The electrical resistance of a graph captures its commute and cover times

Abstract

Abstract. View an n-vertex, m-edge undirected graph as an electrical network with unit resistors as edges. We extend known relations between random walks and electrical networks by showing that resistance in this network is intimately connected with the lengths of random walks on the graph. For example, the commute time between two vertices s and t (the expected length of a random walk from s to t and back) is precisely characterized by the eective resistance R st between s and t: commute time = 2mR st . As a corollary, the cover time (the expected length of a random walk visiting all vertices) is characterized by the maximum resistance R in the graph to within a factor of log n: mR cover time O(mR log n). For many graphs, the bounds on cover time obtained in this manner are better than those obtained from previous techniques such as the eigenvalues of the adjacency matrix. In particular, we improve known bounds on cover times for high-degree graphs and expanders, and give new proofs of known results for multidimensional meshes. Moreover, resistance seems to provide an intuitively appealing and tractable approach to these problems.

R Package gdistance: Distances and Routes on Geographical Grids

“…Commute distance is the number of cells traversed during a random walk from a cell i on the grid to a cell j and back to i (Chandra et al 1996).…”

“…Travel is restricted to the land mass. The third is the commute distance (using the same conductance matrix), which is related to effective resistance between points if we conceive of the grid as an electrical circuit (Chandra et al 1996;McRae 2006). Among the distance measures evaluated until now, the great-circle distance between points turns out to be the best predictor of genetic distance.…”

“…Commute distance represents the random walk commute time, e.g., the average number of edges traversed during a random walk from an starting point on the graph to a destination point and back again to the starting point (Chandra, Raghavan, Ruzzo, Smolensky, and Tiwari 1996). Resistance distance reflects the average travel cost during this walk (McRae 2006).…”

Rural avenues as dispersal corridors for the vulnerable saproxylic beetle Elater ferrugineus in a fragmented agricultural landscape

“…1.1-5 (van Etten 2014) in the R version 3.1.0 (R Development Core Team 2013). The resistance distance is computed as the random-walk commute time between points (Chandra et al 1996;McRae 2006). During calculations, we used a connection scheme where movement was allowed between the nearest 16 cells.…”

“…Since elements in a landscape mosaic may differ in their resistance to movement, measures taking into account diversified viscosity of the different landscape components seem to be superior. Among them, the isolation-by-resistance approach, based on the theory of electrical circuits and graph theory (Chandra et al 1996;McRae 2006), is gaining increasing popularity. This approach takes into account the effect of all alternative pathways between habitat patches, which is especially appropriate when the specific aim is to assess gene flow across a landscape over many generations.…”