The Fiedler connection to the parametrized modularity optimization for community detection

Abstract

This paper presents a comprehensive analysis of the generalized spectral structure of the modularity matrix $B$, which is introduced by Newman as the kernel matrix for the quadratic-form expression of the modularity function $Q$ used for community detection. The analysis is then seamlessly extended to the resolution-parametrized modularity matrix $B(\gamma)$, where $\gamma$ denotes the resolution parameter. The modularity spectral analysis provides fresh and profound insights into the $\gamma$-dynamics within the framework of modularity maximization for community detection. It provides the first algebraic explanation of the resolution limit at any specific $\gamma$ value. Among the significant findings and implications, the analysis reveals that (1) the maxima of the quadratic function with $B(\gamma)$ as the kernel matrix always reside in the Fiedler space of the normalized graph Laplacian $L$ or the null space of $L$, or their combination, and (2) the Fiedler value of the graph Laplacian $L$ marks the critical $\gamma$ value in the transition of candidate community configuration states between graph division and aggregation. Additionally, this paper introduces and identifies the Fiedler pseudo-set (FPS) as the de facto critical region for the state transition. This work is expected to have an immediate and long-term impact on improvements in algorithms for modularity maximization and on model transformations.