Random networks with sublinear preferential attachment: The giant component

TL;DR

This paper analyzes a dynamic random network model with sublinear preferential attachment, providing explicit criteria for the existence and robustness of a giant component, and characterizing its size and component distribution.

Contribution

It establishes necessary and sufficient conditions for the emergence and robustness of a giant component in a sublinear preferential attachment network.

Findings

Criteria for the existence of a giant component

Explicit conditions for robustness under random edge removal

Asymptotic size and distribution of components

Abstract

We study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function f of its current degree. We give a criterion for the existence of a giant component, which is both necessary and sufficient, and which becomes explicit when f is linear. Otherwise it allows the derivation of explicit necessary and sufficient conditions, which are often fairly close. We give an explicit criterion to decide whether the giant component is robust under random removal of edges. We also determine asymptotically the size of the giant component and the empirical distribution of component sizes in terms of the survival probability and size distribution of a multitype branching random walk associated with f.