Local Limits of Small World Networks

TL;DR

This paper studies the local structure of small-world networks like Watts-Strogatz and Kleinberg models, showing how local convergence explains global properties and phase transitions in network behavior.

Contribution

It applies local convergence theory to small-world models, linking local structures to global network measures and phase transitions in decentralized search.

Findings

Network measures converge to limits determined by local structure.

Global phenomena like giant component size can be estimated from local neighborhoods.

Identifies a critical threshold where the behavior of Kleinberg's model changes.

Abstract

Small-world networks, known for high local clustering and short path lengths, are a fundamental structure in many real-world systems, including social, biological, and technological networks. We apply the theory of (marked) local convergence (also known as Benjamini-Schramm convergence) to derive the limiting behavior of the local structures for two commonly studied small-world network models: the Watts-Strogatz and the Kleinberg models. Establishing local convergence enables us to show that key network measures, such as clustering coefficient, PageRank, greedy maximal independent set, number of spanning trees and tree entropy, 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 the size of the giant component under bond percolation and the closely related properties, the size of the epidemic and information cascades, using local information from small neighborhoods. Furthermore, 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.