Dynamics-Driven Evolution to Structural Heterogeneity in Complex Networks
aa r X i v : . [ phy s i c s . s o c - ph ] M a r Dynamics-Driven Evolution to Structural Heterogeneity in Complex Networks
Zhen Shao and Haijun Zhou
Institute of Theoretical Physics, Chinese Academy of Sciences, P. O. Box 2735, Beijing 100190, China (Dated: October 29, 2018)The mutual influence of dynamics and structure is a central issue in complex systems. In thispaper we study by simulation slow evolution of network under the feedback of a local-majority-ruleopinion process. If performance-enhancing local mutations have higher chances of getting integratedinto its structure, the system can evolve into a highly heterogeneous small-world with a global hub(whose connectivity is proportional to the network size), strong local connection correlations andpower law-like degree distribution. Networks with better dynamical performance are achieved ifstructural evolution occurs much slower than the network dynamics. Structural heterogeneity ofmany biological and social dynamical systems may also be driven by various dynamics-structurecoupling mechanisms.
PACS numbers: 89.75.Fb, 87.23.Kg, 05.65.+b
I. INTRODUCTION
The underlying networks of many biological and social complex systems are distinguished from purely randomgraphs. These real-world networks often have the small-world property [1] and scale-free (power-law) vertex degreeprofiles [2]; they have system-specific local structural motifs [3] and often are organized into communities [4, 5] ofdifferent connection densities. In recent years models have been proposed to understand the structural properties ofreal-world complex systems [6, 7]; among them the “rich-get-richer” mechanism of network growth by preferentialattachment [2] gained great popularity. As the connection pattern affects considerably functions of a networkedsystem, there may exist various feedback mechanisms which couple the system’s dynamical performance (efficiency,sensitivity, robustness,...) with the evolution of its structure. But the detailed interactions between dynamics andevolution are often unclear for real-world systems, and understanding complex networks from the angle of dynamics-structure interplay is still a challenging and largely unexplored research topic. Among the few theoretical works ondynamics-driven network evolutions from the physics and the computer science communities (see, e.g., [8, 9, 10, 11,12, 13, 14, 15, 16, 17, 18] and review [19]), the main focus has been on network evolutionary games for which networkdynamics and evolution occur on comparable time scales. A payoff function is defined for the system, and verticeschange their local connections to optimize gains. In many complex systems, however, the dynamical performance of anetwork is a global property which can not be predicted by only looking at the local structures. Most structural changesin such systems, on the other hand, take place locally and relatively randomly, without knowing their consequences tothe system’s dynamical performance. The time-scales of network dynamics and network structural evolution can alsobe very different. Will dynamics-structure coupling mechanisms build highly nontrivial architectures out of random,blind, and local structural mutations?In this work extensive simulations of dynamics-driven network evolution are performed on a simple model system,namely the local majority-rule (LMR) opinion dynamics of complex networks. There are two main motivations for thisstudy. First, earlier analytical and simulation studies [20, 21, 22] revealed that networks with heterogeneous structuralorganizations have remarkably better LMR dynamical performances than homogeneous networks. In complementaryto these studies, we want to know, in this simple LMR dynamical system, to what extent the dynamical performance ofa network can influence the evolution trajectory of the network’s structure. Second, as LMR-like dynamical processesare frequently encountered in neural and gene-regulation networks and other biological or social systems, it is hopedthat a detailed study of dynamics–structure coupling in the model LMR system will also shed light on the structuralevolution and optimization in real-world complex systems.In the simulation, a fitness value is assigned to a network to quantitatively measure its efficiency of LMR dynamicalperformance. A slow (in comparison with the LMR dynamics) mutation-selection process is performed on a popu-lation of networks, and networks of higher fitness values are more likely to remain in the population. The networkpopulation dynamics reaches a steady state after passing through several transient stages. A steady-state networkhas high clustering coefficient [1] and strong local degree-degree correlations, and the fraction P ( k ) of vertices in thenetwork with degree k resembles a power-law distribution of P ( k ) ∝ k − γ with γ ≈
