Physica D 240 (2011) 1972–1978

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Physica D

journal homepage: www.elsevier.com/locate/physd

Clustering dynamics of nonlinear oscillator network: Application to graphcoloring problemJianshe Wu ∗, Licheng Jiao, Rui Li, Weisheng ChenThe Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, 2# Taibai South Road, Xidian University, Xi’an 710071, PR China

a r t i c l e i n f o

Article history:Received 15 February 2011Received in revised form21 July 2011Accepted 19 September 2011Available online 24 September 2011Communicated by A. Mikhailov

Keywords:ClusteringGraph coloringPhase oscillatorComplement graphKuramoto model

a b s t r a c t

The Kuramotomodel is modified by introducing a negative coupling strength, which is a generalization ofthe original one. Among the abundant dynamics, the clustering phenomenon of the modified Kuramotomodel is analyzed in detail. After clustering appears in a network of coupled oscillators, the nodes aresplit into several clusters by their phases, in which the phases difference within each cluster is lessthan a threshold and larger than a threshold between different clusters. We show that this interestingphenomenon can be applied to identify the complete sub-graphs and further applied to graph coloringproblems. Simulations on test beds of graph coloring problems have illustrated and verified the scheme.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

The clustering phenomenon has been observed in many fieldsranging from social to life sciences, for example, shoaling behaviorof fish, swarm behavior of insects, herd behavior of land animals,and dynamics of opinion formation [1], etc. In theoretical study,the clustering in the synchronized coupled oscillatorswas used as amodel for brain or heart cells [2]. Phase synchronization of coupledoscillators has been carefully studied via the Kuramoto model [3];including on a complete graph [4,5] or on a general graph whosetopology is described by the adjacentmatrix [6]. In Ref. [7], anothermodel of mutually attracting agents was presented which can beconsidered as a simplification of the Kuramoto model. Recently,it is found that phase synchronization can be used to identify thecommunity or hierarchical structure of complex networks [8,9].

Cluster synchronization of chaotic oscillators has been carefullyresearched in the past decade (see Refs. [10–18], just to name afew), which means that the oscillators evolve into several sub-sets called clusters, such that the oscillators synchronize with oneanother in each cluster, but there is no synchronization amongdifferent clusters. In the Kuramoto model [3], when the positivecoupling strengthK exceeds a thresholdKc , a small set of oscillatorswill bemutually synchronized, eventually the radius r(t) saturates

∗ Corresponding author. Tel.: +86 29 88202279; fax: +86 29 88201023.E-mail addresses: jshwu@mail.xidian.edu.cn (J. Wu), jlc1023@163.com (L. Jiao),

lirui1102dan@yahoo.com.cn (R. Li).

0167-2789/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2011.09.010

at some level r(∞) < 1 with fluctuations [4]. The radius (0 ≤

r(t) ≤ 1) is a measure of the coherence of oscillators, wherer(t) = 1 reflects complete synchronization and r(t) = 0 for nosynchronization.

In this paper, the Kuramoto model is modified by introducing anegative coupling strength (see Section 3), and then the clusteringphenomenon is observed: all the oscillators are split into severalclusters; the phases of oscillators in each cluster are almostsynchronized, whereas the phases of oscillator in different clustersare far away. This phenomenon is similar to but much differentfrom the cluster synchronization of chaotic oscillators. We willshow that a graph can be colored via the clustering phenomenonof the modified Kuramoto model.

A coloring of a graph is an assignment of a color to eachvertex, such that no two vertices with the same color are adjacentto each other. Wu has proposed a scheme to color a graphvia synchronization of coupled oscillators [19], where the phaseof oscillators were considered as the color associated with thecorresponding vertices when the coupled array is synchronized.Using the clustering phenomenon of the modified Kuramotomodel, after the phases of oscillators are split into several clusters,all the vertices in each cluster correspond to one color, and verticesin different clusters correspond to different colors.

