Microscopic model for the logarithmic size effect on the Curie point in Barab\'asi-Albert networks

TL;DR

This paper presents a microscopic model explaining how the critical temperature of the Ising model on Barabási-Albert networks depends logarithmically on network size, based on the distribution of fully connected clusters.

Contribution

It introduces a microscopic approach linking cluster size distribution to the logarithmic size effect on the Curie point in BA networks.

Findings

Number of fully connected clusters follows exponential distribution with exponent 2/m.

Critical temperature is determined by the largest fully connected cluster.

Logarithmic dependence of critical temperature on network size is explained.

Abstract

We found that numbers of fully connected clusters in Barab\'asi-Albert (BA) networks follow the exponential distribution with the characteristic exponent $\kappa=2/m$. The critical temperature for the Ising model on the BA network is determined by the critical temperature of the largest fully connected cluster within the network. The result explains the logarithmic dependence of the critical temperature on the size of the network $N$.