Derivation of the percolation threshold for the network model of Barabasi and Albert

TL;DR

This paper rigorously derives the percolation threshold for the Barabasi-Albert network model, clarifying its properties and implications for scale-free networks, and introduces an extension to include clustering effects.

Contribution

It provides the first explicit derivation of the percolation threshold for the BA-model and proposes a simple extension to incorporate clustering effects.

Findings

Explicit derivation of the BA-model's percolation threshold

Connection between scale-free networks and linear preferential attachment

Extension of the BA-model to include clustering

Abstract

The percolation threshold of the network model by Barabasi and Albert (BA-model) [Science 286, 509 (1999)] has thus far only been 'guessed' based on simulations and comparison with other models. Due to the still uncertain influence of correlations, the reference to other models cannot be justified. In this paper, we explicitly derive the well-known values for the BA-model. To underline the importance of a null model like that of Barabasi and Albert, we close with two basic remarks. First, we establish a connection between the abundance of scale-free networks in nature and the fact that power-law tails in the degree distribution result only from (at least asymptotically) linear preferential attachment: Only in the case of linear preferential attachment does a minimum of topological knowledge about the network suffice for the attachment process. Second, we propose a very simple and realistic extension of the BA-model that accounts for clustering. We discuss the influence of clustering on the percolation properties.