arXiv:nlin/0112023v1 [nlin.CD] 17 Dec 2001
Synchronization in Small–world Systems
Mauricio Barahona
1, ∗
and Louis M. Pecora
2
1
Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 91125
2
Naval Research Laboratory, Code 6340, Washington, DC 20375
(Dated: February 5, 2008)
We quantify the d ynamical implications of the small-world ph enomenon. We consider the generic
synchronization of oscillator networks of arbitrary topology, and link the linear stability of the
synchronous state to an algebraic condition of the Laplacian of the graph. We show numerically
that the addition of random shortcuts produces improved network syn chronizability. Further, we
use a perturbation analysis to place the synchronization threshold in relation to the boundaries of
the small-world region. Our results also show that small-worlds synchronize as efficiently as random
graphs and hypercubes, and more so than standard constructive graphs.
PACS numbers:
Recently, Watts and Strogatz [1] showed that the ad-
dition of a few long-rang e shortcuts to an otherwise lo-
cally connected lattice (the ”pristine world”) produces
a sharp reduction of the average distance between a rbi-
trary nodes. The e nsuing semi-random lattice was de-
noted a small-world (SW) because the sudden appea r-
ance of short paths occurs early on, while the system is
still rela tively localized. This concept has wide appeal:
the SW pr ope rty seems to b e a quantifiable characteris-
tic of many real-world structures [1, 2, 3, 4], both human
generated (social networks, WWW, power grid), or of
biological origin (neural and biochemical network s).
A spur of ongoing research [2] has concentrated on
static and combinatoric properties [5, 6, 7, 8, 9, 10] of a
tractable SW model [1, 11]. Monas son [11] considered the
SW effect on the distribution of eigenvalues of the con-
nectivity matrix (the graph Laplacian) which specifies
the coupling between nodes—a relevant topic fo r poly-
mer networks [12]. However, despite their central role in
real-world network s, there are fewer studies of dyna mi-
cal processes taking place on SW lattices. Among those,
automata epidemics simulations [13] and We b-browsing
studies [14] have revealed the importance of shortcuts.
Numerical work on synchronization of Kuramoto oscil-
lators [3], discrete maps [15] and Hodgkin-Huxley neu-
rons [16] has shown improved SW synchronizability, as
intuitively expected. However, these numerical examples
are not generic, and fail to provide insight into how the
SW property influences the dynamics.
In this paper , we explicitly link the SW addition of
random shortcuts to the synchronization of networks o f
coupled dynamical systems. This is an example of dy-
namics on networks—leaving aside the distinct problem
of evolution of networks here. By using a gener ic syn-
chronization formulation [17, 18] to factor out the con-
nectivity of the network, we identify the synchroniza-
tion threshold with an algebraic condition of the graph
Laplacian. Through numerics and analysis, we qua ntify
α
1
α
2
-0.8
-0.4
0.0
α
-1.2
λ
max
151050
FIG. 1: Four typical master stability functions (scaled for
clearer visualization) for R¨ossler systems: chaotic (bold) and
periodic (regular lines); with y-coupling (dashed) and x-
coupling (solid lines). Here we consider the x-coupled chaotic
case (solid bold) with a negative well between (α
1
, α
2
).
how the SW scheme improves the synchronizability of the
pristine world, mainly as a result of the steep increase o f
the first-non-zero eigenvalue (FNZE). The synchroniza-
tion threshold is found to lie in the SW re gion [3, 13], but
does not coincide with its onset—it can in fact be linked
to the effective randomizatio n that ends SW. Within this
framework, we show that the synchronization efficiency
of semi-random SW networks is higher than standard de-
terministic graphs, and comparable to bo th fully random
and ideal constructive graphs.
Consider n identical dynamical systems (placed at the
nodes of a graph) that are linearly and symmetrically
coupled (as represented by the edges of the undirected
graph) with global coupling strength σ. The to po logy of
the graph c an be encoded in the Laplacian matr ix G, a
symmetric matrix with zero row-sum and real sp e ctrum
2
{θ
k
}, k = 0, 1, . . . , n − 1. A general linear stability cri-
terion for the synchronized state of the system [17, 18]
is given by the negativity of the master stability func-
tion λ
max
(σ θ
k
) < 0, ∀k. This λ
max
is a characteris tic
of the particular dynamics at the nodes but, crucially,
a large class of oscillatory systems (chaotic, periodic
and quasiperiodic) have master stability functions with
generic features [1 8]. In particular, for several chaotic
systems λ
max
has a single deep well, as depicted in Fig. 1.
