, i.e., minimize or maximize it. Among the
acyclic graphs (trees) with n nodes the linear chain is the
graph having the maximum value of ϒ(G) and the star graph
is the one having the minimum 1. When we analyze the graphs
with 8 nodes and m 7 links we found some remarkable
regularities in the structures of the graphs minimizing or
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FIG. 1. Plot of the average path length (squares), average clustering coefficient (triangles), and Q index (circles) for the graphs with seven
(left) and eight (right) nodes. The values of each index is averaged for all graphs with the same number of links.
maximizing ϒ(G). For instance, the graphs minimizing the
communicability distance index are illustrated in Fig. 2. We
can see that these graphs display very small average path
FIG. 2. Graphs with eight nodes and given number of links (m)
that minimize the Q index.
length (not larger than 2) and they have a very low clustering
coefficient, which is equal to zero for all graphs minimizing
ϒ(G) with m 16 (except for the graph with m = 8). Notice
that most of the graphs that minimize the communicability
distance index are bipartite or “almost bipartite.” We recall
that a bipartite graph does not have any odd-length cycle, and,
consequently, it has zero clustering coefficient. An almost
bipartite graph is a graph that can be transformed into a
bipartite one by removing very few links. All these graphs are
characterized by the fact that the amount of excitation returning
to the nodes is only slightly larger than that transmitted among
the pairs of nodes.
Let us now analyze the graphs that maximize the communicability distance index, which are displayed in Fig. 3. These
graphs display relatively large average path length and high
clustering coefficients. The graphs labeled as Lr,s are known
as the lollipop graphs and are formed by joining a complete
graph K r and the path P s by a bridge. For n > 5 nodes we
have observed that the lollipop graph Ln−2,2 has the maximum
value of ϒ(G) among all graphs having n nodes. These graphs
are characterized by a large clustering coefficient produced by
the large cliquishness in the complete subgraph and a relatively
large average path length due to the fact that there are n−3
pairs of nodes separated at distance 3. As a consequence, if
we consider a pair of nodes p and q such as p ∈ K r and
q ∈ P s , we see that the amount of excitation returning to
the node p is very large (Gpp 1) and it is significantly
larger than that returning to the node q (Gpp Gqq ) as well
as to that transmitted between the two nodes (Gpp Gpq ).
Consequently, ξpq ≈ Gpp 1. However, for any pair of
nodes s,t in K r we have seen before that ξst ≈ 0.85, which
means that the index ϒ(G) is mainly determined by the
communicability distances between K r and P s .
In the previous analysis we have used a simplified picture
of the communicability distance in terms of the shortest
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ERNESTO ESTRADA
PHYSICAL REVIEW E 85, 066122 (2012)
FIG. 3. Graphs with eight nodes and given number of links (m) that maximize the Q index.
path distance and the average clustering coefficient. We
have to remark that the communicability distance is not
trivially related to these network parameters. For instance,
there is no significant correlation between the communicability distance index and a combination of the average
path length and clustering coefficient for the 12 111 graphs
studied here. However, the use of these two parameters in
a more colloquial and less quantitative basis is useful for
understanding the structural meaning of the communicability
distance.
IV. “FLEXIBILITY” OF COMPLEX NETWORKS
We start by considering a simple intuitive model based on
a path Pn . It is easy to realize that the shortest path distance
embeds this graph into a segment of straight line (see Fig. 4) of
length n−1. Let us consider an excitation which is transmitted
through the nodes of Pn by using the shortest path in such a
way that the time taken by the signal to go from vi to vi+1 is
one unit. Then, for the excitation to travel the whole path, it
needs n − 1 time units. Therefore, for an infinitely long path,
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PHYSICAL REVIEW E 85, 066122 (2012)
FIG. 4. Illustration of the embedding of a path having four
nodes by using the shortest path distance (left) and communicability
distance (right).
n → ∞, the time taken by the excitation to travel the whole
path is infinitely large.
