A random graph of moderate density

Tamás F. Móri,Ágnes Backhausz

doi:10.1214/21-ecp444

Abstract

We analyse a randomly growing graph model in which the average degree is asymptotically equal to a constant times the square root of the number of vertices, and the clustering coefficient is rather small. In every step, we choose two vertices uniformly at random, check whether they are connected or not, and we either add a new edge or delete one and add a new vertex of degree two to the graph. This dependence on the status of the connection chosen vertices makes the total number of vertices random after n steps. We prove asymptotic normality for this quantity and also for the degree of a fixed vertex (with normalization n1∕6). We also analyse the proportion of vertices with degree greater than a fixed multiple of the average degree, and the maximal degree.

Highlights

Random graphs have been intensively studied in the last decades [Durrett (2007), van der Hofstad (2016)]
We analyse a randomly growing graph model in which the average degree is asymptotically equal to a constant times the square root of the number of vertices, and the clustering coefficient is rather small
We study a randomly growing graph model which has intermediate edge density, which includes deletion as a step, and which more likely has a small clustering coefficient

Summary

Introduction

Random graphs have been intensively studied in the last decades [Durrett (2007), van der Hofstad (2016)]. We study a randomly growing graph model which has intermediate (moderate) edge density, which includes deletion as a step, and which more likely has a small clustering coefficient. From the point of view of applications, the edge density can be different, for example, the brain as a network can be very different from a proteine-proteine interaction network, or the physical network of the internet can be very different from an online social network from this point of view as well This is one of the reasons why it is important to study random graphs of various densities. For the proofs we are using methods from discrete-time martingale theory

The model

Degree process of a fixed vertex

Useful lemmas

Asymptotic degree distribution

Ni therefore ε ε

Maximal degree