Cut-offs and finite size effects in scale-free networks

TL;DR

This paper investigates how the finite size of scale-free networks influences the degree distribution's cut-off, revealing that topological constraints significantly affect the cut-off behavior and differ from natural extremal limits.

Contribution

It provides an analytical expression for the structural cut-off in finite scale-free networks, accounting for topological constraints, and applies it to the configuration model.

Findings

Structural cut-off is smaller than the natural extremal cut-off.

Topological constraints govern the cut-off behavior in finite networks.

Results are applied to the configuration model to analyze tadpoles and multiple edges.

Abstract

We analyze the degree distribution's cut-off in finite size scale-free networks. We show that the cut-off behavior with the number of vertices $N$ is ruled by the topological constraints induced by the connectivity structure of the network. Even in the simple case of uncorrelated networks, we obtain an expression of the structural cut-off that is smaller that the natural cut-off obtained by means of extremal theory arguments. The obtained results are explicitly applied in the case of the configuration model to recover the size scaling of tadpoles and multiple edges.