Let us now consider the communicability distance between
the two extremes of the path Pn (here denoted with labels 1
and n) at β ≡ 1. It can be seen (see Appendix A 3) that as
n → ∞, ξn1 → 1.7836. That is, using the communicability
distance the path is embedded into a Euclidean space in a way
that it is folded (see Fig. 4) and the distance between the two
extremes is relatively small and independent of the size of the
path for n → ∞. In other words, the time that the excitation
will take to travel from one extreme to another of the path is
just 1.7836 units, in contrast with n−1 time units if the signal
travels through the shortest path of the graph. Therefore, we
can think figuratively that the path is a “flexible” network
because we can embed it in the communicability space in a
more “packed” way than in the shortest path space.
The sum of all shortest-path distances in a graph is known
as the Wiener index W (G) and has been widely studied in the
literature [21,22]. It has been identified with the compactness
of a graph [23] and with the molecular surface area [24].
According to our previous analysis W (G) accounts for the
total time that an excitation needs to visit every node in the
network by traveling through the shortest paths. On the other
hand, the communicability distance index ϒ(G) accounts for
the time that the excitation takes to visit every node if it travels
in a graph embedded in the communicability space. Therefore,
we define the following index:
def
Q = ln(0.8578W/ϒ)
(14)
to account for the ratio of the transmission of an excitation in a
network using shortest paths to that using the communicability
distance. The term 0.8578 corresponds to the communicability
distance between a pair of nodes in Kn at β ≡ 1 and it is
introduced to make Q(Kn ) = 0. The values Q > 0 indicate
that the transmission using shortest paths takes more time than
that using the communicability distance. In other words, if
Q > 0, the network embedded into the shortest path distance
space is not efficiently packed because it can be packed into
a more efficient conformation such as the one given by the
communicability distance. We call these networks flexible
due to the fact that they can be packed into more efficient
conformations. For instance, in the case of the path, we
have that Q ≈ ln[(n − 1)/2.0793], which indicates a large
inefficiency in the use of the shortest-path space, i.e., a very
large flexibility. On the other hand, Q < 0 indicates a good
efficiency in the use of the shortest-path space. That is, an
excitation is transmitted faster using the shortest paths than by
using the communicability distance. We call these networks
rigid in contrast with the previously analyzed ones.
The best way for understanding these differences in the
values of Q is to consider an example. Let us first consider
a planar network with n nodes, i.e., a network that can be
drawn in a plane without any link intersection. By definition
we can represent the nodes and the links of this network on
the surface of a sphere. If we suppose that the diameter of the
planar network is very large so it is the radius of the sphere.
As a consequence of the large diameter of the network, W (G)
is relatively large. Because the network does not contain very
much cliquishness (due to its planarity) we would expect that
ϒ(G) be small and, as a result, Q should be large. Now we
can destroy the planarity by adding a few more links which
travel through the interior of the sphere. As in the small-world
model of Watts and Strogatz [20], this effect decreases W (G).
The addition of these few links would increase Gpp for some
nodes more than the increase of Gpq for the nodes involved.
In such a case it makes that the communicability distance
and consequently ϒ(G) increases significantly. The resulting
effect is that Q decreases.
Therefore, the Q index can be used as an indicator for the
small-world effect in the Watts-Strogatz (WS) model [20]. In
this model we start by defining a circulant graph in which
every node i is linked to the (i + j )th and (i − j )th nodes for
j = 1, . . . ,k/2, where k is the degree of the node. The links
then are rewired with probability p in such a way that when
p → 1 the resulting network resembles a random graph. Let
β ≡ 1 and let Q0 be the value of the index Q for p = 0. Let
us select a value p = c such that 0 < c 1. When 0 < p < c
the average path length, and, consequently, W (G), decays very
fast but ϒ(G) is still large enough because the cliquishness of
the graph remains high. As a consequence, we should expect
that Q0<p<c < Q0 . When p = c the cliquishness of the graph
drops significantly and the combination of small average path
length and small clustering should make that ϒ(G) decreases
significantly. Consequently, we expect that Qc < Q0<p<c <
Q0 . After this point the index ϒ(G) must decay much faster
than W (G), as a consequence of the fact that the cliquishness of
the graph is destroyed as p → 1. Therefore, we would expect
that for the total process the following relation holds: Qp>c >
Qc < Q0<p<c < Q0 . In closing, in the WS model there is a
point c in which the index Q is a minimum. This point indicates
when the “equilibrium” between small average path length and
relatively high clustering coefficient is broken. Therefore, this
point indicates when the network is a “small-world.” In Fig. 5
we illustrate this effect for the WS model using a network of
200 nodes and average degrees k̄ = 8 and k̄ = 10. The results
are the average of 100 random realizations.
