Interpolating between random walks and optimal transportation routes: Flow with multiple sources and targets

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

“…Interestingly, it was found that the model derived by Guex (2016) is exactly equivalent to one of the two models introduced in this work, the one dealing with regular, non-hitting, paths, in the sense that they provide the same routing policy. Therefore, in comparison with (Guex, 2016), the present work reformulates the problem in terms of probabilities and relative entropy over paths in the network, instead of transition probabilities on nodes. It also introduces another algorithm for solving the problem and it derives a new algorithm for dealing with hitting paths.…”

“…Note that the present work is partly a re-interpretation of (Guex, 2016) in which the author already studied a similar optimal transport on a graph problem regularized by an entropic term. There, the entropic term at the node level was defined by considering, on each node, the relative entropy between the desired transition probabilities (the policy) and the reference transition probabilities corresponding to a natural random walk on the graph.…”

“…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).…”

Soft Image Segmentation: On the Clustering of Irregular, Weighted, Multivariate Marked Networks

“…Considering probabilities on paths (Bavaud and Guex [8]); Françoisse et al [18]) instead of probabilities on nodes or pairs of nodes, permits to extend the formalism to random walks based modularities (Devooght et al [15]) or multi-target based clustering (e.g. Sinop and Grady [37]; Guex [24]). This line of research pursues the "electric interpretation" of reversible Markov chains (Doyle and Snell [16]), involving Dirichlet differential equations and computation of the electric potentials, already standard in image segmentation (Grady [20]).…”

“…This line of research pursues the "electric interpretation" of reversible Markov chains (Doyle and Snell [16]), involving Dirichlet differential equations and computation of the electric potentials, already standard in image segmentation (Grady [20]). The possibility, in probabilistic formulations of random walks, to set independently the edge capacities and the edge resistances (Bavaud and Guex [8]; Guex [24]; Fouss et al [17]), seems especially relevant for the clustering of marked networks: it is tempting to identify the capacity contribution as a spatial term enabling transitions between neighbors, and the resistance contribution as a barrier preventing transitions between too dissimilar pixels.…”