Average Distance in a General Class of Scale-Free Networks with Underlying Geometry

TL;DR

This paper demonstrates that a broad class of scale-free networks with underlying geometry, including Chung-Lu and hyperbolic models, share similar average distances, confirming the robustness of the small-world property across these models.

Contribution

It introduces a generalized model encompassing various geometric and non-geometric scale-free networks and proves they all have similar average distances, up to a negligible factor.

Findings

All models in the class have average distance O(log log n).

The models exhibit a giant component with high probability.

The diameter of the networks is polylogarithmic with high probability.

Abstract

In Chung-Lu random graphs, a classic model for real-world networks, each vertex is equipped with a weight drawn from a power-law distribution, and two vertices form an edge independently with probability proportional to the product of their weights. Chung-Lu graphs have average distance O(log log n) and thus reproduce the small-world phenomenon, a key property of real-world networks. Modern, more realistic variants of this model also equip each vertex with a random position in a specific underlying geometry. The edge probability of two vertices then depends, say, inversely polynomial on their distance. In this paper we study a generic augmented version of Chung-Lu random graphs. We analyze a model where the edge probability of two vertices can depend arbitrarily on their positions, as long as the marginal probability of forming an edge (for two vertices with fixed weights, one fixed position, and one random position) is as in Chung-Lu random graphs. The resulting class contains Chung-Lu random graphs, hyperbolic random graphs, and geometric inhomogeneous random graphs as special cases. Our main result is that every random graph model in this general class has the same average distance as Chung-Lu random graphs, up to a factor 1+o(1). This shows in particular that specific choices, such as the underlying geometry being Euclidean or the dependence on the distance being inversely polynomial, do not significantly influence the average distance. The proof also yields that our model has a giant component and polylogarithmic diameter with high probability.