The diameter of the directed configuration model

Typical distances in the directed configuration model

We analyze the distribution of the distance between two nodes, sampled uniformly at random, in digraphs generated via the directed configuration model, in the supercritical regime. Under the assumption that the covariance between the in-degree and out-degree is finite, we show that the distance grows logarithmically in the size of the graph. In contrast with the undirected case, this can happen even when the variance of the degrees is infinite. The main tool in the analysis is a new coupling between a breadth-first graph exploration process and a suitable branching process based on the Kantoro…

Ranking algorithms on directed configuration networks

This paper studies the distribution of a family of rankings, which includes Google's PageRank, on a directed configuration model. In particular, it is shown that the distribution of the rank of a randomly chosen node in the graph converges in distribution to a finite random variable $\mathcal{R}^*$ that can be written as a linear combination of i.i.d. copies of the endogenous solution to a stochastic fixed point equation of the form $$\mathcal{R} \stackrel{\mathcal{D}}{=} \sum_{i=1}^{\mathcal{N}} \mathcal{C}_i \mathcal{R}_i + \mathcal{Q},$$ where $(\mathcal{Q}, \mathcal{N}, \{ \mathcal{C}_i\})…

Girth, words and diameter

We study the girth of Cayley graphs of finite classical groups G on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word w takes the value 1 when evaluated in G in terms of the length of w, which has additional applications. We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.

Diameter and Stationary Distribution of Random $r$-out Digraphs

Let $D(n,r)$ be a random $r$-out regular directed multigraph on the set of vertices $\{1,\ldots,n\}$. In this work, we establish that for every $r \ge 2$, there exists $η_r>0$ such that $\text{diam}(D(n,r))=(1+η_r+o(1))\log_r{n}$. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on $D(n,r)$. In particular, we determine the asymptotic behaviour of $π_{\max}$ and $π_{\min}$, the maximum and the minimum values of the stationary distribution. We show that with high probability $π_{\max} = n^{-1+o(1)}$ and $π_{\min}=n^{-(1…

Universality for the directed configuration model: metric space convergence of the strongly connected components at criticality

We consider the strongly connected components (SCCs) of a uniform directed graph on $n$ vertices with i.i.d. in- and out-degree pairs distributed as $(D^-,D^+)$, with $\mathbb E[D^+]=\mathbb E[D^-]=μ$. We condition on equal total in- and out-degree. A phase transition for the emergence of a giant SCC is known to occur at the critical value $\mathbb E[D^-D^+] = μ$. We study the model at this critical value and, additionally, require that $\mathbb E[(D^-)^i(D^+)^j]<\infty$ for all $i+j\leq 3$, and for $(i,j)=(1,3)$ and $(i,j)=(3,1)$. We show that, under these conditions, the SCCs ranked by decre…

Stationary distribution and cover time of sparse directed configuration models

We consider sparse digraphs generated by the configuration model with given in-degree and out-degree sequences. We establish that with high probability the cover time is linear up to a poly-logarithmic correction. For a large class of degree sequences we determine the exponent $γ\ge 1$ of the logarithm and show that the cover time grows as $n\log^γ(n)$, where $n$ is the number of vertices. The results are obtained by analysing the extremal values of the stationary distribution of the digraph. In particular, we show that the stationary distribution $π$ is uniform up to a poly-logarithmic factor…

Convergence of rank based degree-degree correlations in random directed networks

We introduce, and analyze, three measures for degree-degree dependencies, also called degree assortativity, in directed random graphs, based on Spearman's rho and Kendall's tau. We proof statistical consistency of these measures in general random graphs and show that the directed configuration model can serve as a null model for our degree-degree dependency measures. Based on these results we argue that the measures we introduce should be preferred over Pearson's correlation coefficients, when studying degree-degree dependencies, since the latter has several issues in the case of large network…

Flooding and Diameter in General Weighted Random Graphs

We study in this paper, the first passage percolation on a random graph model, the configuration model. We first introduce, the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists of describing the asymptotic behavior of the diameter and the flooding time, as the number of vertices n tends to infinity, in the case where the weight distributi…

Diameter in ultra-small scale-free random graphs: Extended version

It is well known that many random graphs with infinite variance degrees are ultrasmall. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least $k$ is approximately $k^{-(τ-1)}$ with $τ\in(2,3)$, typical distances between pairs of vertices in a graph of size $n$ are asymptotic to $\frac{2\log\log n}{|\log(τ-2)|}$ and $\frac{4\log\log n}{|\log(τ-2)|}$, respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order $\log\log n$ precisely when the minimal forward…

具有给定度分布的有向随机图

Directed random graphs with given degree distributions

给定非负整数上的两个分布F和G，我们提出了一个由独立同分布样本构造入度和出度序列的算法。分别来自F和G的观察，具有高概率将是图形的，也就是说，可以从中绘制简单的有向图。然后，我们分析了有向版本的配置模型，并表明，提供F和G具有有限的方差，获得一个单图的概率是有界远离零的节点数量的增长。我们表明，条件所得到的图是简单的，在和出度分布是（近似）F和G的大尺寸图。此外，当度分布只有有限的平均值，我们表明，消除自环和重边不会显着改变度分布在所得到的单图。