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 et al. [ 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.

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.

At the verge of the launch of the first commercial fifth generation (5G) system, trends in wireless and optical networks are proceeding toward increasingly dense deployments, supporting resilient interconnection for applications that carry higher and higher capacity and tighter latency requirements. These developments put increasing pressure on network backhaul and drive the need for a re-examination of traditional backhaul topologies. Challenges of impending networks cannot be tackled by star and ring approaches due to their lack of intrinsic survivability and resilience properties, respectively. In support of this re-examination, we propose a backhaul topology design method that formulates the topology optimization as a graph optimization problem by capturing both the objective and constraints of optimization in graph invariants. Our graph theoretic approach leverages well studied mathematical techniques to provide a more systematic alternative to traditional approaches to backhaul design. Specifically, herein, we optimize over some known graph invariants, such as maximum node degree, topology diameter, average distance, and edge betweenness, as well as over a new invariant called node Wiener impact, to achieve baseline backhaul topologies that match the needs for resilient future wireless and optical networks.