Beyond the Steady-State: Analytical Study of Network Growth at Arbitrary Times, for Arbitrary Initial Conditions

TL;DR

This paper develops a method to analyze the degree distribution of growing networks at any time and initial condition, moving beyond traditional steady-state analysis, with validation through simulations.

Contribution

It introduces a framework for deriving degree distributions at arbitrary times for networks with arbitrary initial conditions, extending beyond steady-state assumptions.

Findings

Derived degree distribution formulas for arbitrary initial states and times.

Validated theoretical predictions with Monte Carlo simulations.

Highlighted the importance of finite-time analysis in network growth studies.

Abstract

In studying network growth, the conventional approach is to devise a growth mechanism, quantify the evolution of a statistic or distribution (such as the degree distribution), and then solve the equations in the steady state (the infinite-size limit). Consequently, empirical studies also seek to verify the steady-state prediction in real data. The caveat concomitant with confining the analysis to this time regime is that no real system has infinite size; most real growing networks are far from the steady state. This underlines the importance of finite-size analysis. In this paper, we consider the shifted-linear preferential attachment as an illustrative example of arbitrary-time network growth analysis. We obtain the degree distribution for arbitrary initial conditions at arbitrary times. We corroborate our theoretical predictions with Monte Carlo simulations.