We quantify the dynamical implications of the small-world phenomenon. 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 synchronizability.
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

A. Barrat

A. Schwenk

B. Bollobás

B. Mohar

D. J. Watts

F. R. K. Chung

Louis M. Pecora

Mauricio Barahona

V. Latora

W. E. Boyce

X. Wang

Physical Review Letters

English

arXiv

Barahona, M

Pecora, LM

Spiral - Imperial College Digital Repository

arXiv:nlin/0112023v1  [nlin.CD]  17 Dec 2001Synchronization in Small–world SystemsMauricio Barahona1, ∗ and Louis M. Pecora21Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 911252Naval Research Laboratory, Code 6340, Washington, DC 20375(Dated: February 5, 2008)We quantify the dynamical implications of the small-world phenomenon. We consider the genericsynchronization of oscillator networks of arbitrary topology, and link the linear stability of thesynchronous state to an algebraic condition of the Laplacian of the graph. We show numericallythat the addition of random shortcuts produces improved network synchronizability. Further, weuse a perturbation analysis to place the synchronization threshold in relation to the boundaries ofthe small-world region. Our results also show that small-worlds synchronize as efficiently as randomgraphs 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-range shortcuts to an otherwise lo-cally connected lattice (the ”pristine world”) producesa sharp reduction of the average distance between arbi-trary nodes. The ensuing semi-random lattice was de-noted a small-world (SW) because the sudden appear-ance of short paths occurs early on, while the system isstill relatively localized. This concept has wide appeal:the SW property seems to be a quantifiable characteris-tic of many real-world structures [1, 2, 3, 4], both humangenerated (social networks, WWW, power grid), or ofbiological origin (neural and biochemical networks).A spur of ongoing research [2] has concentrated onstatic and combinatoric properties [5, 6, 7, 8, 9, 10] of atractable SW model [1, 11]. Monasson [11] considered theSW effect on the distribution of eigenvalues of the con-nectivity matrix (the graph Laplacian) which specifiesthe coupling between nodes—a relevant topic for poly-mer networks [12]. However, despite their central role inreal-world networks, there are fewer studies of dynami-cal processes taking place on SW lattices. Among those,automata epidemics simulations [13] and Web-browsingstudies [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, asintuitively expected. However, these numerical examplesare not generic, and fail to provide insight into how theSW property influences the dynamics.In this paper, we explicitly link the SW addition ofrandom shortcuts to the synchronization of networks ofcoupled dynamical systems. This is an example of dy-namics on networks—leaving aside the distinct problemof evolution of networks here. By using a generic 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 graphLaplacian. Through numerics and analysis, we quantifyα1 α2-0.8-0.40.0α-1.2λmax151050FIG. 1: Four typical master stability functions (scaled forclearer visualization) for Rössler systems: chaotic (bold) andperiodic (regular lines); with y-coupling (dashed) and x-coupling (solid lines). Here we consider the x-coupled chaoticcase (solid bold) with a negative well between (α1, α2).how the SW scheme improves the synchronizability of thepristine world, mainly as a result of the steep increase ofthe first-non-zero eigenvalue (FNZE). The synchroniza-tion threshold is found to lie in the SW region [3, 13], butdoes not coincide with its onset—it can in fact be linkedto the effective randomization that ends SW. Within thisframework, we show that the synchronization efficiencyof semi-random SW networks is higher than standard de-terministic graphs, and comparable to both fully randomand ideal constructive graphs.Consider n identical dynamical systems (placed at thenodes of a graph) that