Complex Network Topologies and Synchronization
Paolo Checco
Gabor Vattay
Mario Biey
Dept. of Electronics
Dept. of Electronics
Collegium Budapest,
Politecnico di Torino
Politecnico di Torino
Inst for Advanced Studies,
Turin, Italy
Turin, Italy
Budapest, Hungary
Email: paolo.checco@polito.it Email: mario.biey@polito.it
Email: vattay@elte.hu
Ljupco Kocarev
Institute for Nonlinear Science
University of California
San Diego, USA
Email: lkocarev@ucsd.edu
Abstract- Synchronization in networks with different topologies is studied. We show that for a large class of oscillators there
exist two classes of networks; class-A: networks for which the
condition of stable synchronous state is 7uY2 > a, and class-B:
networks for which this condition reads
Y2 < b, where a
and b are constants that depend on local dynamics, synchronous
state and the coupling matrix, but not on the Laplacian matrix
of the graph describing the topology of the network. Here
7Y = 0 < -Y2 < ... < 'YN are the eigenvalues of the Laplacian
matrix, where N is the order of the graph. Synchronization in
networks whose topology is described by classical random graphs
and power-law random graphs when N -) oc is investigated in
detail.
is shown that for typical systems only three main scenarios
may arise as a function of coupling strength. Then, we study
synchronization in complex networks topologies. Section III
icopexsies.on
synchronization
of
idevoted to the
of
networks whose topology is described by classical random
networks. In Section IV we study synchronization properties
of power-law random graph models. We close our paper with
conclusions (Section V).
I. INTRODUCTION
being a (chaotic) oscillator. Let xi be the m-dimensional
vector of dynamical variables for the i-th node. Let the
dynamics of each node be described by:
Real networks of interacting dynamical systems be they
neurons, power stations or lasers - are complex. Many realworld networks are small-world [1] and/or scale-free networks [2]. The presence of a power-law connectivity distribution, for example, makes the Internet a scale-free network.
The research on complex networks has been focused so far
on the their topological structure [3]. However, most networks
offer support for various dynamical processes. In this paper we
propose to study one aspect of dynamical processes in nontrivial complex network topologies, namely their synchroniza-
tion behaviors.
The general question of network synchronizability, for many
aspects, is still an open and outstanding research problem [4],
[5]. In this context, an important contribution has been given
by Pecora and Carroll in [6], where, for a network of coupled
chaotic oscillators, they derived the so-called Master Stability
Equation (MSE), and introduced the corresponding Master
the stability analyStability Function (MSF). Consequently, '
sis of the synchronous manifold [6] for the network under
consideration can be decomposed in two sub-problems. The
first sub-problem consists of deriving the MSF for the network
nodes, i.e. to study in which region of the complex plane
the MSE admits a negative largest Lyapunov exponent (LE).
The second sub-problem is to verify whether the eigenvalues
of the so-called connectivity matrix [7] of the network, apart
from the zero-eigenvalue, lie in the synchronization region(s)
analsis synchronizaton properties
II. SYNCHRONIZATION REGIONS
Let us consider a network with N identical nodes, each
N
ti
f (xi) + a E, DikXk
0-7803-9390-2/06/$20.00
©C2006 IEEE
=
1, ....N
(1)
jpm describes the oscillator equations,
m
where f
which we assume to admit a chaotic attractor, o is the overall
strength of coupling, while Dik are m x m real matrixes.
Assume that each matrix, Dik, has the form: D
lj H,
where lj is a real number defined in the following and
H is a m x m diagonal matrix, same for all nodes, called
coupling matrix. The coupling matrix H (hij) contains
the information about which variables are utilized in the
1, if the i-th component
coupling and is defined as h
is coupled, and hii = 0, otherwise. Let x = (X1,.., XN)T
)T
the
matrix L ( l f)
be the Laplacan matrix,
connection toojo ofthe netwok:al =
-1rif
oes
~~~~~connection topology of the network: lij Iji I- if nodes
i
k
0 otherwise
other nodes, and
then,dwe,can rit E (1) iatmorec
Thed e
ct omarixes:
representng
F(x) + or (L 0 H) x,
]mN IT, >
F(x)
(see also [6], [7], [8].) This approach is particularly relevant F(x) =(f(xi),.. , f (xN))
because the MSE depends only on the nodes local dynamics
and on the coupling matrix [7].
