Formation of Modularity in a Model of Evolving Networks

Modularity structures are common in various social and biological networks. However, its dynamical origin remains an open question. In this work, we set up a dynamical model describing the evolution of a social network. Based on the observations of real social networks, we introduced a link-creating/deleting strategy according to the local dynamics in the model. Thus the coevolution of dynamics and topology naturally determines the network properties. It is found that for a small coupling strength, the networked system cannot reach any synchronization and the network topology is homogeneous. Interestingly, when the coupling strength is large enough, the networked system spontaneously forms communities with different dynamical states. Meanwhile, the network topology becomes heterogeneous with modular structures. It is further shown that in a certain parameter regime, both the degree and the community size in the formed network follow a power-law distribution, and the networks are found to be assortative. These results are consistent with the characteristics of many empirical networks, and are helpful to understand the mechanism of formation of modularity in complex networks. Real-world complex networks usually have certain universal properties, such as small-world, scale-free, and modularity [1–4]. Scale-free means that the degree of a network follows a power-law distribution, and the modularity refers to the fact that networks typically consist of communities or clusters in which the nodes are more highly connected to each other than that to the rest of the network [1]. In the past decade, the investigation on networked systems is mainly focused on two aspects. On the one hand, the collective behaviors of networked oscillators have been extensively studied. Most of these works are carried out on static networks, aiming at revealing how network topology affects dynamics [5]. On the other hand, many evolving network models have been proposed from the topological perspective, aiming at constructing networks with power-law distribution of degree [3] and modular structure [6,7]. However, from functional perspective, taking biological networks for instance, modularity is generally believed to correspond to certain functional groups, where nodes within the same group may share similar characteristics [8]. This implies that in principle the network topology and dynamics are strongly dependent on each other. In fact, any formed network structure and dynamical pattern are actually the result of coevolution of both network dynamics and topology [9]. For example, in various biological and social networked systems, such as the email network [10] and the mobile communication network [4], individuals are more likely to interact with others “similar” to themselves [11], which usually leads to networks consisting of communities driven by shared activities, attributes, affiliations, and so on. The formation mechanism of modularity and scale-free property is crucial to the understanding of structural and functional/dynamical properties of complex networks. Recently increasing attentions have been paid to the adaptive coevolutionary networks [9,12–24]. For example, Ref. [19] investigated the interaction between link weight and dynamical states on networked phase oscillators. However, so far, it has not been well understood how modular structure and power-law degree distribution concurrently