Local Limits of Small World Networks

TL;DR

This work establishes Benjamini–Schramm local convergence for two canonical small-world models, Watts–Strogatz and Kleinberg, by deriving their local limit objects. The Watts–Strogatz limit is described by a recursive Full/Reduced k–Fuzz structure, enabling immediate convergence of local functionals such as clustering and PageRank. For Kleinberg, the limit exhibits a phase transition at the distance-decay exponent ll=2: ll e 2 yields a patch-based tree-like limit, while ll>2 yields locally bounded shortcuts with lattice-distance kernels, and both regimes admit lattice-to-graph convergence. These local limits imply that many global phenomena, including information cascades and routing efficiency, can be analyzed from finite local neighborhoods, providing a rigorous framework for understanding small-world networks at scale.

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.

Figures (6)

• Figure 1: $WS(12, \phi, 2)$ increasing randomness from left to right: for $\phi = 0$, $0 < \phi < 1$, $\phi = 1$ respectively. Colored edges correspond to shortcuts while black edges correspond to ring edges. Arrows indicate the directions of edges in the Watts-Strogatz process but the final graphs are undirected.
• Figure 2: Local limit structure of a full $k$-fuzz (for $k=2$). The outgoing shortcut from the root (shown in red) connects to a full $k$-fuzz, while the incoming shortcut connects to a reduced $k$-fuzz.
• Figure 3: $K(9, 2, 2, \ell)$ for some $\ell$. For simplicity, we plot the directed edges that only belong to the center node in the lattice and colored edges based on the type.
• Figure 4: (left) Ordered and marked $WS(12, \phi, 1)$ (right) viewing same graph as a tree. Since $WS(12, \phi, 1)$ can be seen as a tree, there is a $1$-to-$1$ correspondence between the edges and the non-root nodes. Thus, we instead ordered and marked the child of the edge in the BFS with the corresponding edge mark. The root has a special mark. Marks: root, outgoing ring edge, incoming ring edge, incoming shortcut, outgoing shortcut.
• Figure 5: Partial ordering of $K(9, 2, 1, \ell)$.

Theorems & Definitions (52)

• Definition 2.1
• Definition 2.2: Metric on marked rooted graphs $\mathscr{G}_*$
• Definition 2.3: Local Convergence in Probability
• Theorem 3.1: Local Limit of the Watts-Strogatz model
• Corollary 3.2
• Theorem 3.3: Local limit of the Kleinberg model ($\ell\leq 2$)
• Remark 3.4: Hidden random tree
• Theorem 3.5: Local limit of the Kleinberg model ($\ell > 2$)
• Corollary 4.1: Clustering Coefficient of Watts-Strogatz Model
• Corollary 4.2: Clustering Coefficient of Kleinberg Model