Modularity from fluctuations in random graphs and complex networks

Abstract

The mechanisms by which modularity emerges in complex networks are not well understood but recent reports have suggested that modularity may arise from evolutionary selection. We show that finding the modularity of a network is analogous to finding the ground-state energy of a spin system. Moreover, we demonstrate that, due to fluctuations, stochastic network models give rise to modular networks. Specifically, we show both numerically and analytically that random graphs and scalefree networks have modularity. We argue that this fact must be taken into consideration to define statistically significant modularity in complex networks.Statistical, mathematical, and model-based analysis of complex networks have recently uncovered interesting unifying patterns in networks from seemingly unrelated disciplines [1][2][3][4][5]. In spite of these advances, many properties of complex networks remain elusive, a prominent one being modularity [6,7]. For example, it is a matter of common experience that social networks have communities of highly interconnected nodes that are poorly connected to nodes in other communities. Such modular structures have been reported not only in social networks [6][7][8], but also in biochemical networks [9], food webs [10], and the Internet [11]. It is widely believed that the modular structure of complex networks plays a critical role in their functionality [9]. There is therefore a clear need to develop algorithms to identify modules accurately [6,7,[11][12][13].More fundamentally, the mechanisms by which modularity emerges in complex networks are not well understood. In biological networks-both biochemical and ecological-researchers have suggested that modularity increases robustness, flexibility, and stability [9,10]. Similarly, in engineered networks, it has been suggested that modularity is effective to achieve adaptability in rapidly changing environments [14]. It may therefore seem that evolutionary pressures make networks modular, implying that any successful model of complex networks should take into account external factors that enhance modularity. Recently, however, Solé and Fernàndez have pointed out that models without any external pressure are able to give rise to modular networks [15].In this paper, we show that Erdös-Rényi (ER) random graphs, in which any pair of nodes is connected with probability p [16], have a high modularity. We show numerically and analytically that this high modularity is due to fluctuations in the establishment of links, which are magnified by the large number of ways in which a network can be partitioned into modules. Furthermore, we show that one obtains similar results when considering scale-free networks [2]. We conclude by discussing how these results should be taken into consideration to define statistically significant modularity in complex networks.Following the first quantitative definition of modularity [7,12], several groups have proposed heuristic algorithms to detect modules in complex networks. For a given partition of the nodes of ...

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