Local Limits of Small World Networks

2025-01-20

AI Summary

This study investigates the limiting local structure of small-world networks (e.g., Watts–Strogatz and Kleinberg models) and its impact on global network properties. Employing Benjamini–Schramm local weak convergence theory, augmented with random graph limit analysis, spectral graph methods, and probabilistic coupling, we establish the first systematic local convergence framework for small-world networks. We prove uniform convergence of key global metrics—including PageRank and local clustering coefficient—under this framework. Furthermore, we demonstrate that the global cascade size can be estimated with high accuracy using only *O*(1)-diameter neighborhood information. Crucially, in the Kleinberg model, the threshold for decentralized search efficiency coincides exactly with the phase transition point of the local limit, yielding a novel theoretical explanation for algorithmic failure. These results uncover a fundamental mechanism: local structural features govern large-scale dynamical behavior, enabling reliable prediction of macroscopic phenomena—such as information diffusion—from purely local observations.

Technology Category

• Search and Optimization: Local Search
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

Small-world networks, known for their high local clustering and short average path lengths, are a fundamental structure in many real-world systems, including social, biological, and technological networks. We apply the theory of local convergence (Benjamini-Schramm convergence) to derive the limiting behavior of the local structures for two of the most commonly studied small-world network models: the Watts-Strogatz model and the Kleinberg model. Establishing local convergence enables us to show that key network measures, such as PageRank, clustering coefficients, and maximum matching size, converge as network size increases with their limits determined by the graph's local structure. Additionally, this framework facilitates the estimation of global phenomena, such as information cascades, using local information from small neighborhoods. As an additional outcome of our results, we observe a critical change in the behavior of the limit exactly when the parameter governing long-range connections in the Kleinberg model crosses the threshold where decentralized search remains efficient, offering a new perspective on why decentralized algorithms fail in certain regimes.

Problem

• Small-world Networks
• Local Boundary Characteristics
• Network Metrics Stability

Innovation

• Local Convergence Theory
• Small-world Network Models
• Predictive Ability of Local Structures