Rigorous Result for the CHKNS Random Graph Model

Rick Durrett

doi:10.46298/dmtcs.3345

Abstract

We study the phase transition in a random graph in which vertices and edges are added at constant rates. Two recent papers in Physical Review E by Callaway, Hopcroft, Kleinberg, Newman, and Strogatz, and Dorogovstev, Mendes, and Samukhin have computed the critical value of this model, shown that the fraction of vertices in finite clusters is infinitely differentiable at the critical value, and that in the subcritical phase the cluster size distribution has a polynomial decay rate with a continuously varying power. Here we sketch rigorous proofs for the first and third results and a new estimates about connectivity probabilities at the critical value.

Highlights

In the last few years, physicists, mathematicians, and computer scientists, motivated by the world wide web (Albert, Jeong, and Barbasi 1999, Huberman and Adamic 1999), metabolic networks (Jeong et al 2000), and other complex structures, have begun to investigate the difference between static random graphs and networks in which the node set grows and connections are added over time
The CHKNS analysis of their model begins by examining Nkt © the expected number of components of size k at time t
Note that this implies that the percolation probability S is infinitely differentiable at the critical value

Summary

Introduction

In the last few years, physicists, mathematicians, and computer scientists, motivated by the world wide web (Albert, Jeong, and Barbasi 1999, Huberman and Adamic 1999), metabolic networks (Jeong et al 2000), and other complex structures (for a survey see Strogatz 2001), have begun to investigate the difference between static random graphs and networks in which the node set grows and connections are added over time. Barbasi and Albert (1999) considered a model in which new vertices are attached preferentially to already well connected sites and found a power-law distribution for vertex degrees. Callaway, Hopcroft, Kleinberg, Newman, and Strogatz (2001) studied the following model without preferential attachment. In the original CHKNS model the number of edges was 1 with probability δ, and 0 otherwise. We will primarily study the situation in which a Poisson mean δ number of vertices are added at each step We prefer this version since in the Poisson case if we let Ai§ j§ k be the event noi j© edge is added at time k P Ai§ j§ k ©. The CHKNS analysis of their model begins by examining Nkt © the expected number of components of size k at time t. To solve for the ak, which gives the limiting number of clusters of size k per site, CHKNS used generating functions. 1 g 1© gives the fraction of sites that belong to clusters whose size grows in time. If you are curious about their exact words, see the paragraph above (17) in their paper

Rigorous derivation of the critical value

Bounding sums below by integrals the first is

Results at the critical value

The subcritical case