Flow-Based Dissimilarities: Shortest Path, Commute Time, Max-Flow and Free Energy

Guex, Guillaume; Bavaud, François

doi:10.1007/978-3-662-44983-7_9

Cited by 10 publications

(9 citation statements)

References 21 publications

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“…The potential and the surprisal distances introduced in this work fall under the same catalogue of distance families. See also [54,37,36] for other, closely related, formulations of families of distances based on free energy and network flows.…”

Section: Related Workmentioning

confidence: 99%

A bag-of-paths framework for network data analysis

et al. 2017

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This work develops a generic framework, called the bag-of-paths (BoP), for link and network data analysis. The central idea is to assign a probability distribution on the set of all paths in a network. More precisely, a Gibbs-Boltzmann distribution is defined over a bag of paths in a network, that is, on a representation that considers all paths independently. We show that, under this distribution, the probability of drawing a path connecting two nodes can easily be computed in closed form by simple matrix inversion. This probability captures a notion of relatedness between nodes of the graph: two nodes are considered as highly related when they are connected by many, preferably low-cost, paths. As an application, two families of distances between nodes are derived from the BoP probabilities. Interestingly, the second distance family interpolates between the shortest path distance and the resistance distance. In addition, it extends the Bellman-Ford formula for computing the shortest path distance in order to integrate sub-optimal paths by simply replacing the minimum operator by the soft minimum operator. Experimental results on semi-supervised classification show that both of the new distance families are competitive with other state-ofthe-art approaches. In addition to the distance measures studied in this paper, the bag-of-paths framework enables straightforward computation of many other relevant network measures.

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Section: Related Workmentioning

confidence: 99%

A bag-of-paths framework for network data analysis

et al. 2017

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“…The path distribution derived by Bavaud and Guex 20 can, however, be shown to equal the path distribution defined in the RSP framework, although this requires some lengthy and uninteresting derivations and is left out of this paper. The relation between the two approaches has also been studied by in the more recent work of Guex and Bavaud 34 . Compared to the work of Bavaud and Guex 20 , our work focuses more on the computational and practical aspects of the methodology.…”

Section: Graph Node Centralitymentioning

confidence: 99%

Two betweenness centrality measures based on Randomized Shortest Paths

Kivimäki

Lebichot

Saramäki

et al. 2016

Sci Rep

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This paper introduces two new closely related betweenness centrality measures based on the Randomized Shortest Paths (RSP) framework, which fill a gap between traditional network centrality measures based on shortest paths and more recent methods considering random walks or current flows. The framework defines Boltzmann probability distributions over paths of the network which focus on the shortest paths, but also take into account longer paths depending on an inverse temperature parameter. RSP’s have previously proven to be useful in defining distance measures on networks. In this work we study their utility in quantifying the importance of the nodes of a network. The proposed RSP betweenness centralities combine, in an optimal way, the ideas of using the shortest and purely random paths for analysing the roles of network nodes, avoiding issues involving these two paradigms. We present the derivations of these measures and how they can be computed in an efficient way. In addition, we show with real world examples the potential of the RSP betweenness centralities in identifying interesting nodes of a network that more traditional methods might fail to notice.

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“…The model proposed in this work builds on and extends previous work dedicated to the bag-ofpaths (BoP) framework (Françoisse et al, 2017;Mantrach et al, 2010), as well as the randomizedshortest-path (RSP) framework (Akamatsu, 1996;Kivimäki et al, 2014;Kivimäki et al, 2016;Saerens et al, 2009;Yen et al, 2008) and their variants (Bavaud & Guex, 2012;Guex & Bavaud, 2015;Guex, 2016); see also Zhang et al (2013) for a related proposition, called path integral. The main motivation for using such models can be understood as follows (Lebichot & Saerens, 2018).…”

Section: Related Workmentioning

confidence: 99%

Randomized optimal transport on a graph: framework and new distance measures

Guex

Kivimäki

Saerens

2019

Net Sci

Self Cite

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The recently developed bag-of-paths (BoP) 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 T) 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 (margins).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 n × n matrix with n being the number of nodes. Interestingly, for regular paths on an undirected graph, the resulting optimal policy interpolates between the deterministic optimal transport policy (T → 0+) and the solution to the corresponding electrical circuit (T → ∞). Two distance measures between nodes and a dissimilarity between groups of nodes, both integrating weights on nodes, are derived from this framework.

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Flow-Based Dissimilarities: Shortest Path, Commute Time, Max-Flow and Free Energy

Cited by 10 publications

References 21 publications

A bag-of-paths framework for network data analysis

A bag-of-paths framework for network data analysis

Two betweenness centrality measures based on Randomized Shortest Paths

Randomized optimal transport on a graph: framework and new distance measures

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