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

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 (for which we fix an exponent $2 < \beta < 3$), and two vertices form an edge independently with probability proportional to the product of their weights. Modern, more realistic variants of this model also equip each vertex with a random position in a specific underlying geometry, which is typically Euclidean, such as the unit square, circle, or torus. The edge probability of two vertices then depends, say, inversely polynomial on their distance. We show 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, by studying a generic augmented version of Chung-Lu random graphs. Specifically, 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, i.e., proportional to the product of their weights. The resulting class contains Chung-Lu random graphs, hyperbolic random graphs, and geometric inhomogeneous random graphs as special cases. Our main result is that this general model has the same average distance as Chung-Lu random graphs, up to a factor $1+o(1)$. The proof also yields that our model has a giant component and polylogarithmic diameter with high probability

Random Regular Graphs are not Asymptotically Gromov Hyperbolic

In this paper we prove that random $d$--regular graphs with $d\geq 3$ have traffic congestion of the order $O(n\log_{d-1}^{3}(n))$ where $n$ is the number of nodes and geodesic routing is used. We also show that these graphs are not asymptotically $\delta$--hyperbolic for any non--negative $\delta$ almost surely as $n\to\infty$.Comment: 6 pages, 2 figure