  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 $n \times n$ 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 ($T \rightarrow 0^{+}$) and the solution to the
corresponding electrical circuit ($T \rightarrow \infty$). Two distance
measures between nodes and a dissimilarity between groups of nodes, both
integrating weights on nodes, are derived from this framework.
