Edge-attractor random walks on dynamic networks

Giulio Iacobelli, Daniel R. Figueiredo

Abstract: We study the behaviour of continuous time random walks (RWs) moving over dynamic networks in the presence of mutual dependencies: the walker influences the network dynamics and the network influences the walker steps. The network dynamics we consider are described by continuous time Markov chains on a finite set of graph configurations. We introduce the edge-attractor RW model in which the network dynamics is biased towards graph configurations displaying higher degree for the vertex currently occupied by the walker. In particular, assuming the walker is in vertex |$i$|⁠, the network transition rate to a possible configuration |$g$| is linearly proportional to the degree of vertex |$i$| in |$g$|⁠. On the other hand, network dynamics changes the edge set of the underlying graph configuration, thus naturally affecting the walker behaviour, preventing/allowing different walker steps over time. We show that the joint process describing the edge-attractor RW on a time-reversible dynamic network is time reversible. We characterize the stationary distribution of the edge-attractor RW which is independent of the walker rate and is proportional to the average degree across the possible network configurations weighted by the stationary distribution of the underlying dynamic network. Our results are extended to edge-attractor RWs on weighted graphs and to couplings where only network transitions that change the current neighbourhood of the walker location are allowed. We also provide an example of a dynamic network induced by a symmetric On–Off edge Markovian process, for which the stationary distribution of the edge-attractor walker is given in simple closed-form.