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

Abstract

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.

Beyond the landscape: Resistance modelling infers physical and behavioural gene flow barriers to a mobile carnivore across a metropolitan area

“…resistancega can optimise categorical and continuous resistance surfaces, as well as multiple resistance surfaces simultaneously (Peterman, 2018). All analyses in this work were based on categorical resistance surfaces and the commute-time geographic resistance distance (Kivimäki, Shimbo, & Saerens, 2014) To check for convergence, each optimisation run in the study was performed twice for each landscape feature or combination of landscape features. Using the ga.prep() function, we set the maximum value to be assessed during optimization of categorical resistance surfaces to 500 and retained all other default parameters of the ga.prep() function.…”

Functional connectivity and home range inferred at a microgeographic landscape genetics scale in a desert‐dwelling rodent

“…During the optimization process, the genetic algorithm searches all possible combinations of these parameters for transforming resistance surfaces, denoted by "r" in the monomolecular and Ricker equations (Peterman, 2018;Peterman et al, 2014). Previous studies using this optimization approach (e.g., Peterman et al, 2014, Ruiz-Lopez et al, 2016, Khimoun et al, 2017 have measured resistance distance using circuitScape (McRae, 2006); however, it is known that commuteDistance is functionally equivalent to circuitScape, with the advantage that it can be run in parallel (Kivimäki, Shimbo, & Saerens, 2014;Peterman, 2018). All these processes were performed using an eight-neighbor connection scheme for assessing connectivity.…”

Absorbing random walks interpolating between centrality measures on complex networks

“…The weighted graph distance, d ij , is the length of the shortest edge path from node i to j, where the length of a given edge (a, b) is d (a,b) = (A ab ) −1 . Defining edge lengths to be inverses of the nonzero weighted adjacency-matrix elements is standard practice when using affinity weights in the adjacency matrix: distance is considered to be the inverse of affinity [5,46,49].…”

“…Their inverse temperature parameter also tunes the preference for geodesics. In [45], RSP is used to interpolate between graph distance and resistance distance, while in [46] it is used to interpolate between random-walk betweenness and a measure similar to standard betweenness centrality. In [47], Bavaud and Guex accomplish a weighting equivalent to RSP through the minimization of a free-energy functional.…”