Average resistance of toroidal graphs

TL;DR

This paper analyzes how the average effective resistance of d-dimensional toroidal grids varies with topology and size, revealing divergence in 2D and boundedness in higher dimensions, with asymptotic estimates.

Contribution

It provides tight estimates of average resistance for toroidal graphs, exploring its asymptotic behavior across different dimensions and grid growth rates.

Findings

Average resistance diverges in 2D grids.

In higher dimensions, resistance remains bounded.

For hypercubes, resistance scales as 1/d for large d.

Abstract

The average effective resistance of a graph is a relevant performance index in many applications, including distributed estimation and control of network systems. In this paper, we study how the average resistance depends on the graph topology and specifically on the dimension of the graph. We concentrate on $d$-dimensional toroidal grids and we exploit the connection between resistance and Laplacian eigenvalues. Our analysis provides tight estimates of the average resistance, which are key to study its asymptotic behavior when the number of nodes grows to infinity. In dimension two, the average resistance diverges: in this case, we are able to capture its rate of growth when the sides of the grid grow at different rates. In higher dimensions, the average resistance is bounded uniformly in the number of nodes: in this case, we conjecture that its value is of order $1/d$ for large $d$. We prove this fact for hypercubes and when the side lengths go to infinity.