Directed Random Graphs with given Degree Distributions

Abstract

Given two distributions F and G on the nonnegative integers we propose an algorithm to construct in-and out-degree sequences from samples of i.i.d. observations from F and G, respectively, that with high probability will be graphical, that is, from which a simple directed graph can be drawn. We then analyze a directed version of the configuration model and show that, provided that F and G have finite variance, the probability of obtaining a simple graph is bounded away from zero as the number of nodes grows. We show that conditional on the resulting graph being simple, the in-and out-degree distributions are (approximately) F and G for large size graphs. Moreover, when the degree distributions have only finite mean we show that the elimination of self-loops and multiple edges does not significantly change the degree distributions in the resulting simple graph.1. Introduction. In order to study complex systems such as the World Wide Web (WWW) 1 or the Twitter network we propose a model for generating a simple directed random graph with prescribed degree distributions. The ability to match degree distributions to real graphs is perhaps the first characteristic one would desire from a model, and although several models that accomplish this for undirected graphs have been proposed in the recent literature [8,10,11,20], not much has been done for the directed case. In the WWW example that motivates this work, vertices represent webpages and the edges represent the links between them; for the Twitter graph vertices represent people and an edge from one vertex to another means that the first person is "following" the second. Empirical studies (e.g., [9,15]) suggest that both the in-degree and out-degree, number of links pointing to a page and the number of outbound links of a page, respectively, follow a power-law distribution, a characteristic often referred to as the scale-free property.

Average nearest neighbor degrees in scale-free networks

“…The configuration model (CM) [7,9,21,29] is an important model for generating graphs G n of size n, with a specific degree sequence. Since it generates graphs with neutral mixing (see e.g.…”

“…To finish the argument we use [9,Lemma 5.3], which states that the probability that the number of erased out-bound and in-bound stubs in the directed configuration model are positive, converges to zero. Their argument has a straightforward extension to the undirected case so that lim n→∞ P (Y 1 > 0) = 0, which implies (47) and hence lim n→∞ P (44) > δ 2 , Λ n = 0, which finishes the proof.…”

“…An important example of such networks are given by the configuration model [7,9,21,29], which generates networks with a given degree sequence and lets the nodes connect randomly to each other. It is well known that the ANND in the configuration model is a constant, given by the empirical second moment of the degrees, divided by the empirical first moment.…”

Asymptotic analysis of network structures: degree-degree correlations and directed paths

“…In this thesis we consider the directed version of the configuration model for the purpose of generating simple graphs with prescribed limiting out-and in-degree distributions, as described and analyzed in [28].…”

“…Since for a bi-degree sequence the sum of the out-degrees should equal the sum of the in-degrees, one cannot sample these independently. In [28] an algorithm for generating bi-degree sequences Chapter 1. Introduction is given, where the degrees are still close to being i.i.d.…”