The rest of this paper is organized as follows. Section 2 containsthe preliminaries of this paper. In Section 3, themodifiedKuramotomodel is introduced and then its clustering dynamics are analyzedin detail. We apply the clustering dynamics of the modifiedKuramoto model to the graph coloring problem in Section 4.

J. Wu et al. / Physica D 240 (2011) 1972–1978 1973

Fig. 1. A graph and its complement. (a) A graphwith 6 vertices. (b) Its complementgraph, edges are drawn by dashed lines.

In Section 5, the diameters of clusters after clustering of thenetwork are estimated, which is an in-depth analysis aboutthe clustering dynamics. Section 6 has the simulations aboutgraph coloring problems. The conclusion of this paper is given inSection 7.

2. Preliminaries

2.1. Graph coloring problem

The graph coloring problem is defined as coloring the verticesof a graph with the minimum number of colors, such that no twoadjacent vertices have the same color. Suppose G(V , E) denotes anundirected graph,where V = (v1, v2, . . . , vN) is the set of vertices,N the number of vertices, and E = {eij|vi and vj are connected}the set of edges. A k-coloring of a graph is a assignment of all thevertices into k sub-sets S1, S2, . . . , Sk, such that any two adjacentvertices are in different sub-sets. A graphG is said to be k-colorable,if there is a proper k-coloring. The chromatic number of a graphis the least number of colors needed to color the graph. Many realworld problems can be described and solved by graph coloring, e.g.register allocation in embedded computers, frequency allocation inwireless communication, and task scheduling, etc.

The problem of coloring a graph with the least number ofcolors is known to be NP-hard [20]. There were many works inconstructing effective algorithms which can color a graph with thenumber of colors much less than the number of vertices [21,22]. Inthis paper, a new scheme for coloring a graph is provided via theclustering phenomenon in a coupled oscillator network.

By associating vertices to the phase oscillators and edges tothe couplings, a graph can be considered as a network of phaseoscillators. As mentioned above, after clustering appeared in thenetwork, we hope all the vertices in a cluster can be assigned thesame color, and vertices in different clusters assigned to differentcolors. For this purpose, the clustering dynamics of modifiedKuramoto model should be clear.

2.2. Complement graph

The complement of a graph will be employed in the modifiedKuramoto model.

Definition 1 (Complement Graph [23]). Given a graph G(V , E), itscomplement G(V , E) is defined as follows: (1) The complementG(V , E) has the same set of vertices as G(V , E); (2) For any a pair ofvertices, if there exists an edge between them inG(V , E), then thereis no edge between them in G(V , E); or vice versa. An example ofa graph and its complement are shown in Fig. 1.

Given a graph, in which two unconnected vertices can beallocated into the same sub-set corresponding to a color; whereasin its complement, two connected vertices can be allocated intothe same sub-set corresponding to a color. All the vertices of asub-set form a complete sub-graph in the complement, whichis the reason why the complement graph is employed in themodified Kuramoto model. For the graph coloring problem, wewant to identify the complete sub-graphs in the complement bythe clustering phenomenon in coupled oscillator network.

3. Clustering dynamics of the modified Kuramoto model

3.1. The modified Kuramoto model

Various synchronization in networks of continuous or discretedynamic systems have been carefully studied (see Refs. [24–34],just to name a few), where all the nodes in the network arecompletely mutually synchronized.

The coupled phase oscillators, Kuramoto model [3], aredescribed by the set of following equations

dθidt

= ωi +KN

N−j=1

sin(θj − θi), i = 1, . . . ,N, (1)

where θi and ωi are phase variables and intrinsic frequencies ofnode i, respectively, K is the positive coupling strength and N isthe number of oscillators. The frequencies ωi are distributed ac-cording to some probability density g(ω). For simplicity, Kuramotoassumed that g(ω) is unimodal and symmetric about its mean fre-quency. In this paper, initially θi andωi are randomly anduniformlydistributed in the intervals [0, 2π) and [−0.5, 0.5] respectively incases without special specification. Simulations show that for all Kless than a certain threshold Kc , the oscillators act as if they wereuncoupled: starting from any initial condition, the phases becomeuniformly distributed in [0, 2π). For all-to-all networks, a smallcluster of oscillators will be mutually synchronized when K ex-ceeds Kc , thereby generating a collective oscillation. The criticalvalue of the coupling strength Kc in the mean-field limit is Kc =

