Watts-Strogatz随机图邻接矩阵的极限谱分布
Limiting spectral distribution for the adjacency matrix of the Watts-Strogatz random graph
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中文总结 AI 辅助
本文研究Watts-Strogatz随机图邻接矩阵的极限谱分布，在K和pK随顶点数趋于无穷时证明其经验特征值分布收敛于半圆律，K和p固定时提出前五阶矩猜想并给出支撑依据。
中文摘要 AI 辅助
具有n个顶点、参数K（正偶数）和p∈[0,1]的Watts-Strogatz随机图模型分两步构造：首先构建n个顶点的环形格点，每个顶点连接两侧各K/2个最近邻；接着每条边以概率p独立重连，将一个端点替换为未相邻的均匀随机顶点。该模型邻接矩阵的元素因重连构造高度依赖，我们研究其经验特征值分布。在K和pK随顶点数n趋于无穷的 regime 中，证明经适当缩放后经验特征值分布收敛于半圆律，证明基于新颖的耦合论证，将邻接矩阵近似为两个独立随机矩阵之和：一个是稀疏Wigner矩阵，另一个是随机带矩阵。在K和p保持固定的情况下，提出极限特征值分布前五阶矩的猜想公式，这些猜想由Watts-Strogatz模型与另一随机图模型的收敛结果及数值模拟支撑。
英文摘要
The Watts-Strogatz random graph model on $n$ vertices with parameters $K$ (a positive even integer) and $p \in [0, 1]$ is constructed in two steps. First, one starts with a ring lattice on $n$ vertices, where each vertex is connected to its $K/2$ nearest neighbors on each side. Each edge in turn is then independently rewired with probability $p$ by replacing one endpoint with a uniformly chosen vertex not already adjacent to it. We study the empirical eigenvalue distribution of the adjacency matrix for this model, whose entries are highly dependent due to the rewiring construction. In the regime where both $K$ and $pK$ grow to infinity with the vertex size $n$, we show that, after appropriate scaling, the empirical eigenvalue distribution converges to the semicircle law. The proof is based on a novel coupling argument that approximates the adjacency matrix by a sum of two independent random matrices, one a sparse Wigner matrix and the other a random band matrix. In the case where $K$ and $p$ remain fixed, we propose conjectural formulas for the first five moments of the limiting eigenvalue distribution. These conjectures are supported by a convergence result relating the Watts-Strogatz model to another random graph model, together with numerical simulations.