are linearly and symmetricallycoupled (as represented by the edges of the undirectedgraph) with global coupling strength σ. The topology ofthe graph can be encoded in the Laplacian matrix G, asymmetric matrix with zero row-sum and real spectrum2{θk}, 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 λmax(σ θk) < 0, ∀k. This λmax is a characteristicof the particular dynamics at the nodes but, crucially,a large class of oscillatory systems (chaotic, periodicand quasiperiodic) have master stability functions withgeneric features [18]. In particular, for several chaoticsystems λmax has a single deep well, as depicted in Fig. 1.(We remark that this analysis is quite general: it can beextended [18] to eliminate the zero row-sum constraint,and to comprise nonlinear coupling and more general syn-chronization criteria.) Stability is thus ensured by tun-ing the coupling σ to try and place the entire spectrumof transverse eigenvalues (times σ) in the deep, stableregion: σ θk ∈ (α1, α2). This leads to an algebraic con-dition for the existence of a linearly stable synchronousstate: a network is synchronizable ifθmax/θ1 < α2/α1 ≡ β, (1)where θ1 is the FNZE and θmax is the maximum eigen-value of the Laplacian G. The figure of merit (β) rangesfrom 5 to 100 for a variety of oscillators (e.g., Lorenz,Rössler, double scroll).Small-worlds are generated from a pristine world: ak-cycle of n nodes and range k, each node coupled to its2 k nearest neighbors for a total of nk edges [11]. TheLaplacian of this graph G0 is a banded circulant matrixwith non-zero elements on the main diagonal and the 2 kadjacent diagonals: G0ii = 2k and G0ij = G0ji = −1 with(i + 1) mod n ≤ j ≤ (i + k) mod n, and 1 ≤ i ≤ n.The SW scheme dopes the pristine world by adding nsedges picked at random from the n(n − 2k − 1)/2 re-maining pairs. Each new edge between nodes l and madds off-diagonal ∆Glm = ∆Gml = −1 and on-diagonal∆Gll = ∆Gmm = 1 contributions to the Laplacian, thuspreserving the null row-sum and the bidirectional cou-pling. The average number of shortcuts per node (s) isrelated to other measures of randomness (p and q) usedpreviously [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 sizeand range. For concreteness, all our numerics have beenperformed for a network of identical x-coupled Rösslerchaotic oscillators with β ≃ 37.85. Similarly to otherlocally connected networks, pristine worlds have a largeeigenratio θmax/θ1 (i.e., they are difficult to synchronize).However, as s is increased the eigenratio falls sharply un-til, at a value ssync, the condition (1) is reached (i.e., theaddition of shortcuts makes it synchronizable). The de-pendence of ssync on the network parameters {n, k} isnotably complicated. First, there appears to be an op-timal range k ≃ 4 for which the SW is most efficient.Moreover, the synchronization threshold ssync lies in thesmall-world region but does not seem to coincide with itsonset. The SW onset (sL) is defined [1, 3] by the decay of0 10 20 30 40 50 60 7010−310−210−11001010   0.5 1   Average shortcuts per node, sRange of pristine world, kn=1000 n=500 n=300 n=400 sC sL Large worldRandomSmall−worldssync sL L(s)/L(0)sC C(s)/C(0)ssync FIG. 2: Synchronizability thresholds ssync(◦) for graphs withn 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 n1/3 <k < kmin(n). For most parameters, ssync lies within the small-world region between the dashed lines (sL < s < sC) de-picted here for n = 1000, but it is distinct from the SWonset sL. Note how synchronization is achievable withoutrandom shortcuts by increasing the deterministic range up tokmin(n) (see Fig. 3). Inset: decay of the average distance L,clustering C, and eigenratio (squares) as shortcuts are addedto a pristine world of range k = 20 and n = 500. We de-fine sL and sC as the points where L and C are 75% of thepristine world value; ssync is the point where the eigenratioθmax/θ1 = β ≡ 37.85.the average graph distance [7] L(s) ≃ (n/k) f(ns), wheref(x) = arg tanh(x/√x2 + 2x)/√4(x2 + 