(We remark that this analysis is quite genera l: it can be
extended [18] to eliminate the zer o row-sum constraint,
and to comprise nonlinear coupling and mor e general s yn-
chronization criteria.) Stability is thus ensured by tun-
ing the coupling σ to try and place the entire spectrum
of transverse eigenvalues (times σ) in the deep, stable
region: σ θ
k
∈ (α
1
, α
2
). This leads to an algebraic con-
dition for the existence of a linearly stable synchronous
state: a network is synchronizable if
θ
max
/θ
1
< α
2
/α
1
≡ β, (1)
where θ
1
is the FNZE and θ
max
is the ma ximum eigen-
value of the Laplacian G. The figure of merit (β) r anges
from 5 to 10 0 for a variety of oscillators (e.g., Lorenz,
R¨ossler, double scroll).
Small-worlds are generated from a pristine world: a
k-cycle of n nodes and range k, each node coupled to its
2 k nearest neighbors for a total of nk edges [11]. The
Laplacian of this graph G
0
is a banded circulant matrix
with non-ze ro elements on the main diagonal and the 2 k
adjacent diagonals: G
0
ii
= 2k and G
0
ij
= G
0
ji
= −1 with
(i + 1) mod n ≤ j ≤ (i + k) mod n, and 1 ≤ i ≤ n.
The SW scheme dope s the pristine world by adding ns
edges picked at random from the n(n − 2k − 1)/2 re-
maining pa irs. Each new edge between nodes l and m
adds off-diagona l ∆G
lm
= ∆G
ml
= −1 and o n-diagonal
∆G
ll
= ∆G
mm
= 1 contributions to the Laplacian, thus
preserving the null row-sum and the bidirectional cou-
pling. The average number of shortcuts per node (s) is
related to other measures of randomnes s (p and q) us e d
previously [1, 11]: s ≡ kp ≡ q(n − 2k − 1)/(2n).
The numerical results in Fig. 2 illustrate the SW ef-
fect on the synchronization of networks of different size
and ra nge. For concreteness, all our numerics have been
performed for a network of identical x-coupled R¨ossler
chaotic oscillators with β ≃ 37.85. Similarly to other
locally connected networks, pristine worlds have a large
eigenratio θ
max
/θ
1
(i.e., they are difficult to synchronize).
However, as s is increased the eigenratio falls sha rply un-
til, at a value s
sync
, the condition (1) is reached (i.e., the
addition of shortcuts ma kes it synchronizable). The de-
pendence of s
sync
on the network parameters {n, k} is
notably complicated. First, there appears to be an op-
timal range k ≃ 4 for which the SW is most efficient.
Moreover, the synchronization threshold s
sync
lies in the
small-world region but does not seem to coincide with its
onset. The SW onset (s
L
) is defined [1, 3] by the decay of
0 10 20 30 40 50 60 70
10
−3
10
−2
10
−1
10
0
10
1
0
0.5
1
Average shortcuts per node, s
Range of pristine world, k
n=1000
n=500
n=300
n=400
s
C
s
L
Large world
Random
Small−world
s
sync
s
L
L(s)/L(0)
s
C
C(s)/C(0)
s
sync
FIG. 2: Synchronizability thresholds s
sync
(◦) for graphs with
n nodes (n = 300, 400, 500, 1000) and range k ∈ [1, 70], nu-
merically averaged over 1000 realizations. The solid lines cor-
respond to the analytical Eq. (8), valid in the range n
1/3
<
k < k
min
(n). For most parameters, s
sync
lies within the small-
world region between the dashed lines (s
L
< s < s
C
) de-
picted here for n = 1000, but it is distinct from the SW
onset s
L
. Note how synchronization is achievable without
random shortcuts by increasing the deterministic range up to
k
min
(n) (see Fig. 3). Inset: decay of the average distance L,
clustering C, and eigenratio (squares) as shortcuts are added
to a pristine world of range k = 20 and n = 500. We de-
fine s
L
and s
C
as the points where L and C are 75% of the
pristine world value; s
sync
is the point where the eigenratio
θ
max
/θ
1
= β ≡ 37.85.
the average graph distance [7] L(s) ≃ (n/k) f (ns), where
f(x) = arg tanh
x/
√
x
2
+ 2x
/
p
4(x
2
+ 2x). Fixing
L(s
L
)/L(0) = 3/4, we obtain s
L
≃ 1.061/n, nk ≫ 1.
The end of the SW region (s
C
) corresp onds to the effec-
tive graph randomization through the loss of tr ansitiv-
ity [1, 3, 19], as given by the decay of the clustering co-
efficient C [6, 20]: C(s)/C(0 ) ≃ (2k −1)/(2k(1 + s/k)
2
−
1), n ≫ 1. Again, fixing C(s
C
)/C(0) = 3/4, we obtain
s
C
≃ k
−1 + [(8 k − 1)/(6k − 3)]
−1/2
≃ 0.155 k. The
synchronization threshold generally lies between these
two boundar ie s which scale differently with n and k:
s
L
≃ 1.061 /n < s
sync
< s
C
≃ 0.155 k.
