Local Limits of Small World Networks
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.
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