2x). FixingL(sL)/L(0) = 3/4, we obtain sL ≃ 1.061/n, nk ≫ 1.The end of the SW region (sC) corresponds to the effec-tive graph randomization through the loss of transitiv-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(sC)/C(0) = 3/4, we obtainsC ≃ k(−1 + [(8k − 1)/(6k − 3)]−1/2)≃ 0.155 k. Thesynchronization threshold generally lies between thesetwo boundaries which scale differently with n and k:sL ≃ 1.061/n < ssync < sC ≃ 0.155 k.We can gain insight into the synchronization thresh-old through an analytical perturbation of the eigenra-tio of the SW Laplacian G = G0 + Gr. Here, G0 isthe deterministic Laplacian of the pristine world, andGr is the stochastic Laplacian for the random shortcuts:Grij = Grji = −ξij (for i+k+1 ≤ j ≤ min{n, n−k+i−1}with 1 ≤ i ≤ n − k + 1); Grij = 0 (otherwise); andGrii = −∑nj=1 Grij (for 1 ≤ i ≤ n). The ξij aren(n − 2k − 1)/2 i.i.d. Bernoulli random variables whichtake the value 1 with probability q/n ≡ 2s/(n − 2k − 1)(and the value 0 with probability 1 − q/n). The cir-culant G0 is Fourier-diagonalizable [11] with spectrumθ0j = (2k+1)−sin[(2k+1)πj/n]/ sin[πj/n], 1 ≤ j ≤ n−1,3(plus θ00 = 0 of any Laplacian). The FNZE and the max-imum eigenvalue of the unperturbed lattice are:θ01 ≃ 2π2k(k + 1)(2k + 1)/(3n2), k ≪ n (2)θ0max ≃ (2k + 1) + csc[3π/22k + 1](3)≃ (2k + 1)[1 + 2/3π], k ≫ 1, (4)where (3) follows from a continuum approximation.Following an “honest” treatment [21] with Gr as theperturbation, we treat the analytical expressions of thedoubly degenerate eigenvalues as random variables to ob-tain their expectations. We postpone the detailed calcu-lations [22] and sketch here the main results. After somestochastic calculus, the expectations of the eigenvalues ofthe SW Laplacian to second order are shown to be:Eθ(1)i ≃ q ±√3πq/4n (5)Eθ(2)i ≃2qnn∑m=1′(θ0i − θ0m)−1, (6)for q/n and k/n small. To improve the accuracy of ssync,we have also obtained an approximation to (6) for FNZE:Eθ(2)1 ≃−2qK3[9nπ2+ K2 −(75+6π2)K − 2π], (7)where K = 2k+1. Eqns. (1),(2),(4),(5), and (7) are thenused to obtain an estimate of ssync as the solution of analgebraic equation involving only n and k:θ0max + Eθ(1)max = β(θ01 + Eθ(1)1 + Eθ(2)1). (8)As shown in Fig. 2, this approximates well our numer-ics for n1/3 < k ≪ n, where the Rayleigh-Schrödingerperturbation expansion is valid.Using (8), we can obtain [22] a first order estimatefor the maximum of the synchronization threshold s∗syncin the valid range. The maximum occurs at k∗ ≃n√(1 + 2/3π)/2π2β with the asymptotic value s∗sync ≃(2 + 4/3π)(1 − 2k∗/n)/3(β − 1). Therefore, ssync <sC [2(1+2/3π)/(2√3−3)(β−1)] is bounded by the end ofthe SW region but linked to it. For k < n1/3, the eigen-value bunching (quasi-degeneracy) in the pristine latticerenders the doubly degenerate perturbation invalid. Weare developing another approximation to quantify the be-havior in this limit, but our numerics [22] indicate thatthe dependence of ssync with n is sub-logarithmic. Thisconfirms that the synchronizability is most effectively im-proved for small-range networks (Fig. 2).How efficient is the addition of random shortcuts forsynchronization? We have compared the semi-randomSW approach with purely random and purely determin-istic schemes. An example of the latter is the synchro-nization of pristine worlds through the increase of the10−210−1100100101102103Fraction of edges of complete graph,   fEigenratio,  θ max / θ 1k=1 k=2 k=4 k=6 k=10 k=14 Random graphβ  Deterministic(Pristine worlds)Small−worldFIG. 