发表机构
Uppsala University（乌普萨拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出带软团约束的指数随机图模型，证明当$n$趋于无穷时，几乎必然从该模型抽取的随机图具有特定渐近结构，且该结构与正权重无关，还将结果推广到多团规模带权重的场景。
AI 中文摘要
设$r\geq3$为固定值，$\mathbf{G}_n$是顶点集为$[n] = \{1,\ldots,n\}$的所有简单图的集合。我们考虑一种指数随机图模型，若图$G \in \mathbf{G}_n$的$r$-团数量少于图$H \in \mathbf{G}_n$，则该模型赋予$G$比$H$更高的概率，但$\mathbf{G}_n$中所有图都具有正概率；赋予少$r$-团的图更高概率的程度由正权重$w$决定。我们证明，当$n \to \infty$时，几乎必然从$\mathbf{G}_n$中抽取的随机图可划分为$r-1$个大小大致相等的顶点部分，各部分之间的边密度接近$1/2$，且对任意$\varepsilon > 0$，任意部分内部的边密度均小于$\varepsilon$；只要$w$为正，该渐近结构性质与权重$w$无关。我们还将该结果推广到多个团规模的场景，每个团规模对应各自的权重。
英文摘要
Let $r\geq3$ be fixed, and let $\mathbf{G}_n$ be the set of all simple graphs with vertex set $[n]=\{1,\ldots,n\}$. We consider an exponential random graph model which gives higher probability to $G \in \mathbf{G}_n$ than to $H \in \mathbf{G}_n$ if $G$ has fewer $r$-cliques than $H$. But all graphs in $\mathbf{G}_n$ have positive probability. The degree to which graphs with fewer $r$-cliques are given higher probability is determined by a positive weight $w$. We prove that, asymptotically almost surely as $n \to \infty$, a random graph from $\mathbf{G}_n$ has a vertex partition into $r-1$ parts of roughly equal size, the density of edges between the parts is close to $1/2$, and for every $\varepsilon > 0$ the density of edges within any part is less than $\varepsilon$. The asymptotic structural properties are independent of the weight $w$ as long as it is positive. We also extend the result to the context of several clique sizes, each one with its own weight.
Comments58 pages, 1 figure