We can g ain insight into the synchronization thresh-
old through an ana lytical perturbation of the eigenra-
tio of the SW Laplacian G = G
0
+ G
r
. Here, G
0
is
the deterministic Laplacian of the pristine world, and
G
r
is the stochastic Laplacian for the rando m shortcuts:
G
r
ij
= G
r
ji
= −ξ
ij
(for i+k+1 ≤ j ≤ min{n, n−k+i−1}
with 1 ≤ i ≤ n − k + 1); G
r
ij
= 0 (otherwise); and
G
r
ii
= −
P
n
j=1
G
r
ij
(for 1 ≤ i ≤ n). The ξ
ij
are
n(n − 2k − 1)/2 i.i.d. Bernoulli random variables which
take the value 1 with probability q/n ≡ 2s/(n − 2k − 1)
(and the value 0 with probability 1 − q/n). The cir-
culant G
0
is Four ie r-diagona lizable [11] with spectrum
θ
0
j
= (2k+1)−sin[(2k+ 1)πj/n]/ sin[πj/n], 1 ≤ j ≤ n−1,
3
(plus θ
0
0
= 0 of any Laplacian). The FNZE and the max-
imum eigenva lue of the unperturbe d lattice are:
θ
0
1
≃ 2π
2
k(k + 1)(2k + 1)/(3n
2
), k ≪ n (2)
θ
0
max
≃ (2k + 1) + csc
3π/2
2k + 1
(3)
≃ (2k + 1)[1 + 2/3π], k ≫ 1, (4)
where (3) follows from a continuum approximation.
Following an “honest” treatment [21] with G
r
as the
perturbation, we treat the ana ly tical expressions of the
doubly degenerate eigenvalues as ra ndom variables to ob-
tain their expectations. We postpo ne the detailed calcu-
lations [22] and sketch her e the main results. After some
stochastic calculus, the expectations of the eigenva lues of
the SW Laplac ian to second order are shown to be:
Eθ
(1)
i
≃ q ±
p
3πq/4n (5)
Eθ
(2)
i
≃
2q
n
n
X
m=1
′
(θ
0
i
− θ
0
m
)
−1
, (6)
for q/n and k/n small. To improve the accuracy of s
sync
,
we have als o obtained an approximation to (6) for FNZE:
Eθ
(2)
1
≃
−2q
K
3
9n
π
2
+ K
2
−
7
5
+
6
π
2
K −
2
π
, (7)
where K = 2k +1. Eqns. (1),(2),(4),(5), and (7) are then
used to obtain an e stimate of s
sync
as the solution of an
algebraic equation involving only n and k:
θ
0
max
+ Eθ
(1)
max
= β
θ
0
1
+ Eθ
(1)
1
+ Eθ
(2)
1
. (8)
As shown in Fig. 2, this approximates well our numer-
ics for n
1/3
< k ≪ n, where the Rayleigh- Schr¨odinger
perturbation expansion is valid.
Using (8), we can obtain [22] a first order estimate
for the maximum o f the synchronization threshold s
∗
sync
in the valid range. The maximum occ urs at k
∗
≃
n
p
(1 + 2/3π)/2π
2
β with the asymptotic va lue s
∗
sync
≃
(2 + 4/3π)(1 − 2 k
∗
/n)/3(β − 1). Therefore, s
sync
<
s
C
[2(1+2/3π)/(2
√
3−3)(β−1)] is bounded by the end of
the SW region but linked to it. For k < n
1/3
, the eigen-
value bunching (quasi-degeneracy) in the pristine lattice
renders the doubly degenerate perturbation invalid. We
are developing another approximation to quantify the be-
havior in this limit, but our numerics [22] indica te that
the dependence of s
sync
with n is sub-logarithmic. This
confirms that the synchronizability is most effectively im-
proved for small-range networks (Fig. 2).
How efficient is the addition of random shortcuts for
synchronization? We have compared the semi-random
SW approach with purely r andom and purely determin-
istic schemes. An example of the latter is the synchro-
nization of pristine worlds through the increase of the
10
−2
10
−1
10
0
10
0
10
1
10
2
10
3
Fraction of edges of complete graph, f
Eigenratio, θ
max
/ θ
1
k=1
k=2
k=4
k=6
k=10
k=14
Random graph
β
Deterministic
(Pristine worlds)
Small−world
FIG. 3: Eigenratio decay in a n = 100 lattice as f edges are
added following purely deterministic, semi-random (SW), and
purely random schemes. Networks become synchronizable be-
low the dashed horizontal line (β). The squares (numerical)
and the solid line (Eq. (9)) show the decrease of the eigenratio
of pristine worlds (k-cycles) through th e deterministic addi-
tion of short-range connections—for n = 100, networks with
k ≥ 7 are synchronizable. The semi-random SW approach
(dots, shown for k = 1, 2, 4, 6, 10, 14) is more efficient in pro-
ducing synchronization. The dot-dashed line corresponds to
purely random graphs (RG, Eq. (10)), which become almost
surely disconnected at f ≃ 2 ln n/(n + 2 ln n) = 0.0843 (thus,
with θ
RG
1
= 0 and unsynchronizable). The merging of the SW
and RG behaviors as f → 1 is the dynamical analogue of the
effective randomization that leads to s
C
.