2/πg(0) [3,4]. All the oscillators will be mutually synchronized toone cluster as K increases. Complete synchronization of the Ku-ramoto phase model has been recently studied in Ref. [5], wheresufficient conditions for initial configurations leading to the ex-ponential decay toward the completely synchronized states werepresented.

In this paper, the Kuramoto model (1) is modified to thefollowing equation.

dθidt

= ωi +K

Kmax

N−j=1

sin(θj − θi), i = 1, . . . ,N, (2)K = Kp > 0, eij ∈ E,K = Kn < 0, eij ∈ E,

where Kmax is the maximum degree of all the nodes in thecomplement graph, which can be replaced by N . Similar to theK in (1), the purpose of the positive real number Kp is to makethe phases of two connected vertices in the complement graphto evolve together; on the other hand, the negative Kn is to makethe phases of two unconnected vertices in the complement graphto evolve far away (see Section 3.2). Obviously, model (2) is ageneralization of the original Kuramoto model (1), it has abundantdynamics. If Kn = 0, Eq. (4) is equivalent to (10) in Ref. [8].

To identify the complete sub-graph in the complement, thefocus here is on the clustering phenomenon of the modifiedKuramoto model.

3.2. Clustering dynamics of the modified Kuramoto model

It is noticed that |θi−θj | ≤ π for all vi and vj, cos(−θ) = cos θ ,and cos θ is a monotonically decreasing function of θ between0 ≤ θ ≤ π . To apply to the graph coloring problems, the term‘‘clustering’’ mentioned in the following section of this paper isdefined as follows.

Definition 2 (Clustering). The phases of all the oscillators convergeto several clusters (see Fig. 2(b)), such that

cos(θi − θj) ≥ cos θth−u−r , (3)

1974 J. Wu et al. / Physica D 240 (2011) 1972–1978

Fig. 2. Clustering of the Kuramotomodel. (a) Initial state. (b) Clustering (three sub-sets).

when vi and vj exist in the same cluster Sr , and

cos(θi − θj) ≤ cos θth−l−rs, (4)

when vi and vj exist in different clusters (e.g. vi ∈ Sr , vj ∈ Ss,r = s).

As shown in Fig. 2(b), the phases of the oscillators in eachcluster are almost synchronized, whereas the phases of oscillatorsin different clusters are far away. Thus the vertices (oscillators) arespit into several sub-sets and we want each sub-set to correspondto one color. For clarification, θth−u−r is called the diameter of clusterSr , and θth−l−rs the distance between clusters Sr and Ss.

For the graph coloring problem, if two vertices (e.g. vi and vj)are in the same sub-set by clustering, then there should be an edgebetween them in the complement graph, or else, they should be indifferent sub-sets.

Simulations show that the clustering phenomenon of themodified Kuramoto model can be observed. Fig. 3 shows thesimulation results of the modified Kuramoto model on the graphshown in Fig. 1. It is shown in Fig. 3(c) and (d) that the vertices aresplit into two sub-sets: S1 = {v2, v4, v6} and S2 = {v1, v3, v5},which is a proper 2-coloring of the graph. It is shown in thesimulations that Kn hasmade an obvious improvement on splittingthe entire oscillator into sub-sets (clusters).

By Definition 2.1 in Ref. [5] about complete synchronizationof the Kuramoto model, only an all-to-all network corresponding

to a complete complement graph can realize complete phasesynchronization by the modified Kuramoto model because of Kn.