In this work, at first, we study the synchronization regions,
using the properties of the MSF. Namely, in Section II it
i
k=1
where
(2)
jmN is defined
f
as
The matrix L, which will be our main concern, is positive
semi-definite and symmetric. Its smallest eigenvalue is tYi 0.°
Denote by tY1 0 <t72 <K. .. < yN the eigenvalues of L. In
particular, /N iS the maximal eigenvalue of L.
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ISCAS 2006
Since L is symmetric, the master stability function, in this
case, has the form [6]
* Jf
H]
(3)
(3)
¢)=
[Jf + ag H] ¢,
where a C ? and Jf is the Jacobian matrix of f(x).
Therefore, in this case the corresponding largest Lyapunov
exponent or MSF, A(a), depends only on one parameter, a.
Master stability function determines the linear stability of the
synchronized state; in particular, the synchronized state is
stable if all eigenvalues of the matrix L are in the region
A(a) < 0. We denote by S C if? the region where the MSF is
negative and call it synchronization region. Discussions in [5]
show in fact that for the system (2), the synchronization region
S may have one of the following forms:
* Si = 0
S2 = (arn : +°°)(5
52
)
** S3=uj(atm,+
S3
+
=Uj(a$ Qam)
Examples of the these scenarios are given in [9], [10]. In the
majority of cases am atj,and aC turn out be positive and,
furthermore, in the case S3 there is only one parameter interval
(a$(j) a(j)) on which A(a) < 0. For this reason, we will limit
ourself to consider only such cases, focusing, in the remaining
of this paper, on the scenarios S2 = (am +o0) and S3 =
(am, CaM). It is easy to see that for S2 the condition of stable
synchronous state is Ory2 > am. For S3, one can easily show
that there is a value of the coupling strength or for which the
synchronization state is linearly stable, if and only if -yNf-y2 <
aM/am. Therefore, for a large class of (chaotic) oscillators
there exist two classes of networks:
1.oClass-A nypetr netwforkswhoshte synchrioniz stio
ginchronoisofype2,frwhchtecodit
stableis
re>
a;
>
kstatet
nch
2. 2Classynchronetous
Class-B networks:
networkshosea;
whose synchronization
rew
is o
giN
YNreY2 < b;
Or-~2
ti
where a a omand b a =M/a.m
are constants that depend on
1 matri L. For and the matrix
f, the synchronousLstate =c
H, but not on the Laplacian matrix L. For typical oscillators
b > 1.
III.
SYNCHRONIZATION IN CLASSICAL RANDOM
NETWORKS
The primary model for the classical random graphs is the
Erdos-Renyi model gq [11], in which each edge is independently chosen with the probability q for some given q > 0.
Let G(N, q) be a random graph on N vertices.
For the model of a random graph we take a sequence
of probability spaces (F(N, q))N, where q is a real number
between 0 and 1, and N is an integer. We shall assume that q
is fixed, but in general it may depend on N. The probability
space F(N, q) consists of all labelled simple graphs on N
vertices, and an edge between an arbitrary pair of vertices
appears with probability q, i.e. F(N, q) has 2M elements,
where M =N(N -1)72, and each graph in F(N, q) with m
edges has the probability equal to qm(1-q)M-m. By PN,q(X)
we will denote the probability of an event X C F(N, q) in
the probability space F(N, q). Let p(G) mean that the graph
G has the property p. We say that the property p happens
asymptotically almost surely (a.a.s)), if
(4)
lim PN,q{G C F(N,q): p(G)} 1.
N--coo
Theorem 3.1: Let G(N, q) be a random graph on N vertices. Then the class-A network G(N, q) asymptotically almost
surely synchronize for arbitrary small coupling or and the
class-B network G(N,q) asymptotically almost surely synchronize for b > 1.
