局部收敛动态随机图上的SIR模型

Real-time growth rate for general stochastic SIR epidemics on unclustered networks

Networks have become an important tool for infectious disease epidemiology. Most previous theoretical studies of transmission network models have either considered simple Markovian dynamics at the individual level, or have focused on the invasion threshold and final outcome of the epidemic. Here, we provide a general theory for early real-time behaviour of epidemics on large configuration model networks (i.e. static and locally unclustered), in particular focusing on the computation of the Malthusian parameter that describes the early exponential epidemic growth. Analytical, numerical and Mont…

Factor models on locally tree-like graphs

We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree $T$, and study the existence of the free energy density $ϕ$, the limit of the log-partition function divided by the number of vertices $n$ as $n$ tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity $ϕ$ subject to uniqueness of a relevant Gibbs measure for the factor model on $T$. By way of example we compute $ϕ$ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic …

随机有向图中的巨型分量是否“几乎”局部？

Are giants in random digraphs `almost' local?

本文将随机无向图中巨型分量“几乎局部”的结果推广到随机有向图，给出所需局部收敛的精确版本，并研究强连通分量数量的界与渐近行为，揭示有向图强连通性的非局部精细本质。

Resolvent of Large Random Graphs

We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdos-Renyi graphs and preferential attachment graphs. We sketch examples of application for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices.

临界渗透概率是局部的

The critical percolation probability is local

本文证明了Schramm猜想：对局部收敛的传递图序列，临界渗透概率连续，并由此推出非一维无穷顶点传递图的临界概率严格小于1。

基于中心性的顶点移除后随机图的连通性

Connectivity of random graphs after centrality-based vertex removal

本文研究移除中心顶点对网络连通性的影响，利用局部收敛技术分析连通分量极限数量并证明巨分量大小上界，在配置模型中证明移除最高度顶点对巨分量破坏最大。

The rank of diluted random graphs

We investigate the rank of the adjacency matrix of large diluted random graphs: for a sequence of graphs $(G_n)_{n\geq0}$ converging locally to a Galton--Watson tree $T$ (GWT), we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function $ϕ_*$ of $T$. In the first part, we show that the adjacency operator associated with $T$ is always self-adjoint; we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0. In the second part, we establish a sufficient condition on $ϕ_*$ for t…

稀疏随机图上接触过程的相变：基于亚稳态和局部极限

Phase transitions for contact processes on sparse random graphs via metastability and local limits

通过亚稳态密度和局部收敛技术，研究稀疏有限图上接触过程的快慢灭绝渐近区域，给出临界阈值与局部极限中存活/灭绝阈值一致的条件，并证明稀疏图上快灭绝可能发生在拉伸指数时间尺度。

The local limit of the uniform spanning tree on dense graphs

Let $G$ be a connected graph in which almost all vertices have linear degrees and let $T$ be a uniform spanning tree of $G$. For any fixed rooted tree $F$ of height $r$ we compute the asymptotic density of vertices $v$ for which the $r$-ball around $v$ in $T$ is isomorphic to $F$. We deduce from this that if $\{G_n\}$ is a sequence of such graphs converging to a graphon $W$, then the uniform spanning tree of $G_n$ locally converges to a multi-type branching process defined in terms of $W$. As an application, we prove that in a graph with linear minimum degree, with high probability, the densit…

Algorithms Using Local Graph Features to Predict Epidemics

We study a simple model of epidemics where an infected node transmits the infection to its neighbors independently with probability $p$. This is also known as the independent cascade or Susceptible-Infected-Recovered (SIR) model with fixed recovery time. The size of an outbreak in this model is closely related to that of the giant connected component in ``edge percolation'', where each edge of the graph is kept independently with probability $p$, studied for a large class of networks including configuration model \cite{molloy2011critical} and preferential attachment \cite{bollobas2003,Riordan2…