Growing a Network on a Given Substrate

TL;DR

This paper analyzes the time evolution of degree distributions in network growth models, focusing on transient behaviors and convergence rates for different attachment mechanisms.

Contribution

It provides a comprehensive analysis of the degree distribution dynamics over time for networks with arbitrary initial conditions, including uniform and preferential attachment models.

Findings

Derived explicit formulas for degree distribution over time

Characterized transient behavior and convergence rates

Applicable to networks with arbitrary initial states

Abstract

Conventional studies of network growth models mainly look at the steady state degree distribution of the graph. Often long time behavior is considered, hence the initial condition is ignored. In this contribution, the time evolution of the degree distribution is the center of attention. We consider two specific growth models; incoming nodes with uniform and preferential attachment, and the degree distribution of the graph for arbitrary initial condition is obtained as a function of time. This allows us to characterize the transient behavior of the degree distribution, as well as to quantify the rate of convergence to the steady-state limit.