论文标题
稀疏指数随机图模型的典型结构
Typical structure of sparse exponential random graph models
论文作者
论文摘要
我们考虑一般指数随机图模型（ERGMS），其中足够的统计数据是固定的简单图集合$ f_k $的同构函数的函数。尽管以前的工作表明在密集的ERGM中存在退化现象，但我们表明可以通过将足够的统计数据提高到分数能力来解决。我们严格地建立了相应Gibbs测量的分区函数的幼稚平均近似值，如果具有消失边缘密度的“铁磁”模型，则表明典型的样品类似于典型的Erdős-rényi图，带有植物集团和/或种植的完整的biptite图形，可用于适当尺寸。我们还为在大偏差制度中以多余的$ f_k $ - 肌形态计数的大偏差制度中的erdős--rényi图的条件结构建立了这种行为。这些结构性结果是通过结合了先前工作中建立的定量大偏差原理，以及在渐近解决方案上[5]的新型稳定性形式而获得的，以解决相关的熵变分问题。独立利益的技术成分是Finner广泛的Hölder不平等现象的稳定形式。
We consider general Exponential Random Graph Models (ERGMs) where the sufficient statistics are functions of homomorphism counts for a fixed collection of simple graphs $F_k$. Whereas previous work has shown a degeneracy phenomenon in dense ERGMs, we show this can be cured by raising the sufficient statistics to a fractional power. We rigorously establish the naïve mean-field approximation for the partition function of the corresponding Gibbs measures, and in case of "ferromagnetic" models with vanishing edge density show that typical samples resemble a typical Erdős--Rényi graph with a planted clique and/or a planted complete bipartite graph of appropriate sizes. We establish such behavior also for the conditional structure of the Erdős--Rényi graph in the large deviations regime for excess $F_k$-homomorphism counts. These structural results are obtained by combining quantitative large deviation principles, established in previous works, with a novel stability form of a result of [5] on the asymptotic solution for the associated entropic variational problem. A technical ingredient of independent interest is a stability form of Finner's generalized Hölder inequality.
