Typical distances in a geometric model for complex networks

TL;DR

This paper demonstrates that in a hyperbolic geometric model for complex networks, typical distances between nodes grow doubly logarithmically with the number of nodes, confirming the ultra-small world property.

Contribution

It establishes the asymptotic behavior of typical distances in hyperbolic random graphs, showing they scale as doubly logarithmic in network size, and links this to the Chung-Lu model.

Findings

Distances are doubly logarithmic in network size

Distance distribution converges to a constant depending on the power law exponent

The ultra-small world property is confirmed in hyperbolic geometric models

Abstract

We study typical distances in a geometric random graph on the hyperbolic plane. Introduced by Krioukov et al.~\cite{ar:Krioukov} as a model for complex networks, $N$ vertices are drawn randomly within a bounded subset of the hyperbolic plane and any two of them are joined if they are within a threshold hyperbolic distance. With appropriately chosen parameters, the random graph is sparse and exhibits power law degree distribution as well as local clustering. In this paper we show a further property: the distance between two uniformly chosen vertices that belong to the same component is doubly logarithmic in $N$, i.e., the graph is an ~\emph{ultra-small world}. More precisely, we show that the distance rescaled by $\log \log N$ converges in probability to a certain constant that depends on the exponent of the power law. The same constant emerges in an analogous setting with the well-known \emph{Chung-Lu} model for which the degree distribution has a power law tail.