We have explored empirically the dependence of the values
of Qc with both the size of the network and the average degree.
Previous analysis of the plot of the average path length versus
probability in the WS model has revealed that there is a drop in
the curve, which occurs at lower and lower values of p as the
size of the network grows (see Fig. 4 in Ref. [25]). When we
calculate the value of pc (defined as the probability at which Qc
appears) for networks having k̄ = 8 and sizes ranging from n =
100 to n = 10 000 we have seen that pc ≈ 0.05 independently
of the size of the network. That is, the rewiring probability at
which the index Q is minimum does not depend on the size
of the network. However, when we analyze the dependence
of pc with the average degree of the network, we find a clear
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PHYSICAL REVIEW E 85, 066122 (2012)
FIG. 5. (Color online) Plot of the average path length (circles), average clustering coefficient (squares), and Q index (triangles) for the
networks having 200 nodes and average degree 8 (left) and 10 (right), which are generated by using the Watts-Strogatz model.
dependence between both parameters. We studied networks
with n = 1000 nodes and having average degrees between 8
and 28, for which we find the values of pc by averaging over
100 random realizations of the same network. In Fig. 6 we plot
the values of pc versus k̄, where a power-law decay is observed
according to pc = 2.135k̄ −1.778 with a correlation coefficient
equal to 0.9974, which is significantly better than that for an
exponential fit. That is, the rewiring probability at which the
value of the index Q is minmum decays as a power-law of the
average degree of the network in the WS model. In closing,
FIG. 6. Plot of the probability at which the value of the index Q
is minimum versus the average degree in the Watts-Strogatz model
of “small-world” networks. Every point is the average of 100 random
realizations of a network with 1000 nodes and the given average
degree.
we can redefine the “small-world” effect as the point in which
a network displays the minimum value of the index Q.
V. REAL-WORLD NETWORKS
In this section we study the communicability distance
for 30 real-world networks. These networks include the
following biological networks: the neural network of C.
elegans (neurons), the transcription networks of yeast (Trans
yeast) and of E. coli (trans E.coli), the PPI networks of human
(PPI human) and yeast (PPI yeast), the brain networks of
macaque visual cortex (Macaque cortex) and cat cortex (Cat
cortex), and three networks representing human brain tissues
in healthy subjects (Brain tissue_1, _2, and _3). Informational
networks representing the following systems: a network of
the Roget thesaurus (Roget) and the citation networks in the
field of “small-world” (Small World citation) and of centrality
measures (Centrality citation). The social and economic
networks considered are the social networks of corporate elite
in USA (Corporate elite), the social networks of injecting drug
users (Drugs), persons with HIV infection during its early
epidemic phase in Colorado Springs (Colorado Spring), and
the world trade network of miscellaneous metal manufactures
in the world in 1994 (World trade). The technological networks
included are three electronic sequential logic circuits parsed
from the ISCAS89 benchmark set (electronic1, electronic2,
and electronic3), the USA airport transportation network
of 1997 (USAir 97), and a version of the Internet at an
autonomous system in 1997 (Internet). For the references and
details of these networks the reader is referred to the Appendix
in Ref. [3].
