  This work elaborates on the important problem of (1) designing optimal
randomized routing policies for reaching a target node t from a source note s
on a weighted directed graph G and (2) defining distance measures between nodes
interpolating between the least cost (based on optimal movements) and the
commute-cost (based on a random walk on G), depending on a temperature
parameter T. To this end, the randomized shortest path formalism (RSP,
[2,99,124]) is rephrased in terms of Tsallis divergence regularization, instead
of Kullback-Leibler divergence. The main consequence of this change is that the
resulting routing policy (local transition probabilities) becomes sparser when
T decreases, therefore inducing a sparse random walk on G converging to the
least-cost directed acyclic graph when T tends to 0. Experimental comparisons
on node clustering and semi-supervised classification tasks show that the
derived dissimilarity measures based on expected routing costs provide
state-of-the-art results. The sparse RSP is therefore a promising model of
movements on a graph, balancing sparse exploitation and exploration in an
optimal way.