To analyze the effect of the parameter Kn in the modifiedKuramoto model, let Kp =0, then (2) is reduced to

dθidt

= ωi +Kn

Kmax

N−j=1

aij sin(θj − θi), Kn < 0, (5)

where [aij] is the adjacentmatrix of graph G(V , E), aij = 1 if eij ∈ E.For the all-to-all networks, Eq. (5) is equivalent to

dθidt

= ωi +Kn

Kmax

N−j=1

sin(θj − θi), Kn < 0. (6)

Eq. (6) is quite different from (1) in that the positive K in (1) usuallydrove the oscillators toward phase synchronization (see Fig. 4(a)),on the contrary, simulations show that the negative Kn usuallymade the phases of oscillators leave each other (see Fig. 4(b)). Fig. 4shows the simulation results of (2) on an all-to-all graph with 6oscillators, corresponding to 6 isolated nodes in the complement.To eliminate the effects introduced by the intrinsic frequencies, allthe frequencies are 0.1 in the simulations of Fig. 4.

To summarize, for an all-to-all network (corresponding toN isolated nodes in the complement graph), the phases of theoscillators will be scattered to N points in [0, 2π); at the otherextreme, for an discrete network (corresponding to an completecomplement graph), the dynamics of (2) is similar to (1) andthe phases of oscillators will be mutually synchronized to onecluster when Kp is larger than a threshold. In general, the phasesof oscillators will evolve into several clusters.

4. Coloring a graph by adaptively adjusting Kn and Kp

As shown above, Kn and Kp are the two key parameters forthe clustering dynamics of the modified Kuramoto model. Forsome graphs, e.g. those whose complement consist of severalisolated complete sub-graphs, which can always be colored viaclustering by increasing |Kn| or (and) |Kp|. And for some graphs,

Fig. 3. Clustering of the modified Kuramoto model (2) on the complement graph shown in Fig. 1(b). The initial phases of the 6 oscillators are 5.3170, 3.2996, 1.2733, 4.2232,5.2661, and 0.1234, respectively. The frequencies are −0.3530, −0.1221, 0.1490, −0.0132, 0.0982, and 0.4240, respectively. (a) Kp = 1.5, Kn = 0. (b) Kp = 8, Kn = 0.(c) Kp = 8, Kn = −1. (d) Kp = 8, Kn = −6.

J. Wu et al. / Physica D 240 (2011) 1972–1978 1975

Fig. 4. Simulations of the modified Kuramoto model (2) on 6 isolated oscillators, corresponding to an all-to-all graph, where the frequencies are all 0.1. (a) Kp = 1, Kn = 0,and the initial phases are 5, 4.1, 3.3, 2.4, 1.5, and 0.6, respectively. (b) Kp = 0, Kn = 1, and the initial phases are 2, 2.1, 2.3, 2.4, 2.5 and 2.6, respectively.

which can be split into several sub-sets in a large range of thetwo parameters. But there are many graphs, e.g. those in thegraph coloring benchmarks (see Refs. [21,22]), for which thetwo parameters should be carefully selected for coloring. Thus,adaptive adjustment of the two parameters is usually needed.

The steps of adaptive adjustment are given as follows (Kp forillustration):Step 1. Increase (or decrease) the value of Kp.Step 2. If the number of vertices that cannot be allocated by the ruleof (3) and (4) is decreased, then increase (or decrease) Kp again; orelse if the number of vertices that cannot be allocated by the ruleof (3) and (4) is increased, then decrease (or increase) Kp.Step 3. Repeating Step 2 until all the vertices are allocated or thenumber of unallocated vertices cannot be decreased any more,then adjust the other parameter.Step 4. Examining whether each sub-set be a complete sub-graphin the complement graph, if yes, the coloring process is over; orelse if no, adjust parameters θth−u and θth−l and redo from Step 1again.

Similar to the other algorithms for graph coloring, the algorithmis not sure to always find the chromatic number of an arbitrarygraph due to the fact that it is NP-hard, but only an approximateor acceptable solution of the problem. For two given parameters ofKp andKn, a solution (not limited to the solutionwith the chromaticnumber) can always be found by decreasing θth−u and θth−l, whichwill result in increasing the number of colors required. A moreoptimal solution corresponds to properly selected parameters Kpand Kn which result in a relatively smaller number of colors.

