A preferential attachment model with random initial degrees

TL;DR

This paper introduces a random graph model with preferential attachment and random initial degrees, showing that the degree distribution follows a power law with an exponent determined by initial degrees and attachment rules.

Contribution

It extends existing models by analyzing the asymptotic degree distribution in a preferential attachment process with random initial degrees, revealing a combined power-law behavior.

Findings

Degree sequence follows a power law with exponent min(τ_W, τ_P)

The model generalizes previous preferential attachment models

Provides a comprehensive analysis of degree distribution in the presence of random initial degrees

Abstract

In this paper, a random graph process ${G(t)}_{t\geq 1}$ is studied and its degree sequence is analyzed. Let $(W_t)_{t\geq 1}$ be an i.i.d. sequence. The graph process is defined so that, at each integer time $t$, a new vertex, with $W_t$ edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on $G(t-1)$, the probability that a given edge is connected to vertex i is proportional to $d_i(t-1)+\delta$, where $d_i(t-1)$ is the degree of vertex $i$ at time $t-1$, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent $\tau=\min\{\tau_{W}, \tau_{P}\}$, where $\tau_{W}$ is the power-law exponent of the initial degrees $(W_t)_{t\geq 1}$ and $\tau_{P}$ the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed.