An Intelligent Hybrid Cross-Entropy System for Maximising Network Homophily via Soft Happy Colouring

Abstract

The Soft Happy Colouring (SHC) problem serves as a rigorous mathematical framework for identifying homophilic structures in complex networks. The SHC seeks to maximise the number of $ρ$-happy vertices, which are those vertices that the proportion of their neighbours sharing colour with them is at least $ρ$. The problem is NP-hard, making optimal solutions computationally intractable for large-scale networks. Consequently, metaheuristic approaches are useful, yet existing methods often struggle with premature convergence. Based on the problem's solution structure and the characteristics of the feasible region, an effective solution method needs to navigate efficiently among promising solutions while utilising information learned from less favourable ones. The Cross-Entropy method is suitable for this because it has a smoothing mechanism that adaptively balances exploration and exploitation, informed by the knowledge accumulated during the search process. This paper introduces a novel intelligent hybrid algorithm, CE+LS, which synergises the adaptive probabilistic learning of the Cross-Entropy method with a fast, structure-aware local search (LS) mechanism. We conduct a comprehensive experimental evaluation on an extensive dataset of 28,000 randomly generated graphs using the Stochastic Block Model as the ground-truth benchmark. Test results demonstrate that CE+LS consistently outperforms existing heuristic and memetic algorithms in homophily maximisation, exhibiting superior scalability and solution quality. Notably, the proposed algorithm remains efficient even in the tight regime, which is the most challenging category of problem instances where comparative algorithms fail to yield effective solutions.

Figures (7)

• Figure 1: System Architecture of the CE+LS Hybrid Algorithm. The diagram illustrates the synergy between the Global Exploration (Adaptive Sampling) of the Cross-Entropy method and the Local Exploitation provided by the Local Search improver.
• Figure 2: Average ratios of $\rho$-happy vertices in the output of the tested algorithms when no condition is imposed on $\rho$.
• Figure 3: Histogram of ratios of $\rho$-happy vertices ($\alpha (\sigma)$) of colouring outputs ($\sigma$) of the tested algorithms. For each of the six diagrams, the dotted vertical line represents the mean value that is also reported in Figure \ref{['fig:ce-happy-bar']}. The number of bins for demonstrating histograms is 100.
• Figure 4: Average ratios of $\rho$-happy vertices in the output of the tested algorithms when $\rho<\mu$, $\mu\le\rho\le \tilde{\xi}$, or $\rho >\tilde{\xi}$.
• Figure 5: Comparison of the tested algorithms for their average ratios of $\rho$-happy vertices considering (a) the number of vertices $n$, (b) the proportion of happiness $\rho$, and (c) the number of colours $k$.