The recently developed bag-of-paths framework consists in setting a Gibbs-Boltzmann distribution on all feasible paths of a graph. This probability distribution favors short paths over long ones, with a free parameter (the temperature @math ) controlling the entropic level of the distribution. This formalism enables the computation of new distances or dissimilarities, interpolating between the shortest-path and the resistance distance, which have been shown to perform well in clustering and classification tasks. In this work, the bag-of-paths formalism is extended by adding two independent equality constraints fixing starting and ending nodes distributions of paths. When the temperature is low, this formalism is shown to be equivalent to a relaxation of the optimal transport problem on a network where paths carry a flow between two discrete distributions on nodes. The randomization is achieved by considering free energy minimization instead of traditional cost minimization. Algorithms computing the optimal free energy solution are developed for two types of paths: hitting (or absorbing) paths and non-hitting, regular paths, and require the inversion of an @math matrix with @math being the number of nodes. Interestingly, for regular paths, the resulting optimal policy interpolates between the deterministic optimal transport policy ( @math ) and the solution to the corresponding electrical circuit ( @math ). Two distance measures between nodes and a dissimilarity between groups of nodes, both integrating weights on nodes, are derived from this framework.
The main motivation for using such models can be understood as follows @cite_0 . Most of the traditional network measures are essentially based on two different paradigms about movement or communication occurring in the network: optimal communication based on shortest paths, and random communication based on a random walk on the graph. For instance, shortest path distance, as well as the standard betweenness centrality @cite_37 are defined from shortest paths, while resistance distance and random walk centrality @cite_36 @cite_6 are based on random walks (which have a strong analogy with electrical flow on the network @cite_75 ).
@cite_37 @cite_36 @cite_6 @cite_0 @cite_75
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Abstract A new model of centrality is proposed for networks. The centrality measure is based on the “information” contained in all possible paths between pairs of points. The method does not require path enumeration and is not limited to the shortest paths or geodesies. We apply this measure to two examples: a network of homosexual men diagnosed with AIDS, and observations on a colony of baboons. Comparisons are made with “betweenness” and “closeness” centrality measures. The processes by which structural changes in networks occur over time are also discussed. We investigate random walks on complex networks and derive an exact expression for the mean firstpassage time (MFPT) between two nodes. We introduce for each node the random walk centrality C, which is the ratio between its coordination number and a characteristic relaxation time, and show that it determines essentially the MFPT. The centrality of a node determines the relative speed by which a node can receive and spread information over the network in a random process. Numerical simulations of an ensemble of random walkers moving on paradigmatic network models confirm this analytical prediction. Centrality measures, or at least popular interpretations of these measures, make implicit assumptions about the manner in which traffic flows through a network. For example, some measures count only geodesic paths, apparently assuming that whatever flows through the network only moves along the shortest possible paths. This paper lays out a typology of network flows based on two dimensions of variation, namely the kinds of trajectories that traffic may follow (geodesics, paths, trails, or walks) and the method of spread (broadcast, serial replication, or transfer). Measures of centrality are then matched to the kinds of flows that they are appropriate for. Simulations are used to examine the relationship between type of flow and the differential importance of nodes with respect to key measurements such as speed of reception of traffic and frequency of receiving traffic. It is shown that the off-the-shelf formulas for centrality measures are fully applicable only for the specific flow processes they are designed for, and that when they are applied to other flow processes they get the “wrong” answer. It is noted that the most commonly used centrality measures are not appropriate for most of the flows we are routinely interested in. A key claim made in this paper is that centrality measures can be regarded as generating expected values for certain kinds of node outcomes (such as speed and frequency of reception) given implicit models of how traffic flows, and that this provides a new and useful way of thinking about centrality. © 2004 Elsevier B.V. All rights reserved. We consider variations of two well-known centrality measures, betweenness and closeness, with a different model of information spread. Rather than along shortest paths only, it is assumed that information spreads efficiently like an electrical current. We prove that the current-flow variant of closeness centrality is identical with another known measure, information centrality, and give improved algorithms for computing both measures exactly. Since running times and space requirements are prohibitive for large networks, we also present a randomized approximation scheme for current-flow betweenness.