Proof: The proof of the theorem follows from the
following result [12]. Let q be a fixed real number between 0
and 1. For almost every graph and every E > 0
{ -2(G) > qN-(2- +)pqNlogN
-y2(G)
and
r YN(G)
< qN-(2--)pqNlogN
> qN +
(2-c)pqN log N
(6)
N (G) < qN + (2 + c)pqN log N.
Therefore, for large N, -Y2 r N, while -Nl/-2 approaches 1.
Now, for class-A networks the condition for synchronization
reads (7 > a/N and or can be chosen arbitrary small. For
class-B networks with b > 1, since aNl/-2 approaches 1, when
N -> oc, it follows that the network almost surely synchronizes.
U
SYNCHRONIZATION
IN
POWER-LAW
NETWORKS
IV.
We consider a random model introduced recently by Chung
and Lu [13], which produces graphs with a given expected
degree sequence. Therefore, this model does not produce the
exact
graph with
sequence.
random
withgiven
graph
givendegree
expected
degree Instead,
sequence.it yields a
We consider the following class of random graphs with a
given expected degree sequence w = (wl, w2,. . , WN). The
vertex vi is assigned vertex weight wi. The edges are chosen
independently and randomly according to the vertex weights
as follows. The probability Pij that there is an edge between
vi and vj is proportional to the product wiwj where i and j
are not required to be distinct. There are possible loops at vi
2
with probability proportional to wi, i.e.,
7
wiwi
(7)
k
pij
=
EkWk
Zk Wk. This assumption ensures
and we assume maxi w2i <
that Pij < 1 for all i and j. We denote a random graph
with a given expected degree sequence w by G(w). For
example, a typical random graph G(N, q) (see the previous
section) on N vertices and edge density q is just a random
graph with expected degree sequence (qN, qN, . . . , qN). The
random graph G(w) is different from the random graphs with
an exact degree sequence such as the configuration model. We
will use dito denote the actual degree of mjina random graph
G in G(w), where the weight Wi denotes the expected degree.
The following proposition is proved in [13].
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Proposition 4.1: With probability 1 - 2/N, all vertices vi
satisfy
2Vw_ logN < di-wi <
(8)
0.6
We consider the model M(N, Q, d, m), where N is the
number of vertices, Q > 2 is the power of the power law,
d is the expected average degree, defined as d EwiIN,
and m is the expected maximum degree (or an upper bound
for the range of degrees that obey the power law), such that
= o(Nd). We assume that the i-th vertex vi has expected
-m
degreewi = c(i+io 1)-B 1, for 1 < i < N. Here cdepends
on the average degree d and io depends on the maximum
expected degree m (see [5] for passages):
1
;3 2
72
04
< 2 logN +
ogN.
(w3og)
0.5 _-^,
=
-d(1 -(2)
= [N
No
- 1
T
m_(_1)
(10)
-.
200
Fig. 1.
300
400
500
600
2 versus N for the model
700
N
800
900
1000
M(N, 3, d, m) with 3
=
1100
3, d
=
1200
7, and
random graph which has the same set of vertices as G2. Then
;2(M) > ;2(Gi).
According to [15] and [16] (see also [17]), we have
It is easy to compute that the number of vertices of
expected degree between k and k + 1 is of order c'k-, where
c' = c ( - 1), as required by the power law.
72(M) > -2(G1) > k - -k2 d(In2 - kIn2). (13)
Let k be the expected minimum degree. Then
/3-F7d/3- Y\ l 1 On the other hand, from the following inequality (see [5])
13 -2
(d
/3 I~~Tn
(/3 -2) '3 12
(/3.-(11)
For the considered model d can be in any range greater
than 1: it does not have to grow with N [13]. We first consider
the case when d grows with N.
Theorem 4.2. Let M(N, Q, d, m) be a random power-law
go with
w N and d/m -> O
wh d,
d grows
graph on N vertices, for which
when N -* ocx. Then class-A network M (N, /, d, m) asymptotically almost surely synchronizes for arbitrary small
coupling ur and class-B network M(N, Q, d, mr) asymptotically
or
almost surely does not synchronize.
Proof: From [5], the following inequalities hold
Nand d/me-la0
graheon N.2:
verti
alm
.dos notsnronize.
