Fixed point theory for community detection in complex networks: contraction mapping and convergence in \(b\)-metric spaces

Abstract

Message-passing algorithms, such as the Label Propagation Algorithm (LPA), are popular due to their computational efficiency but suffer from fundamental drawbacks: the lack of a formal convergence guarantee and susceptibility to oscillations (limit cycles) in ambiguous or noisy graph structures. This study proposes a new theoretical framework utilizing Fixed Point Theory for Community Detection (FP-CD) by formulating the clustering process as the search for a unique fixed point of a continuous update operator \(T\) within a complete \(b\)-metric space. Through a mathematical proof based on the Banach fixed point theorem—adapted for \(b\)-metric spaces—the designed operator is shown to satisfy the contraction mapping condition with a contraction factor of \(k\cdot s<1\), thereby theoretically guaranteeing the existence of a unique, oscillation-free fixed-point solution. An empirical evaluation was conducted using the synthetic LFR Benchmark dataset and four real-world datasets. Python-based simulation results confirm that FP-CD consistently outperforms baseline methods, achieving Normalized Mutual Information (NMI) values of up to \(1.000\) on real-world networks and maintaining an NMI of \(0.865\) under high-noise conditions (\(\mu=0.5\)). Furthermore, convergence tests confirm that the residual \(R(t)\) decays exponentially toward zero (\(R(t)<10^{-5}\)) in fewer than 20 iterations. The framework proves to be efficient and linearly scalable—\(\mathcal{O}(|E|)\)—for application to large-scale complex networks.