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Defining the Dimension of a Complex Network and Zeta Functions

机译：定义复杂网络的维度和Zeta函数

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The concept of dimension has played a key role in mathematics over the ages. While it was originally applied to dense sets, like the points on a line, it has been generalized to apply to discrete objects. In this work the dimension of a complex network is reviewed, with particular reference to the definition based on the complex network zeta function. We look at some definitions which have been proposed for the complex network dimension. The complex network zeta function is introduced and applied to define the complex network dimension. The function is presented for different systems, including discrete regular lattices and random graphs. This definition is particularly appealing for applications in statistical mechanics, since it generalizes the definition based on the scaling of volume with distance. The properties of the complex network zeta function are studied based on the theory of Dirichlet series. The applications to language analysis and to statistical mechanics are presented. The shortcut model is introduced to interpolate between systems with integer dimensions. Algorithms for calculating the complex network zeta function are studied. The similarity of the dimension definition to the dimension of complexity classes in computer science is brought out. This allows us to generalize theorems from computer science, like the entropy characterization of dimension and topological theorems like the theorem relating connectedness to dimension. The complex network zeta function is studied for fractal skeleton branching trees which occur in scale free networks. The study concludes by showing that the complex network zeta function provides a definition of dimension for complex networks which has mathematically robust properties.
机译：多年来，维度的概念在数学中起着关键作用。尽管最初将其应用于密集集（如直线上的点），但已普遍适用于离散对象。在这项工作中，对复杂网络的维度进行了回顾，尤其参考了基于复杂网络zeta函数的定义。我们来看一些为复杂网络维度提出的定义。引入了复杂网络zeta函数并将其应用于定义复杂网络维度。该函数针对不同的系统提供，包括离散的规则晶格和随机图。此定义特别适合统计力学中的应用，因为它基于体积随距离的缩放来概括该定义。基于Dirichlet级数理论研究了复网络ζ函数的性质。介绍了在语言分析和统计力学中的应用。引入了快捷方式模型以在具有整数维的系统之间进行插值。研究了复杂网络zeta函数的计算算法。提出了维度定义与计算机科学中复杂度类维度的相似性。这使我们能够概括计算机科学中的定理，例如维的熵表征和拓扑定理，例如将连通性与维联系起来的定理。研究了在无标度网络中出现的分形骨架分支树的复杂网络zeta函数。该研究的结论是表明，复杂网络zeta函数为具有数学鲁棒性的复杂网络提供了尺寸定义。

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