The Origin of Power-Law Emergent Scaling in Large Binary Networks

TL;DR

This paper derives asymptotic formulas for the conduction properties of large binary networks, revealing how power-law behavior emerges near percolation thresholds and how network size influences this phenomenon.

Contribution

It introduces a combined spectral and averaging approach to analytically describe conduction in binary networks, capturing percolation limits and emergent power-law behavior.

Findings

Formulas accurately predict conduction near percolation thresholds

Power-law behavior emerges between percolation limits

Results match extensive numerical simulations

Abstract

In this paper we study the macroscopic conduction properties of large but finite binary networks with conducting bonds. By taking a combination of a spectral and an averaging based approach we derive asymptotic formulae for the conduction in terms of the component proportions p and the total number of components N. These formulae correctly identify both the percolation limits and also the emergent power law behaviour between the percolation limits and show the interplay between the size of the network and the deviation of the proportion from the critical value of p = 1/2. The results compare excellently with a large number of numerical simulations.