Constrained Fuzzy Community Detection by a New Modularity Optimization Framework

Fuzzy community detection can not only reveal crisp community structures of real-life networks, but also provide fine-grained information of node fuzzy membership. However, existing methods for detecting fuzzy partitions often produce inaccurate fuzzy memberships for bridge nodes that do not align with the prior ground-truth crisp/hard partition, which diminishes the reliability and usefulness of fine-grained community information. To solve this issue, we propose a concept of constrained fuzzy partition and the corresponding constrained fuzzy community detection problem. This problem is formulated as a constrained fuzzy modularity maximization task to approximate the modularity-optimal constrained fuzzy partition, in which the fuzzy memberships can accurately quantify node membership to ground-truth communities and association strength between communities. To improve modularity quality and computational efficiency, a general framework of Constrained Fuzzy Modularity Maximization (CFMM) is proposed, which includes two key components: Fuzzy Membership Constraint (FMC) strategy and Node Dimension Reduction (NDR) operation. FMC constrains fuzzy membership of each node to be consistent with its ground-truth hard community and improves the quality of fuzzy partitions, and NDR reduces the time complexity of modularity evaluation by removing converged fuzzy membership of non-bridge nodes. Experimental results on synthetic and real-life networks prove the effectiveness of CFMM on state-of-the-art algorithms in terms of accuracy, quality, and efficiency, providing more precise and high-quality fuzzy community information.