Synchronization Reveals Topological Scales in Complex Networks
Alex Arenas,
1
Albert Dı
´
az-Guilera,
2
and Conrad J. Pe
´
rez-Vicente
2
1
Departament d’Enginyeria Informa
`
tica i Matema
`
tiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
2
Departament de Fı
´
sica Fonamental, Universitat de Barcelona, Martı
´
i Franque
`
s 1, 08028 Barcelona, Spain
(Received 30 November 2005; published 22 March 2006)
We study the relationship between topological scales and dynamic time scales in complex networks.
The analysis is based on the full dynamics towards synchronization of a system of coupled oscillators. In
the synchronization process, modular structures corresponding to well-defined communities of nodes
emerge in different time scales, ordered in a hierarchical way. The analysis also provides a useful
connection between synchronization dynamics, complex networks topology, and spectral graph analysis.
DOI: 10.1103/PhysRevLett.96.114102 PACS numbers: 05.45.Xt, 89.75.Fb
The science of complex networks has been a subject of
interest for the physics community in recent years [1–3].
Complex networks are found in fields as diverse as the
Internet, the World Wide Web, food webs, and biological
and social organizations (see [4] and references therein).
Although the main characteristics of complex networks
have been properly described at the microscale level
(node properties) and also at the macroscale level (whole
network properties), some of the characteristics of the
mesoscale are still elusive. In particular, the community
detection problem concerning the determination of meso-
scopic structures that have functional, relational, or even
social entities is still controversial, starting from the
‘‘a priori’’ definition of what a community is [5,6].
The community detection problem consists of finding
a ‘‘good’’ partition of the network in subgraphs that rep-
resent communities according to a given definition.
However, in many complex networks the organization of
nodes is not completely represented by a unique partition,
but by a set of nested communities that appear at different
topological scales. Let us consider as a naive example the
network formed by all human acquaintances. Thus, at
some topological scale we can expect to find many com-
munities formed by families and friends and, soon after
this scale, the expected partitions into cities will come up;
beyond this regions, followed by countries, and, finally,
probably continental areas. Here, we aim at giving a
method to reveal these different topological scales.
In a completely different scenario, physicists have
largely studied the dynamics of complex biological sys-
tems, and, in particular, the paradigmatic analysis of large
populations of coupled oscillators [7–9]. The emergence
of synchronization patterns in these systems has been
shown to be closely related to the underlying topology of
interactions. In this Letter we show that, for a suitable
model, the dynamical process towards synchronization
shows different patterns over time intrinsically connected
with the hierarchical organization of communities in com-
plex networks. The ubiquity of synchronization phe-
nomena in the real world makes this approach appealing
from a physical and biological perspective. Moreover, we
will show that the connections with the spectral theory of
the Laplacian matrix of a graph spreads the possibilities of
the analysis to any complex network.
One of the most successful attempts to understand syn-
chronization phenomena was from Kuramoto [9], who
analyzed a model of phase oscillators coupled through
the sine of their phase differences. The model is rich
enough to display a large variety of synchronization pat-
terns and sufficiently flexible to be adapted to many differ-
ent contexts [10]. The Kuramoto model consists of a
population of N coupled phase oscillators where the phase
of the ith unit, denoted by
i
t, evolves in time according
to the following dynamics
d
i
dt
!
i
X
j
K
ij
sin
j
i
i 1; ...;N; (1)
where !
i
stands for its natural frequency and K
ij
describes
the coupling between units. The original model studied by
Kuramoto assumed mean-field interactions K
ij
K; 8 i; j. If the oscillators are identical (!
i
!8 i) there
is only one attractor of the dynamics: the fully synchro-
nized regime where
i
; 8 i. Recently, due to the
realization that many networks in nature have complex
topologies, these studies have been extended to complex
networks with local interaction [11–18].
In particular, it has been shown [19,20] that high densely
interconnected sets of oscillators (motifs) synchronize
more easily that those with sparse connections. This sce-
nario suggests that for a complex network with a nontrivial
connectivity pattern, starting from random initial condi-
tions, those highly interconnected units forming local clus-
ters will synchronize first and then, in a sequential process,
larger and larger spatial structures will do the same up to
the final state where the whole population should have the
same phase. We expect this process to occur at different
time scales if a clear community structure exists. Thus, the
dynamical route towards the global attractor will reveal
different topological structures, presumably those which
represent communities. Therefore, it is the complete dy-
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namical process what unveils the whole organization at all
scales, from the microscale at very early stages up to the
macroscale at the end of the time evolution. On the con-
trary, those systems endowed with a regular topological
structure will display a trivial dynamics with a single time
scale for synchronization.
To study this phenomena, instead of considering a global
observable, we define a local order parameter measuring
the average of the correlation between pairs of oscillators
ij
thcos
i
t
j
ti; (2)
where the brackets stand for the average over initial ran-
dom phases. The main advantage of this approach is that it
allows us to trace the time evolution of pairs of oscillators
and therefore to identify compact clusters reminiscent of
the existence of communities.
