# Note on "Average resistance of toroidal graphs" by Rossi, Frasca and Fagnani

## TL;DR

This paper relates the average resistance of toroidal graphs to random walk properties, showing that in higher dimensions, resistance remains bounded and decreases with increasing dimension.

## Contribution

It connects the average resistance of toroidal grids to random walk hitting times, providing bounds and asymptotic behavior for high-dimensional cases.

## Key findings

- Average resistance is proportional to mean hitting time of random walks.
- For dimensions d ≥ 3, resistance is uniformly bounded and scales as 1/d.
- Resistance decreases as the dimension of the toroidal grid increases.

## Abstract

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 $d$.