(12)
A (M) <. N (M) < 2A (M),
A
N-I
where A(.) denotes the maximum degree of a graph. It follows
< 2 A. Therefore, from (8)
that for large N we have A < KyN
iAm for large N [5]. Equation (11)
we have TyN(M) zAz
can be rewritten as
1
d ;3-ll t3 1
k
+-1.1 dJ I
Since d < m, we have k d. Therefore, when d grows with
N, the minimum expected degree k also grows with N.
It is proven in [14] that the function 7-2 (G) iS non-decreasing
for graphs with the same set of vertices, i.e. 7y2(G1) . 7y2(G2)
if G1 C G2 and G1, G2 have the same set of vertices. Let G2
be our M(N,Q, d, m) random graph and G1 be a k-regular
-
~2Y . NNand (8), it follows that for large N,
N
-2(M)<Nl166k,
(14)
(4
(15)
where d is the minimum degree of the graph. Combining (13)
ihk
1)w idta ~()
k.
can beeapoiae
approximated with
and
(15)
If d grows with N, since 2 also grows with N we conclude
that the class-A nletwork M(N, j, d, m) asymptotically almost
surely synchronize for arbitrary small coupling or. Since b is a
finite number, from -yN/ly2 rn/k, we see that for sufficiently
large N, almost every class-B network M(N, Q, d, m) does
not synchronize.
Now we consider the case d < oc. Since, in this case, we
could not obtain analytical bounds for -Y2 and -YN we provide
numerical examples. Consider the model M(N, Q, d, m) with
Q 3,d 7, and m 30. Figures I to 3 show the 72, YN,
and yN/72 versus N. The figures are obtained by simulating
graphs composed of 200 to 1200 nodes, with a step of 10
nodes. For each case, 10 different simulations are computed
+and the mean value is presented as a dot (solid line is a curve
fitting the dots). Note that the actual maximum degree A may
differ from the expected maximum degree m. Consider now
a class-A network with a =1 and a class-B network with
b =40. From Fig. 1 one can compute the value of 72 for
0.31, and therefore, the network synchroN =1200, 72
nizes for uJ > 3.23. Moreover, from Fig. 3 one can compute
the value of ayN/7y2 for N =1200, which is approximately
withat -2(M
n
N
k
=
=
=
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a
V. CONCLUSION
34
33.8
In this paper we studied synchronization in networks with
33.6
different topologies.
We showed that for a large class of oscillators there exist
two classes of networks; class-A: networks for which the
condition of stable synchronous state is u72 > a, and class-B:
networks for which this condition reads -yN/'y2 < b, where a
and b are constant that depend on local dynamics, synchronous
33.2
state and the coupling matrix, but not on the Laplacian matrix
32.8
of the graph describing the topology of the network.
Let G(N, q) be a classic random graph (Erdos-Renyi model)
on N vertices. We proved that for sufficiently large N,
32.4
_
32.2
2(
300
400
500
600
700
800
900
1000
1100
1200
Fig. 2. -N versus N for the model M(N, 3, d, m) with 3 - 3, d - 7, and
30.
m
110 n,0
100 _
*n
VI. ACKNOWLEDGEMENT
This work was supported in part by Ministero
dell'Istruzione, dell'Universit'a e della Ricerca under PRIN
Project no. 2004092944_004.
REFERENCES
90
80
70
%
0,
_
^
the class-A network G(N, q) almost surely synchronize for
arbitrary small coupling or. For sufficiently large N, almost
every class-B network G(N, q) with b > 1 synchronizes.
Let M(N, 3, d, m) be a random power-law graph on N
vertices
We proved
that for
sufficiently large N, the class-A
vrie.W
rvdta
o ufcetylreN h
lsnetwork M(N, Q, d, m) almost surely synchronize for arbitrary small coupling or. For sufficiently large N, almost every
class-B network M(N, Q, d, m) does not synchronize.
60
7N /2/
[1] D. J. Watts and S. H. Strogatz, "Collective dynamics of 'small-world'
networks," Nature, vol. 393, pp. 440 - 442, 1998.
[2] A. L. Barabasi and R. Albert, "Emergence of scaling in random
networks," Science, vol. 286, pp. 509 - 512, 1999.
