Partial-Overlap Community Matching in Product Manifold Spaces

Abstract

The problem of matching communities across disparate networks is central to multiple domains, from biological module alignment to social network analysis. However, real-world networks rarely exhibit perfect isomorphisms or one-to-one community mappings. Instead, they feature partial-overlap communities where only a subset of nodes and structural motifs are shared. Traditional Euclidean embedding methods often fail to capture the complex, heterogeneous topologies of these networks, leading to sub-optimal alignments. This paper introduces a novel framework for partial-overlap community matching by embedding networks into product manifold spaces, which combine the geometric properties of Euclidean, hyperbolic, and spherical manifolds. By leveraging the distinct capacity of hyperbolic spaces to model hierarchical structures, spherical spaces to model cyclical topologies, and Euclidean spaces for regular grids, our approach preserves the intrinsic geometry of multi-scale communities. We propose a purely geometric alignment mechanism that maps these heterogeneous representations into a shared latent space without relying on exact structural equivalence. Through extensive analysis on both synthetic and real-world multiplex networks, the proposed product manifold approach demonstrates significant improvements in identifying partially overlapping communities compared to homogeneous space models. The results indicate a robust tolerance to topological noise and missing data, providing a scalable foundation for cross-network analysis in highly irregular domains.