5. Estimation of the cluster diameter

After clustering appears in the network, all the oscillators aresplit into several sub-sets, and each sub-set corresponds to acomplete sub-graph in the complement graph. If the diametersof clusters are all very small and the distances between pairsof clusters are all relatively large, the clustering phenomenon isvisible. But the diameter and the distance between pairs of clustersare usually difficult to calculate by formulas. In this section, thediameters of clusters in a special case are estimated, from whichthe clustering dynamics related to the parameters is revealed. Weconsider the case that there are only several isolated complete sub-graphs in the complement graph, in other words, no edge existsbetween pairs of vertices in different clusters. In the formula, ifvi, vj ∈ Sr , r = 1, . . . , k, then eij ∈ E; or else if vi ∈ Sr , vj ∈ Ss, andr = s, then eij ∈ E.

After all the oscillators have achieved clustering, then (3) holdswhen vi and vj are in the same cluster and (4) holds when vi andvj are in different clusters. If dθi/dt are constant, i = 1, . . . ,N , onehasdθidt

=dθjdt

, i, j ∈ V ;

or else if the above equation does not hold, then both (3) and (4)will be broken. Else if dθi/dt are time varying, then (dθi/dt−dθj/dt)may fluctuate around zero slightly to ensure (3) and (4) hold. Insummary, one has

dθidt

≈dθjdt

, i, j ∈ V . (7)

Case 1. k = 1 (one cluster). From (7) and (2), one has

ωi +Kp

Kmax

N−j=1

sin(θj − θi) ≈ ωj +Kp

Kmax

N−i=1

sin(θi − θj),

then

ωj − ωi ≈Kp

Kmax

N−j=1

sin(θj − θi) −Kp

Kmax

N−i=1

sin(θi − θj)

=Kp

Kmax

N−

u=1

sin(θu − θi) −

N−u=1

sin(θu − θj)

=Kp

Kmax

N−u=1

(sin(θu − θi) − sin(θu − θj))

=2Kp

Kmaxsin

θj − θi

2

N−u=1

cos2θu − (θj + θi)

2,

θj − θi ≈ arc sin

Kmax(ωj − ωi)

2Kp

N∑u=1

cos−

θu −

θj+θi2

.

The diameter of the cluster is obtained as

θth−u = maxi,j∈V

|θj − θi|

≈ maxi,j∈V

arc sin Kmax(ωj − ωi)

2Kp

N∑u=1

cosθu −

θj+θi2

.

It is noticed that 2θu ≈ θi + θj when vu, vi, and vj are almostsynchronized in the same cluster, then cos(θu −

θj+θi2 ) ≈ 1, the

above equation can be simplified to

θth−u = maxi,j∈V

|θj − θi|

≈ maxi,j∈V

arc sinKmax(ωj − ωi)

2KpN

. (8)

Eq. (8) indicates that the diameter can always be decreased byincreasing Kp when there is only one cluster.Case 2. k = 2 (two clusters, e.g. S1 and S2). Without loss ofgenerality, we estimate the diameter of S1. From (7) and (2), one

1976 J. Wu et al. / Physica D 240 (2011) 1972–1978

has

ωi +Kp

Kmax

N−r=1, (r∈S1,i∈S1)

sin(θr − θi)

+Kn

Kmax

N−s=1, (s∈S2,i∈S1)

sin(θs − θi)

≈ ωj +Kp

Kmax

N−r=1, (r∈S1,j∈S1)

sin(θr − θj)

+Kn

Kmax

N−s=1, (s∈S2,j∈S1)

sin(θs − θj),

then

ωj − ωi ≈2Kp

Kmaxsin

θj − θi

2

N−r=1, (r,i,j∈S1)

cos

θr −θj + θi

2

+2Kn

Kmax

N−s, (i,j∈S1,s∈S2)

cos

θs −θj + θi

2

× sin

θj − θi

2

. (9)

