Edge ratio and community structure in networks

Finding communities in networks in the strong and almost-strong sense

Finding communities, or clusters or modules, in networks can be done by optimizing an objective function defined globally and/or by specifying conditions which must be satisfied by all communities. Radicchi et al. [Proc. Natl. Acad. Sci. USA 101, 2658 (2004)] define a susbset of vertices of a network to be a community in the strong sense if each vertex of that subset has a larger inner degree than its outer degree. A partition in the strong sense has only strong communities. In this paper we first define an enumerative algorithm to list all partitions in the strong sense of a network of moderate size. The results of this algorithm are given for the Zachary karate club data set, which is solved by hand, as well as for several well-known real-world problems of the literature. Moreover, this algorithm is slightly modified in order to apply it to larger networks, keeping only partitions with the largest number of communities. It is shown that some of the partitions obtained are informative, although they often have only a few communities, while they fail to give any information in other cases having only one community. It appears that degree 2 vertices play a big role in forcing large inhomogeneous communities. Therefore, a weakening of the strong condition is proposed and explored: we define a partition in the almost-strong sense by substituting a nonstrict inequality to a strict one in the definition of strong community for all vertices of degree 2. Results, for the same set of problems as before, then give partitions with a larger number of communities and are more informative.

Finding communities in weighted networks through synchronization

Community detection in weighted networks is an important challenge. In this paper, we introduce a local weight ratio scheme for identifying the community structures of weighted networks within the context of the Kuramoto model by taking into account weights of links. The proposed scheme takes full advantage of the information of the link density among vertices and the closeness of relations between each vertex and its neighbors. By means of this scheme, we explore the connection between community structures and dynamic time scales of synchronization. Moreover, we can also unravel the hierarchical structures of weighted networks with a well-defined connectivity pattern by the synchronization process. The performance of the proposed method is evaluated on both computer-generated benchmark graphs and real-world networks.

Module detection in complex networks using integer optimisation

BACKGROUND

The detection of modules or community structure is widely used to reveal the underlying properties of complex networks in biology, as well as physical and social sciences. Since the adoption of modularity as a measure of network topological properties, several methodologies for the discovery of community structure based on modularity maximisation have been developed. However, satisfactory partitions of large graphs with modest computational resources are particularly challenging due to the NP-hard nature of the related optimisation problem. Furthermore, it has been suggested that optimising the modularity metric can reach a resolution limit whereby the algorithm fails to detect smaller communities than a specific size in large networks.

RESULTS

We present a novel solution approach to identify community structure in large complex networks and address resolution limitations in module detection. The proposed algorithm employs modularity to express network community structure and it is based on mixed integer optimisation models. The solution procedure is extended through an iterative procedure to diminish effects that tend to agglomerate smaller modules (resolution limitations).

CONCLUSIONS

A comprehensive comparative analysis of methodologies for module detection based on modularity maximisation shows that our approach outperforms previously reported methods. Furthermore, in contrast to previous reports, we propose a strategy to handle resolution limitations in modularity maximisation. Overall, we illustrate ways to improve existing methodologies for community structure identification so as to increase its efficiency and applicability.

Column generation algorithms for exact modularity maximization in networks

Finding modules, or clusters, in networks currently attracts much attention in several domains. The most studied criterion for doing so, due to Newman and Girvan [Phys. Rev. E 69, 026113 (2004)], is modularity maximization. Many heuristics have been proposed for maximizing modularity and yield rapidly near optimal solution or sometimes optimal ones but without a guarantee of optimality. There are few exact algorithms, prominent among which is a paper by Xu [Eur. Phys. J. B 60, 231 (2007)]. Modularity maximization can also be expressed as a clique partitioning problem and the row generation algorithm of Grötschel and Wakabayashi [Math. Program. 45, 59 (1989)] applied. We propose to extend both of these algorithms using the powerful column generation methods for linear and non linear integer programming. Performance of the four resulting algorithms is compared on problems from the literature. Instances with up to 512 entities are solved exactly. Moreover, the computing time of previously solved problems are reduced substantially.

Loops and multiple edges in modularity maximization of networks

The modularity maximization model proposed by Newman and Girvan for the identification of communities in networks works for general graphs possibly with loops and multiple edges. However, the applications usually correspond to simple graphs. These graphs are compared to a null model where the degree distribution is maintained but edges are placed at random. Therefore, in this null model there will be loops and possibly multiple edges. Sharp bounds on the expected number of loops, and their impact on the modularity, are derived. Then, building upon the work of Massen and Doye, but using algebra rather than simulation, we propose modified null models associated with graphs without loops but with multiple edges, graphs with loops but without multiple edges and graphs without loops nor multiple edges. We validate our models by using the exact algorithm for clique partitioning of Grötschel and Wakabayashi.