The asymptotic behaviour of the weights and the degrees in an N-interactions random graph model

TL;DR

This paper analyzes the long-term behavior of vertex weights and degrees in an N-interactions random graph model, considering both preferential attachment and uniform vertex selection, with results derived via martingale techniques.

Contribution

It provides a detailed asymptotic analysis of vertex weights and degrees, including limits of maximal values, in a novel N-interactions graph model.

Findings

Asymptotic limits for vertex weights and degrees are established.

Maximal weight and degree converge to specific limits.

Martingale methods are effectively used for proofs.

Abstract

A random graph evolution based on the interactions of N vertices is studied. During the evolution both the preferential attachment method and the uniform choice of vertices are allowed. The weight of a vertex means the number of its interactions. The asymptotic behaviour of the weight and the degree of a fixed vertex, moreover the limit of the maximal weight and the maximal degree are described. The proofs are based on martingale methods.