2. Interestingly a global hub ofdegree proportional to network size N spontaneously emerges in the network. These results bring new insights on theoptimized network organization for LMR dynamics. They are also consistent with the opinion that feedback mecha-nisms from dynamics to structure could be a dominant force driving complex networks into heterogeneous structures[19]. Hopefully this work will stimulate studies on the detailed interactions between dynamics and structure in morerealistic complex systems. II. DYNAMICS AND EVOLUTION
The local-majority-rule dynamics runs on a network of N vertices and M = cN/ c beingthe mean connectivity. The network’s adjacency matrix A has entries A ij = 1 if vertices i and j are connected byan edge or A ij = 0 if otherwise. Each vertex i has an opinion (spin) σ i = ± t of the LMR dynamics, every vertex of the network updates its opinion synchronouslyaccording to σ i ( t + 1) = sign[ h i ( t )], where h i ( t ) ≡ P Nj =1 A ij σ j ( t ) is the local field on vertex i (when h i ( t ) = 0 we set σ i ( t + 1) = σ i ( t )). Starting from an initial configuration ~σ (0) ≡ { σ (0) , σ (0) , . . . , σ N (0) } , the LMR process will derivethe system to a consensus state in which all the vertices share the same opinion. To measure a network’s efficiency ofperforming the LMR process, we follow [20, 22] and choose the initial opinion patterns ~σ (0) to be strongly disordered ,with P Ni =1 σ i (0) = P Ni =1 k i σ i (0) ≡ k i is the degree of vertex i ); in other words a vertex (either randomly chosen orreached by following a randomly chosen edge) has probability one-half to be in the plus-opinion state. For networkscontaining N ≥ G + and G − ) of equal size and assign spin +1 to vertices of group G + and spin − G − ; and as long as P Ni =1 k i σ i = 0, two vertices (one from G + and the other from G − ) are randomly chosenand their positions are exchanged if and only if this exchange does not cause an increase in the value of | P Ni =1 k i σ i | .For a given network G , we generate a total number Ω = 1000 of strongly disordered initial opinion patterns ~σ α (0)and, for each of them we run the LMR dynamics for one time step to reach the corresponding pattern ~σ α (1). It hasbeen shown in [20] that the characteristic relaxation time of the LMR dynamics is determined by the mean escapingvelocity of the network’s opinion pattern from the strongly disordered region. In the present work we calculate afitness parameter f for network G according to f ( G ) = 1Ω Ω X α =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X i =1 σ αi (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (1)For networks with the same size N and mean degree c , we have checked that those with higher fitness values haveshorter mean LMR consensus times [23].The fitness f ( G ) as defined by Eq. (1) can also be evaluated using completely random configurations instead ofrandom strongly disordered configurations as the initial conditions. When using random configurations as the initialconditions, we found that the main results of the present paper do not change, but the vertex-vertex correlationpatterns of the network will be slightly affected (i.e., the R value defined by Eq. (2) will be more close to zero).The network of a complex system is not fixed but changes with time. We focus on situations in which the typicaltime scale of network evolution is much longer than that of the dynamical process. In real-world systems, modificationsof network structure often occur distributedly and locally. In accordance with this, in the simulation a simple localedge-rewiring scheme as demonstrated in Fig. 1 is employed (similar simulation results obtained with other local orglobal mutation schemes will be reported elsewhere [23]). For each vertex i of the network with probability (mutationrate) µ propose the following ( i, j ) → ( i, k ) edge redirection: randomly cut one of its edges ( i, j ) to vertex j andlink this edge to vertex j ’s nearest-neighbors k . This proposal is rejected if (1) edge ( i, k ) already