Random Walks on Directed Networks: the Case of PageRank
Santo Fortunato, Alessandro Flammini
备注：
15 pages, 10 figures. Minor modifications, references added, final version to appear in the Special Issue "Complex Networks' Structure and Dynamics'' of the International Journal of Bifurcation and Chaos
PageRank, the prestige measure for Web pages used by Google, is the stationary probability of a peculiar random walk on directed graphs, which interpolates between a pure random walk and a process where all nodes have the same probability of being visited. We give some exact results on the distribution of PageRank in the cases in which the damping factor q approaches the two limit values 0 and 1. When q -> 0 and for several classes of graphs the distribution is a power law with exponent 2, regardless of the in-degree distribution. When q -> 1 it can always be derived from the in-degree distribution of the underlying graph, if the out-degree is the same for all nodes.
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Partially-observed network data collected by link-tracing based sampling methods is often being studied to obtain the characteristics of a large complex network. However, little attention has been paid to sampling from directed networks such as WWW and Peer-to-Peer networks. In this paper, we propose a novel two-step (sampling/estimation) framework to measure nodal characteristics which can be defined by an average target function in an arbitrary directed network. To this end, we propose a personalized PageRank-based algorithm to visit and sample nodes. This algorithm only uses already visited…
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In the study of small and large networks it is customary to perform a simple random walk, where the random walker jumps from one node to one of its neighbours with uniform probability. The properties of this random walk are intimately related to the combinatorial properties of the network. In this paper we propose to use the Ruelle-Bowens random walk instead, whose probability transitions are chosen in order to maximise the entropy rate of the walk on an unweighted graph. If the graph is weighted, then a free energy is optimised instead of entropy rate. Specifically, we introduce a centrality …
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Many popular measures used in social network analysis, including centrality, are based on the random walk. The random walk is a model of a stochastic process where a node interacts with one other node at a time. However, the random walk may not be appropriate for modeling social phenomena, including epidemics and information diffusion, in which one node may interact with many others at the same time, for example, by broadcasting the virus or information to its neighbors. To produce meaningful results, social network analysis algorithms have to take into account the nature of interactions betwe…
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We present analytical results for the distribution of first hitting times of random walkers (RWs) on directed Erdős-Rényi (ER) networks. Starting from a random initial node, a random walker hops randomly along directed edges between adjacent nodes in the network. The path terminates either by the retracing scenario, when the walker enters a node which it has already visited before, or by the trapping scenario, when it becomes trapped in a dead-end node from which it cannot exit. The path length, namely the number of steps, $d$, pursued by the random walker from the initial node up to its termi…
