Random walk centrality for temporal networks

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.Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. connected to those of interacting particle systems such as stochastic opinion formation models [1,2] and to current flow in electric circuits [3]. Furthermore, random walks have been applied to searching and routing on networks [4][5][6][7][8], detection of network communities [9] and respondentdriven sampling [10,11]. A particularly successful application is on ranking of nodes. The PageRank algorithm used for ranking websites and other entities is equivalent to the stationary density of a random walk [12,13]. Other definitions of centrality (i.e. ranking) of nodes in networks on the basis of the random walk have also been proposed [14][15][16][17][18].Previous research mostly focused on static structures, i.e. snapshots of networks where the links between the nodes are fixed. Nevertheless, various networks in which node ranking is relevant are dynamic, meaning that a link is used only occasionally in time. The structure of the web graph, for instance, is continuously fluctuating with webpages and links being added and removed at every moment [19]. Human interaction networks derived from, for example, face-toface conversations [20,21], sexual contacts [22] and email communication [23] are highly dynamic and follow irregular temporal patterns. As a consequence, the respective interaction matrices vary over time, and a static network representation of such systems becomes defi...

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