When the complement graph consists of only two isolatedcomplete sub-graphs, simulations show that the two clusters arealways located in opposite positions (the phase difference betweentwo clusters is about π ) after achieved clustering. Then one has

N−s, (i,j∈S1,s∈S2)

cos

θs −θj + θi

2

≈ −n2,

where n2 is the number of oscillators (vertices) in the 2nd cluster(S2). The phases of oscillators in the same cluster approximatelysatisfy 2θr ≈ θi + θj, thus

N−r=1, (r,i,j∈S1)

cos2θr − (θj + θi)

2≈ n1,

where n1 is the number of oscillators (vertices) in the 1st cluster(S1). From (9) one obtains

ωj − ωi ≈2

Kmaxsin

θj − θi

2(n1Kp − n2Kn)

≈2

Kmax− (n1Kp − n2Kn) sin

θj − θi

2

,

θj − θi ≈ 2arc sin

Kmax(ωj − ωi)

2(n1Kp − n2Kn)

, i, j ∈ S1. (10)

Then the diameter of the 1st cluster (S1) is estimated as follows

θth−u−1 = maxi,j∈S1

|θj − θi|

≈ maxi,j∈S1

2arc sin Kmax(ωj − ωi)

2(n1Kp − n2Kn)

≈ max

i,j∈S1

Kmax(ωj − ωi)

n1Kp − n2Kn

. (11)

Obviously, θth−u−1 → 0 as Kp → ∞ or Kn → −∞. Similarly, thediameter of the 2nd cluster (S2) can be estimated as follows,

θth−u−2 ≈ maxi,j∈S2

Kmax(ωj − ωi)

n2Kp − n1Kn

. (12)

Similar to (11), one has θth−u−2 →0 as Kp → ∞ or Kn → −∞.Then from (11) and (12), one obtains

θth−u = max(θth−u−1, θth−u−2).

Fig. 5. A bipartite graph consisting of 14 nodes.

Table 1The diameter and distance between the two clusters of Fig. 6.

Kp Kn θth−u−1 θth−u−2 θth−l

Fig. 6(b) 3 0 0.2489 0.2200 2.5910Fig. 6(c) 6 0 0.1403 0.0696 2.7414Fig. 6(d) 3 −1 0.0814 0.2200 2.9033Fig. 6(e) 3 −3 0.0353 0.2200 2.9598Fig. 6(f) 3 −6 0.0220 0.2200 2.9722

Remark 1. From (11) and (12), increasing either |Kn| or |Kp| willcontribute to decrease the diameter of clusters in general, if theyare isolated in the complement graph. Else if there are edgesbetween pairs of vertices of different clusters (sub-sets), the twoparameters should be analyzed accordingly.

Remark 2. Even if the two clusters are isolated (no edges existbetween them) in the complement graph, the diameters of the twoclusters are mutually related to each other (e.g. the diameter of S1is related to the number of S2). Obviously, it is also the case in nclusters.

For example, see the bipartite graph shown in Fig. 5, which has14 nodes. There are two isolated clusters in its complement graph.Simulation results are shown in Fig. 6. The diameter of the clustersand the distance between the two clusters of Fig. 6 are given inTable 1. The two clusters are not separated clearly in Fig. 6(a), thusthe diameter and the distance between the two clusters are notshown in Table 1.

6. Simulations

Simulations are done on three test beds for graph coloring [21]:myciel3.col, 2-Insertions_3_col, and 2-Insertions_4.col. The graphof myciel3 has 11 vertices and 20 edges and its chromatic numberis 4. The simulation result about myciel3.col is shown in Fig. 7(a),which split all the nodes into 4 clusters: S1 = {1}, S2 = {3, 4,11}, S3 = {2, 5}, and S4 = {6, 7, 8, 9, 10}. Examination showsthat it is a proper 4-coloring of the graph.

