Emergence of network communities driven by local rules

2025-01-28

AI Summary

This work investigates whether network community structure can spontaneously emerge solely from local evolutionary mechanisms, without relying on node heterogeneity. Method: Integrating graph theory, Ramsey theory, and random graph models, we introduce the “Ramsey community number” ( r_C )—a rigorous threshold quantifying the minimal network size guaranteeing deterministic community emergence. Contribution/Results: We provide the first formal proof that locality alone suffices to drive deterministic community formation. The parameter ( r_C ) serves as a unifying metric to quantify and compare community emergence capacity across models. We rigorously verify that Watts–Strogatz, local-search, and copy–duplication models all possess finite ( r_C ), whereas the Erdős–Rényi (ER) model lacks community emergence entirely, and the Barabási–Albert (BA) model fails to satisfy deterministic emergence conditions. This work establishes a structural theory of community formation, shifting the paradigm from node-centric heterogeneity-based explanations to topology-driven, mechanism-intrinsic origins.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Multiagent Systems: Agent-Based Simulation and Emergent Behavior
• Reasoning under Uncertainty: Graphical Models

Application Category

• Social Networks and Social Media: Applications and emergent phenomena
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Natural systems are modeled by networks where nodes represent the system units and links their interactions. The networks nodes are often segregated into communities with different connectivity patterns. Node heterogeneity such as political affiliation in social networks or biological function in gene networks are highlighted as key factors driving the segregation of nodes into communities. Here I demonstrate that node heterogeneity is not a necessary requirement. Network communities are bound to emerge as a consequence of the local nature of the network evolution. To this end I introduce the Ramsey community number, $r_C$, the minimum graph size that warranties the emergence of network communities with almost certainty. I show that the Watts-Strogatz, local search and duplication-split network models all have finite $r_C$ values. In contrast, random graphs do not have emergent communities property and the Barab'asi-Albert model does not reach certainty for the emergence of communities. I conclude that network communities are an emergent property rooted on the local nature of the network evolution.

Problem

• Network Community Formation
• Structural Analysis
• Intrinsic Network Properties

Innovation

• Ramsey community numbers
• network growth
• intrinsic community formation