The following spatial networks were also considered: Four
street networks for the cities of Bonheiden (Bonheiden city)
[26], the old part of Cordoba (Cordoba city) [26], Foggia
(Foggia city) [27], and the center of Bologna (Bologna center)
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TABLE I. Values of the index Q for real-world networks. All
values were obtained by considering β ≡ 1.
Rigid network
FIG. 7. Illustration of a network having a crowded center and a
peripheral diversion used as an example of the applicability of the
communicability distance.
[27]. Three networks representing the chambers and galleries
in nests built by the Cubitermes termite (termite1, termite2,
and termite3) [28], and a network of the galleries produced by
200 ants during three days (Ant galleries) [29].
We start by analyzing a typical problem of navigability in
networks. Here we always consider β ≡ 1 if not said otherwise. We consider the case in which a piece of information
needs to be sent from a source A to a destination B in a
network in which other nodes are also emitting packages to
other targets. For instance, let us consider the graph illustrated
in Fig. 7 in which a package is to be sent from node 1 to node
5. The typical shortest-path approach assumes that the best
way of sending the package is by using the route 1–8–6–5.
However, let us consider that the other nodes are also sending
packages, so there is a high probability that there is more
than one package at node 8 when the information departing
from node 1 arrives at it. That is, the package leaving 1
can be delayed at node 8 as a consequence of a traffic jam
produced by the fact that nodes 6, 7, and 9 can submit a
package to it. However, if we send the package following
the route 1–2–3–4–5 which, in principle, is longer, it could
arrive at its destination in shorter time as a consequence that
there are fewer bottlenecks in this trajectory. This fact is well
represented by the communicability distance. For instance,
the communicability distances between every pair of adjacent
nodes in this path are ξ1,8 = ξ5,6 = 1.954 and ξ6,8 = 1.173.
Thus, the total distance using this path is 5.081. However, the
communicability distances between the pairs of nodes in the
path 1–2–3–4–5 sum 4.807, which indicates that in fact this
route is “shorter” than the previous one.
The situation illustrated in this fictitious example is much
more critical when we study real-world networks. For instance,
let us consider the brain network of the macaque visual cortex
[29]. There are a few shortest paths of length 3 from region
1 to region 15, e.g., 1–6–11–15, 1–6–12–15, 1–6–19–15, and
1–6–29–15. Without loss of generality let us consider the first
one. The communicability distances between the pairs of nodes
in this path are 137.792, 143.828, and 21.038, respectively.
Thus, the total distance using this route is 302.658. The shortest
communicability distance path between these two nodes is
1–24–27–30–15, with communicability distances equal to
2.786, 43.658, 13.908, and 76.845, respectively, which sum
to 137.198, a distance which is even shorter than that of
the link 1–6 (137.792). Consequently, if a signal needs to
travel from region 1 to region 15 in the macaque visual cortex
USAir97a
World tradea
Interneta
Cat cortexa
Neuronsa
Corporate elite
Small World citation
Centrality citation
Macaque cortexa
PPI yeast
Drugs
PPI human
Roget
Trans yeast
Trans E. coli
a
b
Q(G)
Flexible network
Q(G)
−16.58
−11.74
−10.17
−8.32
−7.62
−7.05
−6.58
−6.22
−4.09
−3.07
−3.99
−2.34
−1.44
−0.47
−0.22
Bonheiden cityb
Foggia cityb
Bologna centerb
Cordoba cityb
Termite2b
Colorado Spring
Termite1b
Brain tissue_1b
Brain tissue_2b
Electronic3b
Termite3b
Ant galleriesb
Electronic2b
Electronic1b
Brain tissue_3b
2.59
1.81
1.62
1.59
1.22
1.06
1.02
0.92
0.80
0.73
0.70
0.65
0.59
0.46
0.01
Networks embedded into three-dimensional or higher spaces.
Networks embedded into two-dimensional space.
avoiding possible traffic jams, the best way is by using the
route 1–24–27–30–15 and not the shortest path.
