Routes to thermodynamic limit on scale-free networks

TL;DR

This paper investigates how finite size effects influence dynamic processes on scale-free networks, revealing two classes of behaviors and clarifying discrepancies between theoretical predictions and simulations.

Contribution

It identifies two classes of finite size effects and explains different routes to the thermodynamic limit in scale-free networks, clarifying previous theoretical and simulation discrepancies.

Findings

Finite networks exhibit size-dependent behaviors.

Some models depend on the upper cutoff of degree distribution.

The contact process belongs to the class with cutoff dependence.

Abstract

We show that there are two classes of finite size effects for dynamic models taking place on a scale-free topology. Some models in finite networks show a behavior that depends only on the system size N. Others present an additional distinct dependence on the upper cutoff k_c of the degree distribution. Since the infinite network limit can be obtained by allowing k_c to diverge with the system size in an arbitrary way, this result implies that there are different routes to the thermodynamic limit in scale-free networks. The contact process (in its mean-field version) belongs to this second class and thus our results clarify the recent discrepancy between theory and simulations with different scaling of k_c reported in the literature.