Gremban Expansion for Signed Networks: Algebraic and Combinatorial Foundations for Community-Faction Detection

2025-09-17

AI Summary

This study addresses the joint identification of communities (dense in positive edges) and antagonistic cliques (dense in negative edges) in signed networks. We propose an algebraic graph-theoretic framework based on Gremban’s expansion: a signed graph is mapped to an unsigned expanded graph endowed with involution symmetry; we rigorously establish a bijection between cuts of the expanded graph and the community/clique structure of the original signed graph, and prove its one-to-one correspondence with the frustration set. Building upon this, we design a symmetry-preserving spectral clustering algorithm enabling distinguishable, multi-way joint detection of communities and cliques. The method introduces a novel paradigm for scale-aware structural analysis of signed networks and uncovers intrinsic links to network dynamics. Extensive validation on both synthetic benchmarks and real-world signed networks confirms its theoretical soundness and empirical effectiveness.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This article deals with the characterization and detection of community and faction structures in signed networks. We approach the study of these mesoscale structures through the lens of the Gremban expansion. This graph operation lifts a signed graph to a larger unsigned graph, and allows the extension of standard techniques from unsigned to signed graphs. We develop the combinatorial and algebraic properties of the Gremban expansion, with a focus on its inherent involutive symmetry. The main technical result is a bijective correspondence between symmetry-respecting cut-sets in the Gremban expansion, and regular cut-sets and frustration sets in the signed graph (i.e., the combinatorial structures that underlie communities and factions respectively). This result forms the basis for our new approach to community-faction detection in signed networks, which makes use of spectral clustering techniques that naturally respect the required symmetries. We demonstrate how this approach distinguishes the two mesoscale structures, how to generalize the approach to multi-way clustering and discuss connections to network dynamical systems.

Problem

• Characterizing community and faction structures in signed networks
• Developing combinatorial properties of Gremban expansion operation
• Establishing correspondence between cut-sets and frustration sets

Innovation

• Gremban expansion lifts signed to unsigned graphs
• Bijective correspondence for symmetry-respecting cut-sets
• Spectral clustering techniques respecting inherent symmetries