The previous situation is observed in many real-world
complex networks, particularly when the value of ϒ(G)
is relatively high. When ϒ(G) W (G) the chances that
at every node there are many alternative routes, i.e., high
cliquishness, is very high. However, in those cases in which
ϒ(G) W (G) most of the routes existing at every given node
coincide with the shortest paths connecting a source with a
target. Consequently, we can distinguish two different kinds
of networks, mainly those for which Q < 0 (ϒ(G) W (G))
which are characterized by a large number of alternative
routes at every node of the network, and those for which
Q > 0 (ϒ(G) W (G)) which are characterized by very few
alternative routes at every node. The first networks are “rigid”
in the sense that they are already very packed and they cannot
be very much packed by the effect of thermal oscillations when
submerged into a thermal bath. The second class of networks
corresponds to networks which are “flexible” as a result that
they do not use efficiently the available (high-dimensional)
space and are very much affected by thermal oscillations in a
thermal bath.
In Table I we give the values of the index Q for the
30 real-world networks studied here. The networks with the
highest rigidity (largest negative Q values) correspond to those
of the airport network in the US, the World trade network,
Internet at the autonomous system, the brain network of cat
cortex, and the neuron network of C. elegans. Remarkably,
all these networks have a common characteristic. They are
geographically embedded into a limited space, which can be
a geographic territory, or a region of a biological organism.
However, all of them use efficiently this space by departing
dramatically from planarity and using the three-dimensional
or even higher-dimensional spaces to increase the connectivity
of the network. For instance, in the USA air transportation
network the nodes are located on the plane of the US territory.
However, the links from one airport to another use the 3D space
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and because the flights can be intersected in space at different
times we can argue that this network can use a 4D space
to increase its connectivity. The cases of brain and neuronal
networks can be considered excellent examples of an efficient
use of the 3D space.
On the other side of the list, we find a few networks
which are highly flexible, which indicates that they do not
use efficiently the higher-dimensional space available. These
networks include the four city street networks, the electronic
circuits, the tissue networks, the termite nests, and the ant
galleries. City street networks are planar or nearly planar
graphs, which indicate that these networks need to be extended
into the 2D space without the possibility of efficiently using the
available 3D space. A similar situation is found in electronic
circuits where planarity is a necessary condition to avoid
short-circuits, and despite the fact that few layers can be used
to create a circuit, the resulting graphs are nearly planar and
basically extended through the 2D space. The ant galleries
network is just another example of planar network because of
the way in which the galleries were constructed avoids any use
of the 3D space available. Cellular network are obtained from
a cut of a tissue in thin layers (see Ref. [3] Chapter 17), which
are not necessarily planar at the cellular level but which are
mainly extended into the 2D space more than into the 3D one.
The case of the termite nests represents an interesting example.
These networks represent the 3D galleries and chambers which
are formed inside termite nests. Termites use all the available
3D space that they can as revealed by the fact that some of
these nests are meters high [28]. However, it is surprising that
the networks of galleries in termite nests are almost planar.
In fact, by removing only 10% of the links these networks
become planar. This situation is probably due to the fact that
the nests are not built by considering an optimization of the 3D
space. In fact, mounds are usually 6 m high for insects which
are less than 1 cm long. However, these mounds are also used
as storage of plant material as well as to evacuate the excess of
CO2 and heat produced inside them. Consequently, technically
speaking, these nests can be built on a plane, but termites prefer
to build them in 3D to guarantee other ecological conditions
as well as the previously mentioned ones.
Finally, we analyze the effects of the temperature on the
values of the index Q on complex networks. Suppose that
Q(G) < 0 for a given network, that is, the network is rigid.
