On the Permanence of Vertices in Network Communities

Despite the prevalence of community detection algorithms, relatively less work has been done on understanding whether a network is indeed modular and how resilient the community structure is under perturbations. To address this issue, we propose a new vertex-based metric called "permanence", that can quantitatively give an estimate of the community-like structure of the network. The central idea of permanence is based on the observation that the strength of membership of a vertex to a community depends upon the following two factors: (i) the distribution of external connectivity of the vertex to individual communities and not the total external connectivity, and (ii) the strength of its internal connectivity and not just the total internal edges. In this paper, we demonstrate that compared to other metrics, permanence provides (i) a more accurate estimate of a derived community structure to the ground-truth community and (ii) is more sensitive to perturbations in the network. As a by-product of this study, we have also developed a community detection algorithm based on maximizing permanence. For a modular network structure, the results of our algorithm match well with ground-truth communities.