条件二星指数随机图的混合时间
Mixing Time of Conditional Two Star Exponential Random Graphs
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中文总结 AI 辅助
本文研究条件二星指数随机图模型，刻画其复制对称区域并证明Kawasaki算法的亚稳态快速混合，进而推导最优阶浓度不等式。
中文摘要 AI 辅助
二星指数随机图模型（ERGMs）是通用ERGM和平均场伊辛模型的一个有趣特例。本文研究在边密度$p$条件下的二星ERGM。我们首先对复制对称区域进行解析刻画，在该区域内，条件模型在切割距离上接近Erdős--Rényi随机图$G(n,p)$。我们证明该区域是无条件模型对应区域的两倍大。为研究条件模型的精细性质，我们随后分析用于从中采样的全局Kawasaki算法。在复制对称区域内，并在附加条件$4\beta p(1-p)<0.5$（当$|p-1/2|\lesssim 0.4632$时，其中$\beta$为模型参数）下，我们证明Kawasaki算法的亚稳态快速混合。作为推论，我们获得一个弱Poincaré不等式，并利用它推导条件子图计数的最优阶浓度不等式。附加条件$4\beta p(1-p)<1/2$（若$|p-1/2|\lesssim 0.4632$）源于我们的压缩耦合证明技术。这一瓶颈在先前将压缩耦合技术应用于无条件ERGM的研究中未曾出现。
英文摘要
Two star exponential random graph models (ERGMs) are an interesting special case of both general ERGMs and mean-field Ising models. In this paper, we study two star ERGMs conditioning on the edge density $p$. We begin with an analytic characterization of the replica symmetric region, where the conditional model is close in cut distance to the Erdős--Rényi random graph $G(n,p)$. We prove that this region is twice as large as the corresponding region for the unconditional model. To study refined properties of the conditional model, we then analyze the global Kawasaki algorithm for sampling from it. Within the replica symmetric region, and under the additional condition that $4βp(1-p)<0.5$ when $|p-1/2|\lesssim 0.4632$, where $β$ is the model parameter, we prove metastable fast mixing of the Kawasaki algorithm. As corollaries, we obtain a weak Poincaré inequality and use it to deduce concentration inequalities of optimal order for conditional subgraph counts. The additional condition $4βp(1-p)<1/2$ if $|p-1/2|\lesssim 0.4632$ comes from our proof technique of contractive coupling. This bottleneck did not appear in previous studies applying the contractive coupling technique to unconditional ERGMs.
发表机构
- The Chinese University of Hong Kong（香港中文大学）
- Dalian University of Technology（大连理工大学）
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