exists or (2) thedegree of vertex j is less than a minimal value k after cutting edge ( i, j ). We set k = 3 or 5 in the simulation. Theonly motivation of setting a cutoff k > k = 1. The local edge rewiring process reserves the number of connected components ofthe network. If initially the network is connected, it will remain to be connected.During each time step T of the evolution, the network is mutated by the above-mentioned random local scheme andthe fitness difference δf between the new and the old network is estimated by Eq. (1). For the dynamics-structurecoupling, a simple simulation rule could be to accept this mutation with probability unity if δf ≥ βδf ) if otherwise, with β controlling the strength of dynamics-structure coupling. When there is onlymutation but no selection ( β = 0), it has already been known that the steady-state network’s vertex-degree profiledecays exponentially for large degrees [24] (see also the solid lines of Fig. 2). Here we focus on the other limit of strongfitness selection ( β ≫
0) and carry out the network evolution process through a population dynamics simulation ofmutation and selection [15]: Starting with a set of P = 25 networks uniformly sampled from the ensemble of randomnetworks of size N and mean connectivity c , at each round of the evolution each parent network generates E = 3slightly mutated offsprings, resulting in an expanded population of ( E + 1) P networks; the fitness values of thesenetworks are estimated and the P ones with the highest values survive and become parents for the next generation.The population dynamics runs for many steps until the system reaches a final steady-state. We have checked that thesteady-states of the simulation are not affect with larger values of population parameters P and E [23]. kkj ii j FIG. 1: Local edge redirection process. The edge rewiring ( i, j ) → ( i, k ) is accepted only if before the rewiring vertex i and k is not connected and that vertex j has vertex-degree higher than k . III. RESULTS
Figure 2 shows the vertex-degree distribution of a network with size N = 2000 and mean degree c (= 20 or 10) atevolution time T = 2 × under mutation rate µ = 0 .
01. One remarkable feature is that the steady-state networkhas a global hub whose connectivity is proportional to N . This global hub samples the opinions of a finite populationof the network (especially those of the low-degree vertices, see below) and serves as a global indicator of the system’sstate; its emergence is completely due to the dynamics-structure coupling. To some extent this global hub balancesthe influences of the minority high-degree vertices (see below) and those of the majority low-degree vertices. Such aglobal hub may correspond to news agencies and public medias in modern societies and to global transcription factorsin biological cells.At mutation rate µ = 0 .
01 the mean clustering coefficient [1] of the steady-state networks is about 0 . ± . N = 2000 and c = 10), considerably larger than the mean value of 0 . ± . . ± . R defined by R = k gnn − h k gnn i rand h k gnn i rand − k , (2)where k gnn is the mean degree of nearest-neighbors of the global hub and h k gnn i rand is the averaged value of this meandegree over an ensemble of randomly shuffled networks. The steady-state value is R ≈ − . N = 1000, c = 10and µ = 0 .
01 (Fig. 3). By this preference the ‘voices’ of low-degree vertices have a larger chance to be heard by thewhole system. Second, there are strong positive degree-degree correlations among vertices of a steady-state network(excluding the global hub). To measure the extent of these correlations, we calculate the assortative-mixing index r of the global hub-removed subnetwork following Ref. [25]. A steady-state assortative-mixing index of r ≈ . µ = 0 .
01 (Fig. 3) suggests that in the subnetwork high-degree vertices (except the global hub) are morelikely than random to connect with other high-degree vertices.The steady-state vertex degree distributions for networks under dynamics-structure coupling deviate remarkablyfrom those of the networks under only mutation (see Fig. 2). Besides the emergence of a global hub, the stead-statevertex degree distribution resembles a power-law form of P ( k ) = Ck − γ , k ≥ k , (3)where C is a normailzation constant. For the data-set of Fig. 2 with mean degree c = 20 and minimal degree k = 3,the fitting reports a decay exponent of γ ≃ .
92, while for the data-set with of c = 10 and k = 5 the fitting gives γ ≃ .
