An $\epsilon$-expansion for Small-World Networks

TL;DR

This paper develops an epsilon-expansion method for analyzing diffusion on small-world networks, enabling calculation of Green's functions and fluctuations that align with numerical results, and applicable to broader diffusion problems.

Contribution

Introduces a well-defined epsilon-expansion technique for diffusion on small-world networks, allowing analytical calculation of Green's functions and fluctuations.

Findings

Analytical results agree with numerical simulations.

Method applicable to other diffusion problems in random media.

First non-leading order calculations provide accurate insights.

Abstract

I construct a well-defined expansion in $\epsilon=2-d$ for diffusion processes on small-world networks. The technique permits one to calculate the average over disorder of moments of the Green's function, and is used to calculate the average Green's function and fluctuations to first non-leading order in $\epsilon$, giving results which agree with numerics. This technique is also applicable to other problems of diffusion in random media.