Energy of Convex Sets, Shortest Paths, and Resistance

AbstractLet us assign independent, exponentially distributed random edge lengths to the edges of an undirected graph. Lyons, Pemantle, and Peres (1999, J. Combin. Theory Ser. A86 (1999), 158–168) proved that the expected length of the shortest path between two given nodes is bounded from below by the resistance between these nodes, where the resistance of an edge is the expectation of its length. They remarked that instead of exponentially distributed variables, one could consider random variables with a log-concave tail. We generalize this result in two directions. First, we note that the variables do not have to be independent: it suffices to assume that their joint distribution is log-concave. Second, the inequality can be formulated as follows: the expected length of a shortest path between two given nodes is the expected optimum of a stochastic linear program over a flow polytope, while the resistance is the minimum of a convex quadratic function over this polytope. We show that the inequality between these quantities holds true for an arbitrary polytope provided its blocker has integral vertices

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 has its foundation in statistical physics. We show that the randomized shortest path dissimilarity can be easily computed 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 defines a metric, in addition to which it satisfies the graph-geodetic property. The derivation and computation of the free energy distance are also straightforward. We then 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 and classification. The comparison, in general, shows that the parametrized distances perform well in the tasks. In particular, we see that the results obtained with the free energy distance are among the best in all the experiments.Comment: 30 pages, 4 figures, 3 table