Permanence and Community Structure in Complex Networks

The goal of community detection algorithms is to identify densely connected units within large networks. An implicit assumption is that all the constituent nodes belong equally to their associated community. However, some nodes are more important in the community than others. To date, efforts have been primarily made to identify communities as a whole, rather than understanding to what extent an individual node belongs to its community. Therefore, most metrics for evaluating communities, for example modularity, are global. These metrics produce a score for each community, not for each individual node. In this article, we argue that the belongingness of nodes in a community is not uniform. We quantify the degree of belongingness of a vertex within a community by a new vertex-based metric called permanence. The central idea of permanence is based on the observation that the strength of membership of a vertex to a community depends upon two factors (i) the extent of connections of the vertex within its community versus outside its community, and (ii) how tightly the vertex is connected internally. We present the formulation of permanence based on these two quantities. We demonstrate that compared to other existing metrics (such as modularity, conductance, and cut-ratio), the change in permanence is more commensurate to the level of perturbation in ground-truth communities. We discuss how permanence can help us understand and utilize the structure and evolution of communities by demonstrating that it can be used to -- (i) measure the persistence of a vertex in a community, (ii) design strategies to strengthen the community structure, (iii) explore the core-periphery structure within a community, and (iv) select suitable initiators for message spreading. We further show that permanence is an excellent metric for identifying communities. We demonstrate that the process of maximizing permanence (abbreviated as MaxPerm) produces meaningful communities that concur with the ground-truth community structure of the networks more accurately than eight other popular community detection algorithms. Finally, we provide mathematical proofs to demonstrate the correctness of finding communities by maximizing permanence. In particular, we show that the communities obtained by this method are (i) less affected by the changes in vertex ordering, and (ii) more resilient to resolution limit, degeneracy of solutions, and asymptotic growth of values.