3: Eigenratio decay in a n = 100 lattice as f edges areadded following purely deterministic, semi-random (SW), andpurely 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 eigenratioof pristine worlds (k-cycles) through the deterministic addi-tion of short-range connections—for n = 100, networks withk ≥ 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 topurely random graphs (RG, Eq. (10)), which become almostsurely disconnected at f ≃ 2 lnn/(n + 2 ln n) = 0.0843 (thus,with θRG1 = 0 and unsynchronizable). The merging of the SWand RG behaviors as f → 1 is the dynamical analogue of theeffective randomization that leads to sC .range k. From (2) and (4), the eigenratio of a pristinelattice isθ0maxθ01≃ 3π + 22π3n2k(k + 1). (9)Therefore, n nodes can be synchronized in a k-cycle ifk > kmin ≃ n√(3π + 2)/2π3β. (Note in Fig. 2 the con-sistency of our analytical approximation (8): ssync = 0precisely at kmin.) For purely random graphs Gn,f [23],θRGmaxθRG1≃ nf −√2f(1 − f)n lnnnf +√2f(1 − f)n lnn, (10)where f is the number of edges measured as a fractionof the complete graph. These are compared with SWgraphs in Fig. 3.We remark on several observations regarding Fig. 3.First, the SW addition of shortcuts is more efficient thanthe deterministic addition of short-range layers. Second,the effective randomization of SW lattices with edge addi-tion translates into converging synchronization behaviorsof random and SW networks at large f . The f → 1 regionis thus robustly stable: cutting connections from the fullyconnected graph has very little effect on synchronizationstability—not until over 90% are cut (for k small) does40 1000 2000 3000 4000 500010−310−210−1100Number of nodes,   nFraction of edges of complete graph,  fk−wheels k−cyclesk−Moebius ladders Best bipartiteHypercubesRandom graphsSmall−world   (k=1,2,4)FIG. 4: “Cost” of synchronization measured as the number ofedges needed to synchronize a lattice of n chaotic Rössler sys-tems arranged in different topologies: deterministic graphs (k-wheels, k-cycles, k-Möbius 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 randomgraphs but demand less (algorithmic) storage memory.the eigenratio begin to change drastically. Moreover, ifwe interpret the number of edges needed to synchronizen nodes as a simple measure of “cost”, adding many con-nections buys little extra stability beyond the small-worldregime. Finally, it can be shown [22] that the generaltrends of the eigenratio in Fig. 3 (namely, “hyperbolic”dependence for f small, and near independence for flarge) can be predicted with the naive perturbation resultthat both θ1 and θmax change linearly with the numberof added connections.We have also compared the synchronization cost (inedges) for SW systems and regular (constructive) lat-tices (Fig. 4). For x-Rössler systems in a k-cycle, thiscost tends to f = 2kmin/(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-Möbius lad-ders), and f = 0.053 (for the most economical bipartitegraph). In all those cases, the cost of synchronizationis high: the necessary number of edges scales like ∼ n2,just like the complete graph. At the other end of deter-ministic graphs lies the quasi-optimal (though virtuallyunrealizable) hypercube, which is always synchronizablewith a number of edges f ∼ log2 n/n. This behavior issimilar to that of random graphs: from Eq. (10) almostsure synchronization of Gn,f is asymptotically achievedwhen f ∼ lnn/n. Remarkably, Fig. 4 shows that the costof synchronizing small-k SW networks is low, i.e., com-parable to that of random graphs and hypercubes [25].These results hint at research that could deepen theconnections between topology and dynamics on 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, andconvergence of Markov chains) [26]. Moreover, othermeasures 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 workto incorporate more general concepts of stability [27], andbroader definitions of small-world lattices [19].We thank Steve Strogatz for his deep and insightfulinvolvement in this work, and Mark Newman for sharingcomputer code and unpublished results.∗ Present address: Dept. of Bioengineering, Imperial Col-lege, London SW7 2BX, UK.[1] D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998).[2] S. H. Strogatz, Nature 410, 268 (2001).[3] D. J. Watts, Small Worlds (Princeton Univ. Press,Princeton, 1999).[4] H. Jeong et al., Nature 407, 651 (2000).[5] M. Barthelemy and L. Amaral, Phys. Rev. Lett. 82, 3180(1999).[6] A. 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