Averaging dynamics, mortal random walkers and information aggregation on graphs

Abstract

Many models of dynamics on graphs implicitly capture some tendency towards alignment or synchronisation between neighbouring nodes. This mechanism often leads to a tension between independent dynamics at each node preserving local variation, and alignment between neighbours encouraging global smoothing. In this paper, we explore some of the intuition behind this phenomenon by considering a simplified set of dynamics where the states of agents are determined by a combination of private signals and averaging dynamics with their neighbours. We show that outcomes of this mechanism correspond to the behaviour of mortal random walkers on the graph, and that steady state outcomes are captured precisely in an object called the fundamental matrix, which summarises expected visitation between pairs of nodes. The bulk of the paper approximates the elements of the fundamental matrix as a function of the topology of the graph, in the case of undirected and unweighted graphs. In doing so we show intuitively how features such as degree distribution, community structure and clustering impact the trade-off between local variation and global smoothing in the outcomes, and can shed light on more complex instances of dynamics on graph. We consider as an application how the results can be used to predict and better understand the steady state outcomes of an information aggregation process.