24. Scale-free networks with decay exponent γ < . N [20, 22]. Thiswork indicates that such heterogeneous optimal network structures might be reachable without employing any centralplanning and any intelligence. The system only needs to accumulate decentralized and local structural changes underthe selection of dynamics-structure coupling. Networks with pronounced power-law degree distributions also emergedin other model systems with comparable dynamicl and evolutionary time scales [16, 17]. In real-world systems, it was Vertex Degree -2 P r o b a b ili t y k_0 = 3, c = 1010 -2 k_0 = 5, c = 109651136 FIG. 2: (Color Online) The degree distribution for a steady-state network (under dynamics-structure coupling) of N = 2000and c = 20, k = 3 (down triangles) and c = 10, k = 5 (up triangles). The dashed lines are the best power-law fit of the pointswith γ = 1 . ± .
02 (main panel) and γ = 2 . ± .
04 (inset). The solid lines are the corresponding degree distributions (asaveraged over 200 samples) for steady-steady networks under only mutation but no selection. The mutation rate µ = 0 . noticed by Aldana [26] that a major fraction of scale-free complex networks has their decay exponent γ in the tinyrange of γ ∈ [2 . , . N = 1000 and mean degree c = 10 .
0, Fig. 3 shows that the evolution can be divided into fourstages. In the first ‘dormant’ stage which lasts for about 5000 evolution steps for muatation rate µ = 0 .
01, the degreedistribution of the networks changes gradually into the form shown by the solid line in the inset of Fig. 2. The fitnessvalues of the networks are small and increase only very slowly, the maximal vertex degrees of the networks are alsosmall, and the degree-degree correlations in the network are weak. This dormant stage is followed by a ‘reforming’stage which lasts for about 10 ,
000 steps for µ = 0 .
01. A global hub emerges and its degree rapidly exceeds those of allthe other vertices of the network, the subnetwork assortative index r also increases rapidly, and the degree distributionof the network becomes power law-like at the end of this stage. This reforming stage has rapid increase in the meanfitness value; it is follows by a long ‘structural fine-tunning’ stage (lasts for about 20 ,
000 steps at µ = 0 .
01) of slowincrease in network fitness, maximal degree, and assortative mixing. Finally the network reaches the steady-state inwhich the network’s fitness value saturates but its local structures are being modified continuously.We have performed simulations with different initial conditions and confirmed that the steady-states are not affected[23]. For example, Fig. 3 demonstrates that the steady-state networks obtained from two different initial conditionsshare the same dynamical performances and the same structural properties. (If the network initially has a globalhub of degree N − µ . For the same network size r Evolution Time (x 10,000) µ = 0.01 µ = 0.05 FIG. 3: (Color Online) The evolution of the mean fitness value, the mean maximal vertex-degree, the correlation index R [Eq. (2)], and the assortative-mixing index r (of the global hub-removed subnetwork) as a function of simulation steps. Thenetwork size is N = 1000, mean vertex degree is c = 10 .
0, and minimal degree is k = 5. The mutation rate is µ = 0 .
01 (blackand red curves, maximal evolution time to 6 × steps) or µ = 0 .
