Developments in the theory of randomized shortest paths with a comparison of graph node distances

There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into account the global structure of the graph. In this article, we develop the theory of one family of graph node distances, known as the randomized shortest path dissimilarity, which we show to be easily computable in closed form for all pairs of nodes of a graph. Moreover, we come up with a new definition of a distance measure that we call the free energy distance. The free energy distance can be seen as an upgrade of the randomized shortest path dissimilarity as it satisfies several nice properties for a distance. In addition, the derivation and computation of the free energy distance are quite straightforward. We also make a comparison between a set of generalized distances that interpolate between the shortest path distance and the commute time, or resistance distance. This comparison focuses on the applicability of the distances in graph node clustering.