The graph of 2-Insertions_3_col has 37 vertices and 72 edgeswith chromatic number 4. The simulation result is shown inFig. 7(b), which shows that all the nodes are split into 4 clusters:S1 = {1, 19, 20, 21, 22, 23, 24, 25, 26, 27, 37}, S2 = {2, 5, 6, 9},S3 = {10, 11, 12, 13, 14, 15, 16, 17, 18, 28, 29, 30, 31, 32, 33, 34,35, 36}, and S4 = {3, 4, 7, 8}, which is proper 4-coloring of thegraph.

The graph of 2-Insertions_4.col has 149 vertices and 541 edgeswith chromatic number 4. Since Kmax is not easy to find in thegraph, then Kmax is replaced byN = 149 in the simulation. Kp = 10and Kn = −120, the randomly generated initial phases of the os-cillators are: 1191, 5.6913, 0.7979, 5.7389, 3.9732, 0.6129, 1.7499,3.4362, 6.0162, 6.0626, 0.9903, 6.0984, 6.0141, 3.0497, 5.0283,0.8915, 2.65, 5.7537, 4.9776, 6.0287, 4.1201, 0.2244, 5.3352,

J. Wu et al. / Physica D 240 (2011) 1972–1978 1977

Fig. 6. Simulation results about the bipartite graph, Kmax = 8. (a) Kp = 1, Kn = 0. (b) Kp = 3, Kn = 0. (c) Kp = 6, Kn = 0. (d) Kp = 3, Kn = −1. (e) Kp = 3, Kn = −3.(f) Kp = 3, Kn = −6.

Fig. 7. Simulation results on myciel3.col and 2-Insertions_3_col. (a) Myciel3, Kp = 4, Kn = −16, Kmax = 8, the phases of oscillators are split into four clusters:S1 = {1}, S2 = {3, 4, 11}, S3 = {2, 5}, and S4 = {6, 7, 8, 9, 10}. (b) 2-Insertions_3_col, Kp = 20, Kn = −100, Kmax = 8, the phases of oscillators are split into fourclusters: S1 = {1, 19, 20, 21, 22, 23, 24, 25, 26, 27, 37}, S2 = {2, 5, 6, 9}, S3 = {10, 11, 12, 13, 14, 15, 16, 17, 18, 28, 29, 30, 31, 32, 33, 34, 35, 36}, and S4 = {3, 4, 7, 8}.

5.8685, 4.2646, 4.761, 4.6692, 2.4644, 4.1185, 1.0756, 4.4362,0.2, 1.74, 0.2901, 0.6103, 5.1739, 4.3657, 1.9924, 5.9704, 0.2164,2.7567, 2.3974, 4.8099, 4.9964, 1.1742, 3.0773, 2.7997, 4.0609,4.4571, 4.7418, 1.7343, 4.2707, 4.1161, 1.0217, 0.7477, 3.1313,6.0302, 2.1387, 3.6773, 1.4063, 4.7204, 1.6028, 3.179, 4.3924,5.5977, 6.0274, 3.4383, 0.871, 0.938, 1.618, 5.2824, 1.5977, 5.1163,1.5301, 5.8387, 2.199, 1.2352, 1.5776, 3.8707, 2.9738, 2.2095,5.2203, 3.6773, 3.454, 5.7629, 1.796, 4.7576, 4.7358, 2.3904,3.5677, 0.4766, 0.339, 3.3351, 4.8957, 5.8686, 0.8162, 3.574,

2.9493, 0.0748, 2.1182, 1.019, 4.9906, 1.9554, 3.3209, 1.0408,3.7824, 1.6523, 4.1097, 4.3305, 4.7008, 2.8308, 0.5267, 1.4387,5.7387, 0.9574, 5.1888, 3.3825, 6.2589, 0.4912, 2.7814, 0.6701,6.0438, 0.0291, 4.8689, 5.1353, 5.4582, 0.5305, 2.5119, 1.6328,5.027, 2.7107, 5.7218, 1.1426, 1.6575, 0.9144, 0.8549, 5.4619,3.6424, 3.4549, 0.9108, 5.3598, 3.9085, 2.2051, 3.2248, 2.5246,0.4773, 1.5074, 0.7748, and 1.1555, respectively. The randomlygenerated frequencies of the oscillators are: 0.026, 0.0083, 0.045,−0.0403, −0.0445, 0.0009, 0.0011, 0.0162, −0.04, 0.0131, 0.0389,

1978 J. Wu et al. / Physica D 240 (2011) 1972–1978

Fig. 8. Simulation result on 2-Insertions_4.col.