05 (blue and green curves, maximal evolution time to 4 × steps). The population dynamics starts from an ensemble of random Poissonian networks (red and green curves, which initiallyare the two lower curves) or an ensemble of modified random Poissonian networks with a single vertex of degree N − N and mean connectivity c , the steady-state networks obtained with a lower network mutation rate µ have betterdynamical performances (Fig. 3). As the network topology becomes heterogeneous, most local structural changes willtend to deteriorate the dynamical performance. When the mutation rate is relatively large, in one evolution stepthe probability for the combination of L = N µ local and distributed mutations to enhance the network’s dynamicalperformance will decrease rapidly with L . The balance between structural entropy (randomness) and dynamicalperformance then makes the system cease to be further optimized. For the dynamics-structure interaction to workmost efficiently, it is therefore desirable that the time scale of network evolution be much slower than the time scaleof network dynamics. IV. CONCLUSION
In this paper, we have studied the evolution and optimization of complex networks from the perspective of dynamics-structure mutual influence. Through extensive simulation on a simple prototypical model process, the local-majority-rule dynamics, we showed that if there exist feedback mechanisms from a network’s dynamical performance to itsstructure, the network can be driven into highly heterogeneous structures with a global hub, strong local correlationsin its connection pattern, and power law-like vertex-degree distributions. The steady-state networks reached by thisdynamics-driven evolution will have better dynamical performance if network evolution occurs much slowly than thedynamical process on the network.For the LMR dynamics specifically, this work confirmed and extended previous studies [20, 22] by showing thatscale-free networks with decay exponent γ < . Acknowledgement
We thank Kang Li and Jie Zhou for helpful discussions, Erik Aurell and Peter Holme for their suggestions on themanuscript, and Zhong-Can Ou-Yang for support. We benefited from the KITPC 2008 program “Collective Dynamicsin Information Systems”. The State Key Laboratory for Scientific and Engineering Computing, CAS, Beijing is kindlyacknowledged for computational facilities. [1] D. J. Watts and S. H. Strogatz, Nature , 440 (1998).[2] A.-L. Barab´asi and R. Albert, Science , 509 (1999).[3] R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon, Science , 824 (2002).[4] E. Ravasz, A. L. Somera, D. A. Mongru, Z. N. Oltvai, and A. Barab´asi, Science , 1551 (2002).[5] M. Girvan and M. E. J. Newman, Proc. Natl. Acad. Sci. USA , 7821 (2002).[6] R. Albert and A.-L. Barab´asi, Rev. Mod. Phys. , 47 (2002).[7] S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. , 1079 (2002).[8] B. Skyrms and R. Pemantle, Proc. Natl. Acad. Sci. USA , 9340 (2000).[9] A. Fabrikant, A. Luthra, E. Maneva, C. H. Papadimitriou, and S. Shenker, in Proc. th annual Symp. on Principl. Dis-tri. Comput. (ACM, New York, 2003), pp. 347–351.[10] A. Barrat, M. Barth´elemy, and A. Vespignani, Phys. Rev. Lett. , 228701 (2004).[11] M. G. Zimmermann, V. M. Eguiluz, and M. San Miguel, Phys. Rev. E , 065102 (2004).[12] J. J. Schneider and S. Kirkpatrick, J. Stat. Mech., P08007 (2005).[13] G. C. M. A. Ehrhardt, M. Marsili, and F. Vega-Redondo, Phys. Rev. E , 036106 (2006).[14] P. Holme and M. E. J. Newman, Phys. Rev. E , 056108 (2006).[15] P. Oikonomou and P. Cluzel, Nature Phys. , 532 (2006).[16] D. Garlaschelli, A. Capocci, and G. Caldarelli, Nature Phys. , 813 (2007).[17] Y.-B. Xie, W.-X. Wang, and B.-H. Wang, Phys. Rev. E , 026111 (2007).[18] B. Kozma and A. Barrat, Phys. Rev. E , 016102 (2008).[19] T. Gross and B. Blasius, J. R. Soc. Interface , 259 (2008).[20] H. Zhou and R. Lipowsky, Proc. Natl. Acad. Sci. USA , 10052 (2005).[21] C. Castellano and R. Pastor-Satorras, J. Stat. Mech.: Theor. Exp., P05001 (2006).[22] H. Zhou and R. Lipowsky, J. Stat. Mech.: Theor. Exp., P01009 (2007).[23] Z. Shao and H. Zhou, manuscript in preparation.[24] M. Baiesi and S. S. Manna, Phys. Rev. E , 047103 (2003).[25] M. E. J. Newman, Phys. Rev. Lett. , 208701 (2002).[26] M. Aldana, Physica D185