Scaling in Small-World Resistor Networks

TL;DR

This paper analyzes how small-world network structures influence the effective resistance, showing that random links suppress large resistances unless conductance decays with link length, where resistance diverges.

Contribution

It provides an analytic characterization of the asymptotic resistance behavior in small-world resistor networks, including effects of link conductance decay.

Findings

Average resistance approaches a finite limit with non-zero random link density.

Standard deviation of resistance also approaches a finite value.

Resistance diverges when conductance decays as a power law with link length.

Abstract

We study the effective resistance of small-world resistor networks. Utilizing recent analytic results for the propagator of the Edwards-Wilkinson process on small-world networks, we obtain the asymptotic behavior of the disorder-averaged two-point resistance in the large system-size limit. We find that the small-world structure suppresses large network resistances: both the average resistance and its standard deviation approaches a finite value in the large system-size limit for any non-zero density of random links. We also consider a scenario where the link conductance decays as a power of the length of the random links, $l^{-\alpha}$. In this case we find that the average effective system resistance diverges for any non-zero value of $\alpha$.