range k. From (2) and (4), the eigenratio of a pristine
lattice is
θ
0
max
θ
0
1
≃
3π + 2
2π
3
n
2
k(k + 1)
. (9)
Therefore, n nodes can be synchronized in a k-cycle if
k > k
min
≃ n
p
(3π + 2)/2π
3
β. (Note in Fig. 2 the con-
sistency of our analy tical approximation (8): s
sync
= 0
precisely at k
min
.) For purely random graphs G
n,f
[23],
θ
RG
max
θ
RG
1
≃
nf −
p
2f(1 − f)n ln n
nf +
p
2f(1 − f)n ln n
, (10)
where f is the number of edges measured as a fraction
of the complete graph. These are compared with SW
graphs in Fig. 3.
We remark o n several observations regarding Fig. 3.
First, the SW addition of shortcuts is more efficient than
the deterministic addition of short-range layers. Second,
the effective randomization o f SW lattices with edge addi-
tion tra nslates into converging synchronization behaviors
of random and SW networks at large f . The f → 1 region
is thus robustly stable: cutting co nnectio ns from the fully
connected graph has very little effect on synchronization
stability—not until over 90% a re cut (for k small) does
4
0 1000 2000 3000 4000 5000
10
−3
10
−2
10
−1
10
0
Number of nodes, n
Fraction of edges of complete graph, f
k−wheels
k−cycles
k−Moebius ladders
Best bipartite
Hypercubes
Random graphs
Small−world
(k=1,2,4)
FIG. 4: “Cost” of synchronization measured as the number of
edges needed to synchronize a lattice of n chaotic R¨ossler sys-
tems arranged in different topologies: deterministic graphs (k-
wheels, k-cycles, k-M¨obius ladders, bipartites, hypercubes),
random graphs, and small-worlds (). Small-worlds scale fa-
vorably compared to deterministic structures (and compara-
bly to the ideal and largely unrealizable hypercubes). Also,
SW graphs with small range k are as cost-efficient as random
graphs but demand less (algorithmic) storage memory.
the eigenratio begin to change drastically. Moreover, if
we interpre t the number of edges needed to sy nchronize
n nodes as a simple measure of “cost”, adding many con-
nections buys little extra stability beyond the small-wor ld
regime. Finally, it can be shown [22] that the genera l
trends of the eigenratio in Fig. 3 (namely, “hyperbolic”
dependence for f small, and near independence for f
large) can be predicted with the naive perturba tion result
that both θ
1
and θ
max
change linearly with the number
of added connections.
We have also compared the synchronization cost (in
edges) for SW systems and regular (constructive) lat-
tices (Fig. 4). For x-R¨ossler systems in a k-cycle, this
cost tends to f = 2k
min
/(n − 1) ≃ 0.140. Other con-
structive lattices [22, 24] also tend to constant fractions:
f = 0.252 (for k-wheels), f = 0.070 (for k-M¨obius lad-
ders), and f = 0.053 (for the most economical bipartite
graph). In all those cases, the cost of synchronization
is high: the necessary number of edges scales like ∼ n
2
,
just like the complete graph. At the other end of deter-
ministic graphs lies the quasi-optimal (though virtually
unrealizable) hypercube, which is always synchronizable
with a number of edges f ∼ log
2
n/n. This behavior is
similar to that of random graphs: from Eq. (10) almost
sure synchronization of G
n,f
is asymptotically achieved
when f ∼ ln n/n. Remarkably, Fig. 4 shows that the cost
of synchronizing small-k SW networks is low, i.e., com-
parable to that of random graphs and hypercubes [25].
These results hint at resea rch that could deepen the
connections between topology and dynamics o n net-
works. With a view to improved design, the dy-
namic eigenratio criterion can be related to other graph-
theoretical properties (e.g., connectivity, diameter , and
convergence of Mar kov chains) [26]. Moreover, other
measures of cost (e.g., robustness under edge deletion)
should be considered as possible design constraints. Fi-
nally, recent results could lead to extensions of this work
to incorporate more general co nce pts of stability [27], and
broader definitions of small-world la ttices [19].
We thank Steve Strogatz for his deep and insightful
involvement in this work, and Mark Newman for sharing
computer code and unpublished results.
∗
Present address: Dept. of Bioengineering, I mperial Col-
lege, London SW7 2BX, UK.
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