广义网络结构：单纯复形的组态模型和正则系综
Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes
摘要：
单纯形复合体是能够编码发生在两个以上节点之间的相互作用的广义网络结构。单纯复合体描述了从大脑网络到社交和协作网络的各种复杂的交互系统。在这里，我们使用它们的广义度来描述单纯复形的结构，这些广义度捕获了一个，两个，三个或更多个链接节点的基本属性。此外，我们还引入了单纯复形的组态模型和正则系综，分别强制节点的广义度序列和节点的期望广义度序列。我们评估这些合奏的熵，发现配置模型中的单纯形复合物的数量的渐近表达式。我们提供了构造属于构型模型的单纯复形和单纯复形的正则系综的算法。我们给出了单纯复形的结构截止的表达式，对于维数为$d=1$的单纯复形，它可以简化为简单网络的结构截止。最后，我们提供了一个数值分析的自然相关性出现在配置模型的单纯复合体没有结构截止。
英文摘要：
Simplicial complexes are generalized network structures able to encode interactions occurring between more than two nodes. Simplicial complexes describe a large variety of complex interacting systems ranging from brain networks, to social and collaboration networks. Here we characterize the structure of simplicial complexes using their generalized degrees that capture fundamental properties of one, two, three or more linked nodes. Moreover we introduce the configuration model and the canonical ensemble of simplicial complexes, enforcing respectively the sequence of generalized degrees of the nodes and the sequence of the expected generalized degrees of the nodes. We evaluate the entropy of these ensembles, finding the asymptotic expression for the number of simplicial complexes in the configuration model. We provide the algorithms for the construction of simplicial complexes belonging to the configuration model and the canonical ensemble of simplicial complexes. We give an expression for the structural cutoff of simplicial complexes that for simplicial complexes of dimension $d=1$ reduces to the structural cutoff of simple networks. Finally we provide a numerical analysis of the natural correlations emerging in the configuration model of simplicial complexes without structural cutoff.
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