XLoCoFC: A Fast Fuzzy Community Detection Approach Based on Expandable Local Communities Through Max-Membership Degree Propagation

Uttam K. Roy, Pranab K. Muhuri, Sajib K. Biswas

Abstract: Fuzzy community detection (FCD) aims to reveal the community structure by allocating quantitative values to nodes across different communities. This article proposes a fast FCD approach called the Expandable Local Community based Fuzzy Community (XLoCoFC) detection method based on max-membership degree propagation (max-MDP) and normalized peripheral similarity index ($ \boldsymbol{n}\mathbf{P}\mathbf{S}\mathbf{I}$). Initially, nodes having comparatively higher $ \boldsymbol{n}\mathbf{P}\mathbf{S}\mathbf{I}$ values are considered as topologically dominating nodes and selected as seeds. For an initial community, called local community, seed’s $ \boldsymbol{n}\mathbf{P}\mathbf{S}\mathbf{I}$ values from the respective neighbors’ peripheries are utilized as the neighbors’ membership degrees. Then an iterative process propagates max-membership degrees from nodes to nodes, and $ \boldsymbol{n}\mathbf{P}\mathbf{S}\mathbf{I}$ values are used as factors in the propagation. In this propagation, local communities having more dominating nodes expand and others contract. The propagation process converges very quickly. Such simplicity in its design makes our proposed XLoCoFC approach to be very fast in finding community structures on large networks. Time complexity of the proposed approach is $ \boldsymbol{O}\left(\boldsymbol{n}\boldsymbol{d}^{2}\times \mathbf{lo}\mathbf{g}_{2} \boldsymbol{d}+\mathbf{k}\mathbf{l}\mathbf{q}\right)$ which is significantly less than the majority of the FCD algorithms, for whom it is either $ \boldsymbol{O}\left(\boldsymbol{n}^{2}\right)$ or more. Moreover, XLoCoFC has no dependence on any network feature. It does not require tuning of any parameter which may impact its output. To demonstrate the working of the proposed XLoCoFC approach, we conduct extensive performance analysis comparatively by executing a set of existing approaches on several popular real-life and synthetic networks with number of nodes ranging from 24 to 1134 890. Evaluation of the results considering the accuracy and quality metrics as well as a group MCDM technique clearly establishes the superiority of our approach over others.