Connectivity of Poissonian Inhomogeneous random Multigraphs

Remarks about inhomogeneous pair correlations

Given an infinite subset $\mathcal A \subseteq\mathbb N$, let $A$ denote its smallest $N$ elements. There is a rich and growing literature on the question of whether for typical $α\in[0,1]$, the pair correlations of the set $αA \pmod 1\subset [0,1]$ are asymptotically Poissonian as $N$ increases. We define an inhomogeneous generalization of the concept of pair correlation, and we consider the corresponding doubly metric question. Many of the results from the usual setting carry over to this new setting. Moreover, the double metricity allows us to establish some new results whose singly metric …

Empirical spectral distributions of sparse random graphs

We study the spectrum of a random multigraph with a degree sequence ${\bf D}_n=(D_i)_{i=1}^n$ and average degree $1 \ll ω_n \ll n$, generated by the configuration model, and also the spectrum of the analogous random simple graph. We show that, when the empirical spectral distribution (ESD) of $ω_n^{-1} {\bf D}_n $ converges weakly to a limit $ν$, under mild moment assumptions (e.g., $D_i/ω_n$ are i.i.d. with a finite second moment), the ESD of the normalized adjacency matrix converges in probability to $ν\boxtimes σ_{\rm sc}$, the free multiplicative convolution of $ν$ with the semicircle law.…

PageRank on inhomogeneous random digraphs

We study the typical behavior of a generalized version of Google's PageRank algorithm on a large family of inhomogeneous random digraphs. This family includes as special cases directed versions of classical models such as the Erdös-Rényi model, the Chung-Lu model, the Poissonian random graph and the generalized random graph, and is suitable for modeling scale-free directed complex networks where the number of neighbors a vertex has is related to its attributes. In particular, we show that the rank of a randomly chosen node in a graph from this family converges weakly to the attracting endogeno…

似然比分析方法向Skorokhod $M_1$拓扑的推广（应用于泊松平滑变点模型）

Extension of Likelihood Ratio Analysis Method to Skorokhod $M_1$ Topology (with Application to Poissonian Smooth Change-Point Model)

将似然比分析方法推广至Skorokhod $M_1$拓扑，处理极限似然比过程不连续的模型，并应用于快速机制下的泊松平滑变点模型，证明极大似然估计量具有与纯变点模型相同的渐近行为。

一种关于随机图的生成边和剩余边的动力学方法

A dynamical approach to spanning and surplus edges of random graphs

研究有限非齐次随机图，通过扩展同步广度优先遍历编码纳入剩余边数据，分析乘法合并的两种图表示，证明一种变换可编码增强乘法合并的完整动态。

通过边到达高效生成期望度图

Efficient generation of expected-degree graphs via edge-arrivals

本文提出一种基于边到达的高效生成方法，用于生成具有指定期望度序列的随机图，其运行时间与生成边数成正比，并简化了传统方法。

Successive minimum spanning trees

In a complete graph $K_n$ with edge weights drawn independently from a uniform distribution $U(0,1)$ (or alternatively an exponential distribution $\operatorname{Exp}(1)$), let $T_1$ be the MST (the spanning tree of minimum weight) and let $T_k$ be the MST after deletion of the edges of all previous trees $T_i$, $i<k$. We show that each tree's weight $w(T_k)$ converges in probability to a constant $γ_k$ with $2k-2\sqrt k <γ_k<2k+2\sqrt k$, and we conjecture that $γ_k = 2k-1+o(1)$. The problem is distinct from that of Frieze and Johansson (2018), finding $k$ MSTs of combined minimum weight, and…