Asymptotic degree distributions in random threshold graphs

TL;DR

This paper analyzes the asymptotic degree distributions in random threshold graphs, revealing that the empirical degree distribution may not reflect the true limiting distribution and challenging their use as scale-free network models.

Contribution

It introduces new limiting results for degree distributions in random threshold graphs under weak assumptions on fitness distribution, highlighting differences from traditional models.

Findings

Degree distribution converges in distribution to a limiting pmf.

Fraction of nodes with a fixed degree converges in distribution to a random variable.

Random threshold graphs with exponential fitness are not scale-free models like Barabási-Albert.

Abstract

We discuss several limiting degree distributions for a class of random threshold graphs in the many node regime. This analysis is carried out under a weak assumption on the distribution of the underlying fitness variable. This assumption, which is satisfied by the exponential distribution, determines a natural scaling under which the following limiting results are shown: The nodal degree distribution, i.e., the distribution of any node, converges in distribution to a limiting pmf. However, for each $d=0,1, \ldots $, the fraction of nodes with given degree $d$ converges only in distribution to a non-degenerate random variable $\Pi(d)$ (whose distribution depends on $d$),and not in probability to the aforementioned limiting nodal pmf as is customarily expected. The distribution of $\Pi(d)$ is identified only through its characteristic function. Implications of this result include: (i) The empirical node distribution may not be used as a proxy for or as an estimate to the limiting nodal pmf; (ii) Even in homogeneous graphs, the network-wide degree distribution and the nodal degree distribution may capture vastly different information; and (iii) Random threshold graphs with exponential distributed fitness do not provide an alternative scale-free model to the Barab\'asi-Albert model as was argued by some authors; the two models cannot be meaningfully compared in terms of their degree distributions!