Asymptotic behavior of the node degrees in the ensemble average of adjacency matrix

TL;DR

This paper introduces a new framework using the ensemble average of adjacency matrices to analyze the asymptotic degree distribution in growing networks, providing practical numerical approximations.

Contribution

It proposes an explicit representation of averaged adjacency matrices for growing networks, enabling easier analysis of degree distributions compared to traditional methods.

Findings

Good approximation of degree distribution asymptotically

Applicable to models with monotone increasing degree functions

Facilitates numerical calculations over complex theoretical analysis

Abstract

Various important and useful quantities or measures that characterize the topological network structure are usually investigated for a network, then they are averaged over the samples. In this paper, we propose an explicit representation by the beforehand averaged adjacency matrix over samples of growing networks as a new general framework for investigating the characteristic quantities. It is applied to some network models, and shows a good approximation of degree distribution asymptotically. In particular, our approach will be applicable through the numerical calculations instead of intractable theoretical analysises, when the time-course of degree is a monotone increasing function like power-law or logarithm.