稀疏指数随机图模型的典型结构
Typical structure of sparse exponential random graph models
AI总结：
本文研究稀疏指数随机图模型，通过将充分统计量提升到分数次幂克服稠密模型中的退化现象，并证明典型样本在铁磁情形下类似于植入团或完全二部图的 Erdős–Rényi 图。
摘要：
我们考虑一般的指数随机图模型（ERGMs），其中充分统计量是一组固定简单图 $F_k$ 的同态计数的函数。以往的工作已经表明稠密 ERGMs 中存在退化现象，而我们证明可以通过将充分统计量提升到分数次幂来治愈这一问题。我们严格建立了相应吉布斯测度配分函数的朴素平均场近似，并且在具有消失边密度的“铁磁”模型情形下，证明典型样本类似于一个典型的 Erdős–Rényi 图，其中植入了一个适当大小的团和/或一个适当大小的完全二部图。我们还在超额 $F_k$-同态计数的大偏差区域中，为 Erdős–Rényi 图的条件结构建立了同样的行为。这些结构性结果是通过将先前工作中建立的定量大偏差原理与文献[5]中关于相关熵变分问题渐近解的结果的一种新的稳定性形式相结合而得到的。一个具有独立兴趣的技术成分是 Finner 广义 Hölder 不等式的一种稳定性形式。
英文摘要：
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.
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