Let us then heat the network such that β → 0 (T → ∞) in
order to “expand” it as an effect of heating to the conformation
displayed by it in the shortest path space. In the general context
of complex networks, “heating” means to produce a global
perturbation to the network that increases the deviation of
the nodes from their equilibrium position. For instance, in
a social networks context it corresponds to an increase of
the social agitation as produced by a social conflict, while in
biological networks it could refers to the level of physiological
stress produced by the lack of nutrients, drastic changes in pH,
temperature, and so on. On the other hand, “cooling” refers to
the analogous situations in which such external perturbations
are minimized. By the definition of the communicability
distance we can easily show that ξpq (β) → 2 as β → 0, which
means that the communicability index tends to a constant
value, ϒ(β) → n(n − 1). Consequently, if W (G) > n(n−1)
,
0.8578
the index Q(G) will tend to a positive value as β → 0.
This means that for these networks there is a temperature
at which Q(G) = 0. We call this temperature βc , which is the
temperature at which a network changes the sign of the Q
index. In those networks with Q(G) < 0 and W (G) < n(n−1)
0.8578
FIG. 8. Effect of the temperature over the index Q for real-world networks. On the left, networks with Q < 0: Drugs (empty circles),
neurons (full circles), USAir97 (stars), cat brain cortex (squares), and macaque brain cortex (rhombus). On the right, city networks with Q > 0:
Bohneiden (circles), Cordoba (squares), Foggia (stars), and Bologna (solid circles).
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such βc does not exist and the Q index tends to a t-negative
constant value as β → 0.
Let us now consider the analogous case in which the
network is flexible Q(G) > 0 and let us cool the network
such that β → ∞ (T → 0) and the network becomes rigid.
In this case it can be shown that the communicability distance
between a pair of nodes tends to the value given by
ξpq (β) → eβλ1 [ϕ1 (p) − ϕ1 (q)]2 .
(15)
This means that for sufficiently large β, ξpq (β) → ∞ and
so the communicability index. Consequently, Q → −∞ as
β → ∞, which indicates the existence of a temperature βc at
which Q(G) = 0.
The two previous situations are well observed when we
study real-world complex networks at different temperatures.
For instance, in Fig. 8 (left) we illustrate five networks for
which Q(G) < 0. They represent the systems of injecting drug
users (Drugs), the neuronal network of C. elegans (neurons),
the USA airport transportation network (USAir97), and the cat
(Cat cortex) and macaque visual cortices (Macaque cortex).
As can be seen, as β → 0 all values of Q change from
negative to positive values, with the exception of that for the
Cat cortex network, which never reaches the value Q(β) = 0.
This network has W = 2169 and n(n − 1)/0.8578 = 91070,
so it corresponds to the previously mentioned case where
W (G) < n(n−1)
. The values of βc for the remaining networks in
0.8578
the Fig. 8 (left) are Macaque cortex (0.111), USAir97 (0.156),
neurons (0.272), and drugs (0.494). This indicates that the
macaque cortex network is the most “packed” network or,
in other words, the one using more efficiently the available
high-dimensional space. This is because it is necessary to
heat the network at higher temperatures to unpack it to the
conformation displayed by the network in the shortest path
space.
In Fig. 8 (right) we illustrate the cooling of four flexible networks corresponding to the street city networks of Bonheiden,
Foggia, Bologna, and Cordoba. As can be seen Q → −∞
as β → ∞. Therefore, we have the following values of
βc : Bologna (2.265), Foggia (2.341), Cordoba (2.439), and
Bonheiden (3.404). This indicates that the city of Bonheiden
is the most flexible one, as we need to cool it to the lowest
temperature to obtain an embedding as the one produced by the
communicability distance. In plain words, from the four cities
analyzed here Bonheiden appears to be the most extended one
into a 2D area.
