指数随机图的几何性质及其应用
On the geometry of exponential random graphs and applications
AI总结：
本文从生成多项式的几何视角出发，研究马尔可夫随机图与指数随机图模型（ERGMs）的稳定性和洛伦兹性，刻画了不同类别的性质边界，并探索其在真实网络中的应用价值。
摘要：
在2009年的一篇开创性论文中，Borcea、Brändén和Liggett阐述了概率分布与其生成多项式几何性质之间的关联。具体而言，他们证明了稳定生成多项式对应具有最强形式负相关性的分布。这一结论启发我们研究其他可能具备该性质的分布，本文的研究重点是随机图模型。\n本文为从几何视角研究马尔可夫随机图，以及更广义的指数随机图模型（exponential random graph models, ERGMs）奠定了基础。核心工作是判定这类模型对应的生成多项式何时满足稳定性和/或洛伦兹（Lorentzian）性。洛伦兹性于2020年由Brändén和Huh首次提出，Anari、Oveis-Gharan和Vinzant也独立提出了该性质，后者将其称为完全对数凹性质。稳定多项式理论的出现早于洛伦兹性，通常被视为实根性的多变量推广。Brändén和Huh已证明稳定多项式一定是洛伦兹的。尽管稳定性是一个很强的条件，但验证其成立并不总是可行的。本文刻画了几类马尔可夫随机图何时具备稳定性，以及何时仅具备洛伦兹性，随后进一步探讨了这些性质在真实网络中的应用。
英文摘要：
In a seminal paper in 2009, Borcea, Brändén, and Liggett described the connection between probability distributions and the geometry of their generating polynomials. Namely, they characterized that stable generating polynomials correspond to distributions with the strongest form of negative dependence. This motivates us to investigate other distributions that can have this property, and our focus is on random graph models. In this article, we will lay the groundwork to investigate Markov random graphs, and more generally exponential random graph models (ERGMs), from this geometric perspective. In particular, by determining when their corresponding generating polynomials are either stable and/or Lorentzian. The Lorentzian property was first described in 2020 by Brändén and Huh and independently by Anari, Oveis-Gharan, and Vinzant where the latter group called it the completely log-concave property. The theory of stable polynomials predates this, and is commonly thought of as the multivariate notion of real-rootedness. Brändén and Huh proved that stable polynomials are always Lorentzian. Although it is a strong condition, verifying stability is not always feasible. We will characterize when certain classes of Markov random graphs are stable and when they are only Lorentzian. We then shift our attention to applications of these properties to real-world networks.
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