Near critical preferential attachment networks have small giant components

TL;DR

This paper investigates how the giant component in near-critical preferential attachment networks rapidly diminishes as the edge density approaches the critical point, revealing an exponential decay unlike polynomial decay in similar models.

Contribution

It provides an explicit exponential decay rate for the giant component size near criticality in preferential attachment networks with >3, contrasting with polynomial decay in other scale-free models.

Findings

Giant component size decays as ^{-c/1 ho-1_c} near criticality

Decay is exponential, not polynomial, in the edge density approaching the critical point

Results rely on local neighborhood approximations and branching random walk theory.

Abstract

Preferential attachment networks with power law exponent $\tau>3$ are known to exhibit a phase transition. There is a value $\rho_{\rm c}>0$ such that, for small edge densities $\rho\leq \rho_c$ every component of the graph comprises an asymptotically vanishing proportion of vertices, while for large edge densities $\rho>\rho_c$ there is a unique giant component comprising an asymptotically positive proportion of vertices. In this paper we study the decay in the size of the giant component as the critical edge density is approached from above. We show that the size decays very rapidly, like $\exp(-c/ \sqrt{\rho-\rho_c})$ for an explicit constant $c>0$ depending on the model implementation. This result is in contrast to the behaviour of the class of rank-one models of scale-free networks, including the configuration model, where the decay is polynomial. Our proofs rely on the local neighbourhood approximations of [Dereich, Morters, 2013] and recent progress in the theory of branching random walks [Gantert, Hu, Shi, 2011].