Abstract
Nodes can be ranked according to their relative importance within a network. Ranking algorithms based on random walks are particularly useful because they connect topological and diffusive properties of the network. Previous methods based on random walks, for example the PageRank, have focused on static structures. However, several realistic networks are indeed dynamic, meaning that their structure changes in time. In this paper, we propose a centrality measure for temporal networks based on random walks under periodic boundary conditions that we call TempoRank. It is known that, in static networks, the stationary density of the random walk is proportional to the degree or the strength of a node. In contrast, we find that, in temporal networks, the stationary density is proportional to the in-strength of the so-called effective network, a weighted and directed network explicitly constructed from the original sequence of transition matrices. The stationary density also depends on the sojourn probability q, which regulates the tendency of the walker to stay in the node, and on the temporal resolution of the data. We apply our method to human interaction networks and show that although it is important for a node to be connected to another node with many random walkers (one of the principles of the PageRank) at the right moment, this effect is negligible in practice when the time order of link activation is included.
Highlights
Random walks of various types are prototypical dynamical processes on networks
We examine the stationary density of this random walk and argue that the local inflow considered in the so-called effective network, explicitly constructed from the original network, is sufficient for accurately approximating the stationary density, or the centrality, of the nodes in the temporal networks
We proposed the TempoRank, a node centrality measure for temporal networks
Summary
Random walks of various types are prototypical dynamical processes on networks. Random walk models are objects of pure theoretical interest but the study of their dynamics enlightens general properties of diffusive processes. By simulating random walk dynamics in real temporal networks, Starnini and colleagues analyzed the coverage and the mean first-passage time of a random walk model on temporal network data. They found that the diffusion was slower on the temporal network in comparison to the aggregate version [33]. Ribeiro and colleagues connected temporal network data to the stationary density of the random walk [34] They obtained the degree (or weighted degree, called the strength) of the aggregate network from the data to determine the Poissonian node activity of an evolving network model. We show that the stationary density depends on the sojourn probability, which regulates the tendency of the walker to stay in the current node, and on the temporal resolution of the network data
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