To give evidence of the aforementioned facts, we have
analyzed the dynamics towards synchronization—the time
evolution of
ij
t—in computer-generated graphs with a
hierarchical community structure. In [21] the authors pro-
posed models of networks with a well-defined community
structure that has been used as a benchmark for different
community detection algorithms [6]. Here, we propose a
generalization of this model that includes two hierarchical
levels of communities. The graphs we generate are as
follows: we prescribe, in a set of 256 nodes, 16 compart-
ments that will represent our first community organiza-
tional level, and 4 compartments containing each one four
different compartments of the above first level, which
define the second organizational level of the network.
The internal degree of nodes at first level z
in
1
and the
internal degree of nodes at second level z
in
2
keep an
average degree z
in
1
z
in
2
z
out
18. From now on, net-
works with two hierarchical levels are indicated as z
in
1
-z
in
2
,
e.g., a network with 13-4 means 13 links with the nodes of
its first hierarchical level community (more internal),
4 links with the rest of communities that form the second
hierarchical level (more external), and 1 link with any
community of the rest of the network.
In Fig. 1 we represent
ij
t at the same time t for two
slightly different hierarchical networks 13-4 and 15-2. In
the two figures we can identify the two levels of the
hierarchical distribution of communities. The network
13-4 (left) is very close to a state in which the four large
groups are almost synchronized, whereas the network 15-
2 (right) still presents some of the smaller groups of
synchronized oscillators, and the larger group starting to
synchronize, coherently with their topological structure.
The visualization of the correlation matrix of the system
helps in elucidating the topology of the network. To extract
the quantitative information it is useful to introduce some
threshold T to convert the correlation matrix into a binary
matrix, that will be used to determine the borders between
different groups. We define a dynamic connectivity matrix
D
t
T
ij
1 if
ij
t >T
0 if
ij
t <T
; (3)
that depends on both the underlying topology and the
collective dynamics. For a fixed time t, by moving the
threshold T, we obtain different representations of D
t
T
that inform us about the structure of the dynamic correla-
tions. When the threshold is large enough, the representa-
tion of D
t
T becomes a set of disconnected clumps or
communities. Decreasing T a hierarchical structure of
communities is devised. Note that since the function
ij
t is continuous and monotonic (because the existence
of a unique attractor of the dynamics), we can redefine
D
T
t, i.e., fixing the threshold and evolving in time. We
obtain the same information about the structure of the
dynamic connectivity matrix at different time scales. Let
us show that these time scales unravel the topological
structure of the connectivity matrix at different topological
scales.
From the eigenvalue spectrum of D
T
t, SD
T
t, one
can extract the number of disconnected components of the
system as the number of null eigenvalues. The evolution of
SD
T
t traces the hierarchy of communities as follows: at
short times, all units are uncorrelated and then we have N
disconnected sets, N being the number of nodes in the
network; as time goes on, nodes become synchronized in
groups according to their topological structure. In
Fig. 2 (top) we plot, for the two networks analyzed in
Fig. 1, the number of disconnected components as a func-
tion of time, for a fixed threshold T [22]. We can observe
the relative stability of the two partitions for the two net-
works, corresponding to the two prescribed hierarchical
levels. For the 13-4 network the synchronization of the
4 groups of 64 nodes each is much more stable than the
16 groups of 16 nodes, i.e., the community structure at the
second hierarchical level is stronger, whereas the opposite
can be inferred for network 15-2.
Another interesting link between dynamics and topol-
ogy can be highlighted from the analysis of the whole
spectrum of the Laplacian matrix of the network graph L
[23]. The Laplacian matrix is defined as L
ij
k
i
ij
a
ij
,
FIG. 1 (color online). Average of the correlation between pairs
of oscillators. The structure networks are 13-4 (left) and 15-2-
(right). See text for a description of the networks. The colors are
a gradation between blue (0) and red (1).
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where k
i
is the degree of node i,
ij
is the Kronecker delta,
and a
ij
is the element of the adjacency matrix (1 if nodes i
and j are connected and 0 otherwise). The spectral infor-
mation of the Laplacian matrix has been used to understand
the structure of complex networks [24], and, in particular,
to detect the community structure [25,26]. Recent studies
have also focused on the spectral information of the
Laplacian matrix and the synchronization dynamics [11–
18]. The common approach is to take advantage of the
master stability equation [27] to determine the relation
between the relative stability of the synchronized state
(via the ratio
N
=
2
) and the heterogeneity of the topology,
although sometimes some language abuse appears and
authors talk about better or worse synchonizability instead
of stability of the synchronized state. Our approach differs
from these works in the following: we are interested in the
transient towards synchronization because it is this whole
process which will reveal the topological structure at dif-
ferent scales. For this reason our analysis focuses on the
whole eigenvalue spectrum of the Laplacian matrix SL.
To characterize this spectrum, we rank the eigenvalues
of L using an index i in ascending order 0
1
2
...
i
...