[3] R. Albert and A. L. Barabasi, "Statistical mechanics of complex networks," Rev. Mod. Phys., vol. 74, pp. 47 - 97, 2002.
[4] S. Strogatz, Sync: The Emerging Science of Spontaneous Order. New
York: Hyperion, 2003.
[5] L. Kocarev, P. Checco, G. M. Maggio, and M. Biey, "Synchronization
in complex network topologies," IEEE Trans. Circ. Syst. I, submitted
for publication.
[6] L. Pecora and T. Carroll, "Master stability functions for synchronized
coupled systems," Phys. Rev. Letters, vol. 80, pp. 2109 - 2112, 1998.
[7] K. S. Fink, G. Johnson, T. Carroll, D. Mar, and L. Pecora, "Three
coupled oscillators as a universal probe of synchronization stability in
coupled oscillator arrays," Phys. Rev. E, vol. 61, pp. 5080 - 5090, 2000.
M. Barahona and L. M. Pecora, "Synchronization in small-world sys[8] tems,"
Phys. Rev. Letters, vol. 89, pp. 054 101 (1-4), 2002.
50
40
20
10_
200
Fig. 3.
and m
300
400
500
600
700
800
N
'YN/I72 versus N for the model M(N,
=
30.
900
1000
1100
3, d, m) with 3
=
3, d
1200
=
7,
107. Consequently, since b < 107, the class-B
'YN/'y2
network does not synchronizes.
Let us write
Ic
a/->2
and
bL
'TNI/~2.
m=ay
07 and
bc
ritical values
synchonie
areare critical
values for whichthe
thenetwork
network may synchronize,
in other words, if (7 > (7c (b > bc), then the class-A
(class-B) network synchronizes. The proof of Theorem 4.2
for
which
may beapproxiated
suggestssuggeststhat
that theth criticl
critical value
values may
as
be approximated as
a/k and bc = n/k provided that k and d are close
07c
to each other. For example, consider a network composed
-
by N = 1200 nodes with d = 20, m = 200, and aQ= 3,
1 and
for which k
9.99. Then we have (with a
0.10 and bc
20.02. Simulating such a
b
40) orc
network, the following actual eigenvalues have been obtained:
=
act) _7.61, jac)
196.43, and (yN/ffyn2)(acL)
b(ct) __25.83. In this case, since b
class-B networks synchronize.
25.83. It
[9] P. Checco, L. Kocarev, G. M. Maggio, and M. Biey, "On the synchronization region in networks of coupled oscillators," in Proc. IEEE ISCAS,
vol. IV, Vancouver, Canada, May 23 - 26, 2004, pp. 800 - 803.
[o0] T. Stojanovski, L. Kocarev, U. Parlitz, and R. Harris, "Sporadic driving
[I
of dynamical systems," Phys. Rev. E, vol. 55, pp. 4035 - 4048, 1997.
P. Erd6s and A. R6nyi, "On random graphs," Publ. Math Debrecen,
[1 1] vol.
6, pp. 290 291, 1959.
-
[12] M. Juvan and B. Mohar, "Laplace eigenvalues and bandwidth-type
invariants of graphs," J. Graph Theory, vol. 17, pp. 393 - 407, 1993.
[13] F: Chung and L. Lu, "Connected components in random graphs with
given expected degree sequences," Annals of Combinatorics, vol. 6, pp.
125 - 145, 2001.
[14] M. Fiedler, "Algebraic connectivity of graphs," Czech. Math. J., vol. 23,
pp. 298 - 305, 1973.
[15] B. Bollobas, "The isoperimetric number of random regular graphs,"
Euro. .J. Combinatorics, vol. 9, pp. 241 - 244, 1988.
[16] B. Mohar, "Isoperimetric numbers of graphs," J. Combinatorial Theory
follows~~~~~~~~~~~~~~~~~C WhaWhcuaciia,ausaeo"Synchronization
3ad[7 u
in arrays of coupled nonlinear systems:
=40, both class-A and
Passivity, circle criterion, and observer design," IEEE Trans. Circuits
Syst. I, vol. 48, pp. 1257 - 1261, 2001.
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