Abstract
Summary: The Soft Happy Colouring (SHC) problem, a mathematical framework for identifying homophilic network structures, seeks to maximise the number of $ρ$-happy vertices, i.e., vertices with at least a proportion $ρ$ of neighbours that share the same colour. Because this NP-hard problem makes finding exact solutions intractable for large networks, probabilistic metaheuristics such as the Cross-Entropy (CE) method are suitable candidates. However, pure CE frequently suffers from stagnation of the probability distributions and non-convergence in high-dimensional spaces. To address this, we introduce {\sf CE+LS}, synergising CE’s adaptive learning with a fast, structure-aware local search ({\sf LS}). By restricting the search exclusively to local optima, {\sf CE+LS} learns from high-quality structural characteristics rather than raw random samples. We mathematically and empirically demonstrate that this search space reduction resolves CE’s stagnation, yielding a convergent algorithm. Evaluating {\sf CE+LS} across 28,000 Stochastic Block Model graphs, validated by non-parametric statistical testing, demonstrates that it consistently outperforms existing heuristic and memetic algorithms. Furthermore, benchmarking against the commercial exact solver, CPLEX, on real-world networks confirms that {\sf CE+LS} identifies near-optimal configurations in a fraction of the required computational time for CPLEX. Crucially, {\sf CE+LS} remains highly efficient even in the tight constraint regime, where comparative algorithms usually fail.
Dhananjay Thiruvady
Associate Professor in Mathematics and Optimisation
My research interests include …