Dynamics of random hyperbolic graphs

Abstract

Random hyperbolic graphs (RHGs) have been shown to be adequate models of real-world complex networks, as they naturally and simultaneously possess many of their common structural characteristics. However, existing work on RHGs has mainly focused on structural properties of network snapshots, i.e., of static graphs, while little is known about the dynamical properties of RHGs. In this talk, we will consider the simplest possible model of dynamic RHGs in the cold regime (network temperature T < 1) and derive its most basic dynamical properties, namely the distributions of contact and intercontact durations. These distributions decay as power laws in the model with exponents that depend only on the network temperature T and are consistent with (inter)contact distributions observed in some real systems. Interestingly, these results hold irrespective of the nodes' expected degrees, suggesting that broad (inter)contact distributions in real systems are due to node similarities, instead of popularities. We will also see that several other properties, such as weight and strength distributions, group size distributions, abundance of recurrent components, etc., are also consistent with real systems, justifying why epidemic and rumour spreading processes perform remarkably similar in real and modelled networks. Furthermore, we will discuss a recent generalization of the model that incorporates link persistence, as well as results from dynamic RHGs in the hot regime (network temperature T > 1). In the hot regime, the intercontact distribution is nonnormalizable, meaning that hot RHGs (including the configuration model that emerges for T to ∞) cannot be used as null models for real temporal networks, in stark contrast to cold RHGs. We will conclude with future work directions.