N
. The structure of this sequence brings to
light many aspects of the topological structure: (i) the
number of null eigenvalues gives trivially the number of
disconnected components, (ii) the gaps between consecu-
tive eigenvalues tell us about the relative differences of
time scales, and (iii) large eigenvalues in the last part of the
series stand for the existence of hubs in the network (we
will turn to these points later). In Fig. 2 (bottom) we have
plotted the eigenvalues of the Laplacian matrix for the 13-4
and 15-2 structures. We observe three groups of eigenval-
ues separated by gaps. Each gap separates a community
either of 256, 16, or 4 group’s elements or the whole
population. Notice that for the 13-4 graph the plateau of
16 communities is shorter than the plateau for 4 commun-
ities and the contrary is true for the 15-2 case, indicating
that the 16 clusters community is less well defined in the
former case. Indeed, the ratio between the eigenvalues is a
good quantitative measure of the stability of the structure
(which is measured in terms of modularity in other studies
[6]) and is related to the length of the plateaus observed in
Fig. 2 (top).
We visualize the formation of the connected groups of
synchronized oscillators in time by constructing a dendo-
gram in which we draw lines between groups of oscillators
when they merge. Applying this technique to the above
defined networks we can see two different topological
scales disclosed by synchronization and the relative stabil-
ity of them. The networks investigated so far are homoge-
neous in degree. At this point we ask about the effect when
inhomogeneities in degree are considered. We have applied
this procedure to the network structure proposed by Ravasz
and Barabasi [28] with a hierarchical structure in two
levels and a scale-free degree distribution. As can be
seen from the dendogram depicted in Fig. 3, the commun-
ities synchronize at different times, depending on their role
in the hierarchy, and it also shows the remarkable effect of
hubs in the synchronization process.
Finally we would like to shed some light about the
intriguing relationship between the eigenvalues of the
Laplacian and the dynamic structures that emerge towards
synchronization. To understand this correspondence let us
analyze the linearized dynamics of the Kuramoto model
(i.e., the dynamics close to the attractor of synchronization)
in terms of the Laplacian matrix,
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
time
FIG. 3. Left: Ravasz-Barabasi network of 25 labeled nodes
with two hierarchical levels. Right: Time evolution of the
synchronization process between labeled oscillators. The length
of the dendogram branches indicate the relative stability of the
different structures.
100
time
10
100
i
15-2
0.1 1
1/λ
i
10
100
i
15-2
100
time
10
100
i
13-4
0.1 1
1/λ
i
10
100
i
13-4
FIG. 2 (color online). Top: Number of disconnected synchro-
nized components (equivalent to number of null eigenvalues of
SD
T
t) as a function of time for the two networks of Fig. 1 at
T 0:99. Bottom: Rank index i (see text) versus the inverse of
the corresponding eigenvalues of the Laplacian matrix L. The
shadow regions indicate the stability plateaus for 16 (dark) and
4 (light) communities. The same representation is used for the
plateaus in the eigenvalue spectrum corresponding to indices 16
and 4.
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d
i
dt
k
X
j
L
ij
j
i 1; ...;N; (4)
whose solution in terms of the normal modes ’
i
t reads
’
i
t
X
j
B
ij
j
’
i
0e
i
t
i 1; ...;N; (5)
where
i
are the eigenvalues of the Laplacian matrix, and B
is the eigenvectors matrix.
This set of equations has to be satisfied at any time t.If
we rank the system of equations in descending order of the
eigenvalues (i.e., starting from
N
), the right-hand side
system of Eq. (5) will approach zero in a hierarchical
way. This fact is equivalent in the dynamics to group
oscillators surpassing the synchronization threshold form-
ing communities. The gaps in the spectrum SL clearly
represent different time scales between modes revealing
different topological scales. The collective modes, the
solution of the system represented by Eq. (5), denote two
types of behaviors. Some modes provide information about
reorganization of the phases in the whole network, while
the others inform us about synchronization between pairs
or groups of oscillators. The presence of hubs in the
topology gives rise to large eigenvalues that decay very
fast and are related to the first type of modes, those
representing ‘‘synchronization’’ between the hub and the
topological average of the phases of rest of oscillators. The
rest of the modes relate oscillators that have similar pro-
jections on the corresponding eigenvectors, thus giving rise
to communities at a given topological scale. Indeed, this
fact supports the success of the identification of commun-
ities using spectral analysis [25].
Summarizing, we have analyzed the synchronization
dynamics in complex networks and show how this process
unravels its different topological scales. We have also
reported a connection between the spectral information
of the Laplacian matrix and the hierarchical process of
emergence of communities at different time scales.
We thank M. A. Mun
˜
oz, Y. Moreno, and R. Guimera
`
for
helpful comments. This work has been supported by DGES
of the Spanish Government Grant No. BFM-2003-08258
and EC-FET Open Project No. IST-2001-33555.
Note added in proof.—M. Sales, R. Guimera
`
, and
L. A. N. Amaral have a paper about a closely related sub-
ject: the determination of community hierarchies in com-
plex networks. We are aware of this work by personal
communication although the paper has not been publicly
available.
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