指数随机单纯复形
Exponential Random Simplicial Complexes
摘要：
指数随机图模型在过去的几十年里吸引了大量的研究关注。这些模型是在图的观测量的期望值等于给定值的约束下的最大熵集成。在这里，我们将这些最大熵集成随机单纯复形，这是更充分和通用的结构，在许多应用中的复杂系统建模。我们发现，许多随机单纯复形模型在文献中考虑可以铸造的最大熵合奏在一定的约束。我们介绍并分析了最一般的随机单纯复形系综$\mathbfΔ$与统计独立的单纯。我们的分析是简化的观察，任何分布$\mathbb{P}(O)$的任何集合的对象$\mathcal{O}=\{O\}$，包括图形和单纯复形，是最大熵的约束下，期望值的$-\ln \mathbb{P}(O)$是等于熵的分布。借助这一观察，我们证明了系综$\mathbfΔ$是最大熵下的两种类型的约束，固定的预期数量的单纯形和它们的边界。
英文摘要：
Exponential random graph models have attracted significant research attention over the past decades. These models are maximum-entropy ensembles under the constraints that the expected values of a set of graph observables are equal to given values. Here we extend these maximum-entropy ensembles to random simplicial complexes, which are more adequate and versatile constructions to model complex systems in many applications. We show that many random simplicial complex models considered in the literature can be casted as maximum-entropy ensembles under certain constraints. We introduce and analyze the most general random simplicial complex ensemble $\mathbfΔ$ with statistically independent simplices. Our analysis is simplified by the observation that any distribution $\mathbb{P}(O)$ on any collection of objects $\mathcal{O}=\{O\}$, including graphs and simplicial complexes, is maximum-entropy under the constraint that the expected value of $-\ln \mathbb{P}(O)$ is equal to the entropy of the distribution. With the help of this observation, we prove that ensemble $\mathbfΔ$ is maximum-entropy under two types of constraints that fix the expected numbers of simplices and their boundaries.
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