Abstract
Abstract Community detection in temporal networks must reconcile two competing goals: reacting to genuine mesoscale changes while avoiding spurious label fluctuations caused by transient activity. This paper proposes an adaptive spectral framework that places memory in the modularity objective and smoothness in the labels. At time $ \boldsymbol{t} $, an adaptive modularity operator is formed by exponentially aggregating past modularity residuals with a decay rate $ \boldsymbol{\lambda} $ and history weight $ \boldsymbol{\beta} $, so that persistent community evidence accumulates relative to the configuration-model baseline. The partition is initialized by the leading eigenvector of the adaptive operator and refined by recursive bipartitioning with a Kernighan–Lin-style refinement step that directly optimizes a regularized modularity objective, where label persistence is rewarded only on the shared node set $ \boldsymbol{V_{t}\cap V_{t-1}} $. A matrix-free implementation with history truncation enables scalable eigensolver iterations without materializing dense modularity matrices. Experiments on dynamic stochastic block model benchmarks yield robust accuracy across evolution regimes (e.g. TNMI $ \boldsymbol{=0.92,0.87,0.76} $ for transition probabilities $ \boldsymbol{0.1,0.2,0.5} $), and real temporal networks (MIT Reality Mining and Drosophila gene co-expression) show improved temporal consistency while preserving modular structure. Overall, objective-level memory on modularity residuals, coupled with label-level regularization, provides an interpretable and effective tool for tracking evolving mesoscale organization in temporal networks.