VI. CONCLUSIONS
We propose here a characterization of complex networks
by using a Euclidean distance based on the communicability
function. The communicability distance between a pair of
nodes characterizes the difference in the amount of “perturbation” which is absorbed by the two nodes in relation to
that which is transmitted between them. This new metric
embeds a network into a Euclidean space. The sum of all
communicability distances in a network can be significantly
larger than the similar sum of shortest paths distances. In
this case we consider that the network is more packed in
the space of communicability distances than in that of the
shortest path ones. In general, these networks correspond to
those with a high cliquishness, which does not allow them to be
embedded into a two-dimensional space. These are the cases
of airport transportation networks, Internet, brain networks,
etc., where the network uses efficiently the three-dimensional
or higher space to allow the high connectivity between the
nodes. On the other hand, we can find networks where the
sum of all communicability distances is smaller than the sum
of all shortest paths distances. These “unpacked” networks in
the communicability space correspond to those that can be
embedded into two-dimensional space, such as networks of
cities, electronic circuits, cellular tissues, and so on. We have
introduced here an index that allows differentiating among
these two kinds of networks.
We have provided hints in this work about the fact that the
communicability distance can help in deciding better routes
for distribution of “packages” in complex networks. This is
mainly due to the fact that the communicability distance has
contributions from the number of links separating a couple of
nodes but also from their cliquishness. That is, if there is more
than one alternative route for going from one node to another,
the communicability distance will favor the one combining
short topological (shortest path) distances and relatively low
cliquishness of the nodes involved in the route. This optimal
route according to the communicability distance could be
longer in term of the number of links, but it should be less
crowded than the shortest path connecting both nodes, which
may results in a faster delivery of a package from one point to
another.
We have given here clear indications on possible areas in
which the communicability distance can play an important
role for the analysis of complex networks. For instance,
the communicability distance can be important in detecting
regions of “small-worldness” in random or real networks, on
navigability in crowded communication networks, as well as
in the classification and analysis of networks with different
efficiency of spatial uses. However, more theoretical work
is needed in order to relate this distance with other network
properties and more experimental work is needed to identify
the areas in which the communicability distance would play
such a fundamental role.
ACKNOWLEDGMENTS
The author thanks partial financial support from the
Professor’s New Fund, University of Strathclyde. A. Perna,
O. Sporn, and E. Strano are thanked for sharing data sets used
in this paper.
APPENDIX
1. Communicability as the thermal Green’s function
The details of the following analysis can be found in
Ref. [16]. Let us consider a network as a balls-and-springs
system. Every node is represented by a ball of mass m and
every link is a spring with the spring constant mω2 connecting
two balls. The network is considered to be submerged into
a thermal bath at temperature T . The balls in the complex
network then oscillate under thermal disturbances. Consider
that the vibrational system represented by the network obeys
the laws of quantum mechanics. That is, the momenta pj
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and the coordinates xi are not independent variables but they
are operators that satisfy the commutation relation, [xi ,pj ] =
ih̄δij . In this case, the Hamiltonian of the network is written
in terms of the creation and annihilation operators as
H =
i
2. Communicability distance in a complete network
Let Kn be the complete graph of n nodes. The communicability between any pair of nodes in Kn is given by
h̄ω2 †
†
†
h̄
ai ai + 12 −
(a + ai )Aij (aj + aj ),
4
i,j i
eβn − 1
.
neβ
Similarly,
=
(A1)
†
where ai and ai are, respectively,
√ the boson creation and
annihilation operators and = K/mω, where K is a
constant satisfying K maxi ki (ki is the degree of the node
i). We have previously shown that the Hamiltonian (A1) can
be decoupled as
H =
Hμ .
n−1
eβ(n−1)
+
,
n
neβ
which by substitution on (5) gives
√
ξp,q (β) = 2e−β/2 .
Gp,p =
Within this approximation, where bμ = i Oμi ai =
†
†
T
†
T
i ai (O )iμ , bμ =
i Oμi ai =
i ai (O )iμ , and O is an
orthogonal matrix that diagonalizes the adjacency matrix, i.e.,
= O(KI − A)O T ,where is the diagonal matrix with the
eigenvalues λμ of (KI − A) on the diagonal.
Under these approximations the quantum-mechanical partition function has the form
ω2
βh̄
1+
exp −
(λμ − K) . (A4)
Z=
2
2
2
μ
The thermal Green’s function is given in the framework of
quantum mechanics by
(A6)
If we consider h̄ω2 = 2
, then the thermal Green’s function
of the network is given by the exponential of the adjacency
matrix up to a constant factor,
Gpq = eβh̄
[exp(βA)]pq .
