论文标题
对不均匀的erdős-rényi随机图的最大特征值的大偏差原理
Large deviation principle for the maximal eigenvalue of inhomogeneous Erdős-Rényi random graphs
论文作者
论文摘要
我们考虑一个不均匀的erdős-rényi随机图$ g_n $ with vertex set $ [n] = \ {1，\ dots，n \} $，这对tertices $ i，j \ in [n] $，$ i \ neq j $，与可能性相连$ r（\ tfrac {i} {n}，\ tfrac {j} {n}）$，独立于其他成对的顶点。在这里，$ r \ colon \，[0,1]^2 \ to（0,1）$是扮演参考图的角色的对称函数。令$λ_n$为$ g_n $的邻接矩阵的最大特征值。众所周知，$λ_n/n $满足$ n \ to \ infty $的大偏差原理。相关的速率函数$ψ_r$由涉及Graphon空间上大偏差原理的速率函数$ I_R $的变量公式给出。我们分析了此变量公式，以确定$ψ_R$的属性，特别是当参考Graphon为等级1时。
We consider an inhomogeneous Erdős-Rényi random graph $G_N$ with vertex set $[N] = \{1,\dots,N\}$ for which the pair of vertices $i,j \in [N]$, $i\neq j$, is connected by an edge with probability $r(\tfrac{i}{N},\tfrac{j}{N})$, independently of other pairs of vertices. Here, $r\colon\,[0,1]^2 \to (0,1)$ is a symmetric function that plays the role of a reference graphon. Let $λ_N$ be the maximal eigenvalue of the adjacency matrix of $G_N$. It is known that $λ_N/N$ satisfies a large deviation principle as $N \to \infty$. The associated rate function $ψ_r$ is given by a variational formula that involves the rate function $I_r$ of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of $ψ_r$, specially when the reference graphon is of rank 1.