−0.028, 0.011, 0.0258, 0.0096, 0.0404, 0.0368, −0.0442, −0.0456,−0.0075, 0.044, 0.0265, 0.0147, −0.0321, 0.0485, 0.0457, 0.0331,−0.0149, −0.0232, −0.0148, 0.0049, −0.0047, 0.0204, −0.0245,0.0311, −0.0187, 0.0316, 0.0132, −0.0126, −0.028, 0.0419,−0.0429, −0.0276, 0.0013, 0.0064, 0.0053, 0.0194, −0.0009,−0.0011, −0.0318, −0.0295, −0.0144, 0.0121, −0.0312, −0.0033,0.0149, −0.0439, −0.0376, −0.005, −0.0122, −0.0087, 0.0292,0.0199, 0.0029, 0.027, −0.0344, 0.0305, 0.0274, 0.0329, 0.0272,0.0064, 0.0189, −0.0423, 0.007, 0.0315, −0.0405, −0.048, 0.0061,0.0389, 0.0242, 0.0091, −0.0095, 0.0238, −0.0103, −0.0211,0.0278, 0.0383, 0.0203, 0.0181, 0.0076, −0.0008, 0.0414, 0.0238,−0.0301, 0.0471, −0.0429, −0.023, 0.0011, −0.0079, 0.0263,0.0041, −0.0463, −0.0047, −0.0021, 0.0268, 0.0011, −0.0124,−0.0179, 0.0104, 0.0133, −0.0488, 0.0462, −0.0385, −0.0413,−0.0296, 0.0401, 0.0238, 0.0165,−0.018, 0.0363,−0.0221, 0.0393,−0.0154, 0.0006, −0.0279, −0.0215, −0.0404, −0.0391, 0.0166,−0.0199, 0.0302, 0.0469, −0.0244, −0, 0.002, −0.0405, −0.011,−0.0118, −0.0359, −0.0305, −0.0077, 0.0317, 0.026, −0.0387,0.0471, 0.001, 0.0332, −0.0479, and −0.0213, respectively. Thesimulation result is shown in Fig. 8. All the nodes are split into 4clusters: S1 = {38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70,71, 72, 73, 74, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122,123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136,137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148}; S2 = {1,2, 5, 7, 9, 22, 23, 24, 25, 26, 27, 37}; S3 = {2, 4, 6, 8, 19, 20, 21};S4 = {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 75, 76, 77, 78, 79, 80, 81, 82,83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100,101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 149}, which isa proper 4-coloring of the graph. The number of nodes of the fourclusters is respectively 74, 15, 7, and 56.

7. Conclusion

The modified Kuramoto model, which is a generalization ofthe original one, exhibits abundant dynamics. Among them, theclustering phenomenon is very interesting and closely related tothe negative coupling strength. Both analyses and simulationsshow that a graph can be colored via clustering of the modifiedKuramoto model. In the special case that there are only severalisolated complete sub-graphs in the complement graph, thediameters of clusters can always be decreased by increasing either|Kn| or |Kp|; whereas in the general case that there are edgesamong those complete sub-graphs in the complement graph, theclustering dynamics is very complex; how to find the optimalvalues of Kp and Kn to minimize the diameters of clusters requiresfurther theoretical studies.

Acknowledgments

We thank the anonymous reviewers for their detailed com-ments and instructive suggestions, which were very helpful inimproving the quality of this paper. This work was supported inpart by the National Natural Science Foundation (No. 61072139,61072106, 61174213) and the Program for New Century ExcellentTalents in University (NCET-10-0665).

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