分形网络几何中的量子统计
Quantum statistics in Network Geometry with Fractional Flavor
摘要：
当节点与描述节点获取新链接的内在能力的适应度值相关联时，不断增长的网络模型已经显示出涌现的量子统计。最近已经表明，量子统计也出现在一个不断增长的单纯复杂模型中，称为具有风味的网络几何，它允许描述节点之间的多体相互作用。该模型依赖于一个称为flavor的外部参数，它负责单纯复形的底层拓扑。当味值为$s=-1$时，$d$维单纯复形是一个流形，其中每个$(d-1)$维面只能有一个关联数$n_α\in\{0,1\}$ .在这种情况下，单纯复形的面自然地由玻色-爱因斯坦分布、玻尔兹曼分布和费米-狄拉克分布描述，这取决于它们的维度。本文将味网络几何的研究扩展到味$s=-1/m$的分数值，其中每个$(d-1)$维面只能有关联数$n_α\in\{0,1,2,\dots, m\}$。我们表明，在这种情况下的统计特性的面的单纯复杂的玻色-爱因斯坦或费米-狄拉克分布只描述。最后，我们评论的频谱特性的网络构成的基础结构的考虑单纯复形。
英文摘要：
Growing network models have been shown to display emergent quantum statistics when nodes are associated to a fitness value describing the intrinsic ability of a node to acquire new links. Recently it has been shown that quantum statistics emerge also in a growing simplicial complex model called Network Geometry with Flavor which allow for the description of many-body interaction between the nodes. This model depend on an external parameter called flavor that is responsible for the underlying topology of the simplicial complex. When the flavor takes the value $s=-1$ the $d$-dimensional simplicial complex is a manifold in which every $(d-1)$-dimensional face can only have an incidence number $n_α\in\{0,1\}$. In this case the faces of the simplicial complex are naturally described by the Bose-Einstein, Boltzmann and Fermi-Dirac distribution depending on their dimension. In this paper we extent the study of Network Geometry with Flavor to fractional values of the flavor $s=-1/m$ in which every $(d-1)$-dimensional face can only have incidence number $n_α\in\{0,1,2,\dots, m\}$. We show that in this case the statistical properties of the faces of the simplicial complex are described by the Bose-Einstein or the Fermi-Dirac distribution only. Finally we comment on the spectral properties of the networks constituting the underlying structure of the considered simplicial complexes.
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