Abstract.A common analysis performed on dynamic networks is community structure detection, a challenging problem that aims to track the temporal evolution of network modules. An emerging area in this field is evolutionary clustering, where the community structure of a network snapshot is identified by taking into account both its current state as well as previous time points. Based on this concept, we have developed a mixed integer non-linear programming (MINLP) model, SeqMod, that sequentially clusters each snapshot of a dynamic network. The modularity metric is used to determine the quality of community structure of the current snapshot and the historical cost is accounted for by optimising the number of node pairs co-clustered at the previous time point that remain so in the current snapshot partition. Our method is tested on social networks of interactions among high school students, college students and members of the Brazilian Congress. We show that, for an adequate parameter setting, our algorithm detects the classes that these students belong more accurately than partitioning each time step individually or by partitioning the aggregated snapshots. Our method also detects drastic discontinuities in interaction patterns across network snapshots. Finally, we present comparative results with similar community detection methods for time-dependent networks from the literature. Overall, we illustrate the applicability of mathematical programming as a flexible, adaptable and systematic approach for these community detection problems.

Abstract:Networks constitute powerful means of representing various types of complex systems, where nodes denote the system entities and edges express the interactions between the entities. An important topological property in complex networks is community structure, where the density of edges within subgraphs is much higher than across different subgraphs. Each of these subgraphs forms a community (or module). In literature, a metric called modularity is defined that measures the quality of a partition of nodes into different mutually exclusive communities. One means of deriving community structure is modularity maximisation. In this paper, a novel mathematical programming-based model, DiMod, is proposed that tackles the problem of maximising modularity for directed networks.