Note on "Average resistance of toroidal graphs" by Rossi, Frasca and Fagnani
This note provides a theoretical connection between average resistance and random walk hitting times for toroidal grids, offering insight for researchers in graph theory and random walks.
The authors show that the average effective resistance of d-dimensional toroidal grids is proportional to the mean hitting time of a simple random walk, and for d≥3, the average resistance is uniformly bounded and scales as 1/d for large d.
In our recent paper W.S. Rossi, P. Frasca and F. Fagnani, "Average resistance of toroidal graphs", SIAM Journal on Control and Optimization, 53(4):2541--2557, 2015, we studied how the average resistances of $d$-dimensional toroidal grids depend on the graph topology and on the dimension of the graph. Our results were based on the connection between resistance and Laplacian eigenvalues. In this note, we contextualize our work in the body of literature about random walks on graphs. Indeed, the average effective resistance of the $d$-dimensional toroidal grid is proportional to the mean hitting time of the simple random walk on that grid. If $d\geq3 $, then the average resistance can be bounded uniformly in the number of nodes and its value is of order $1/d$ for large $d$.