(A10)
Let Pn be a path of n nodes labeled in consecutive order
starting by one, such that p q. The pth entry of the j th
eigenvector
of the adjacency matrix of Pn is given by ϕj (p) =
2
n+1
(A7)
jpπ
sin n+1
(j = 1, . . . ,n) and the corresponding eigenvalue
jπ
by λj = 2 cos n+1
. Then, the communicability between any
two nodes in Pn is given by
n
jπ
j qπ
2 jpπ
sin
e2β cos( n+1 )
sin
Gp,q (β) =
n + 1 j =1
n+1
n+1
n j π (p − q)
1 cos
=
n + 1 j =1
n+1
jπ
j π (p + q)
e2β cos( n+1 ) .
(A11)
− cos
n+1
We can now make the following approximation
1 π
cos[θ (p − q)]e2β cos θ dθ
Gp,q (β) ≈
π 0
1 π
−
cos[θ (p + q)]e2β cos θ dθ, (A12)
π 0
where we have use the change of variable θ = πj/(n + 1).
Then, we have
Gp,q (β) ≈ Ip−q (2β) − Ip+q (2β),
(A5)
which indicates how much an excitation at the node p
propagates throughout the network before going to node
q and being annihilated. After the appropriate algebraic
manipulations it can be written as
βh̄ω2
Gpq (β) = e−βh̄
exp
A
.
2
pq
(A9)
3. Communicability distance in a linear chain
Then, if we consider the approximation that each mode of
oscillation does not get excited beyond the first excited state,
we can restrict ourselves to the space spanned by the ground
state (the vacuum) |vac and the first excited states bμ† |vac.
Then
ω2
1
†
. (A3)
(λ
−
K)
b
b
+
Hμ = h̄
1 +
μ
μ μ
2
2
2
1
vac|ap e−βHA aq† |vac,
Z
(A8)
(A2)
μ
Gpq (β) =
1
eβ(n−1)
en−1
+ e−β
− β
ϕj (p)ϕj (q) =
n
n
ne
j =2
n
Gp,q =
(A13)
where Iγ (z) is the Bessel function of the first order.
The expression for the self-communicability of a given node
is
n
jπ
2 2 j πp
Gp,p (β) =
e2β cos( n+1 )
sin
n + 1 j =1
n+1
n jπ
2j πp
1 1 − cos
e2β cos( n+1 ) ,
=
n + 1 j =1
n+1
(A14)
which can be approximated by
1 π 2β cos θ
1 π
Gp,p (β) ≈
e
dθ −
cos(2pθ )e2β cos θ dθ,
π 0
π 0
(A15)
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when we used again θ = πj/(n + 1). Due to the symmetry of the path around the point n/2 (n even) or (n + 1)/2 (n
odd) we have
Gp,p (β) ≈ I0 (2β) − I2r(p) (2β),
where
r(p) =
p
if
n − p + 1 if
p n/2 (n even)
p > n/2 (n even)
or
or
p (n + 1)/2 (n odd)
.
p > (n + 1)/2 (n odd)
By substitution we finally obtain
ξp,q (β) = 2I0 (2β) − I2r(p) (2β) − I2r(q) (2β) + 2Ip+q (2β) − 2Ip−q (2β),
where Iγ (z) is the Bessel function of the first order and
i
if p i/2 (n even)
r(i) =
n − i + 1 if p > i/2 (n even)
or
or
(A16)
(A17)
i (n + 1)/2 (n odd)
.
i > (n + 1)/2 (n odd)
Now, if we consider a the two endpoints of a path we can see that
Gn,1 (β) ≈ In−1 (2β) − In+1 (2β),
which means that G1,n (β) → 0 as n → ∞. Consequently,
ξn,1 (β) ≈ 2I0 (2β) − 2I2r(1) (2β).
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