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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Complex network zeta function ： ウィキペディア英語版 Complex network zeta function Different definitions have been given for the dimension of a complex network or graph. For example, metric dimension is defined in terms of the resolving set for a graph. Dimension has also been defined based on the box covering method applied to graphs.〔K.-I. Goh, G. Salvi, B. Kahng and D. Kim, Phys. Rev. Lett. 96, 018701 (2006).〕 Here we describe the definition based on the complex network zeta function. This generalises the definition based on the scaling property of the volume with distance. The best definition depends on the application. ==Definition== One usually thinks of dimension for a set which is dense, like the points on a line, for example. Dimension makes sense in a discrete setting, like for graphs, only in the large system limit, as the size tends to infinity. For example, in Statistical Mechanics, one considers discrete points which are located on regular lattices of different dimensions. Such studies have been extended to arbitrary networks, and it is interesting to consider how the definition of dimension can be extended to cover these cases. A very simple and obvious way to extend the definition of dimension to arbitrary large networks is to consider how the volume (number of nodes within a given distance from a specified node) scales as the distance (shortest path connecting two nodes in the graph) is increased. For many systems arising in physics, this is indeed a useful approach. This definition of dimension could be put on a strong mathematical foundation, similar to the definition of Hausdorff dimension for continuous systems. The mathematically robust definition uses the concept of a zeta function for a graph. The complex network zeta function and the graph surface function were introduced to characterize large graphs. They have also been applied to study patterns in Language Analysis. In this section we will briefly review the definition of the functions and discuss further some of their properties which follow from the definition. We denote by $\backslash textstyle\; r\_$ the distance from node $\backslash textstyle\; i$ to node $\backslash textstyle\; j$, i.e., the length of the shortest path connecting the first node to the second node. $\backslash textstyle\; r\_$ is $\backslash textstyle\; \backslash infty$ if there is no path from node $\backslash textstyle\; i$ to node $\backslash textstyle\; j$. With this definition, the nodes of the complex network become points in a metric space.〔 Simple generalisations of this definition can be studied, e.g., we could consider weighted edges. The graph surface function, $\backslash textstyle\; S(r\; )$, is defined as the number of nodes which are exactly at a distance $\backslash textstyle\; r$ from a given node, averaged over all nodes of the network. The complex network zeta function $\backslash textstyle\; \backslash zeta\_G\; (\; \backslash alpha\; )$ is defined as :$\backslash zeta\_G\; (\; \backslash alpha\; )\; :=\; \backslash frac\backslash sum\_i\; \backslash sum\_r^\_,$ where $\backslash textstyle\; N$ is the graph size, measured by the number of nodes. When $\backslash textstyle\; \backslash alpha$ is zero all nodes contribute equally to the sum in the previous equation. This means that $\backslash textstyle\; \backslash zeta\_(0)$ is $\backslash textstyle\; N-1$, and it diverges when $\backslash textstyle\; N\; \backslash rightarrow\; \backslash infty$. When the exponent $\backslash textstyle\; \backslash alpha$ tends to infinity, the sum gets contributions only from the nearest neighbours of a node. The other terms tend to zero. Thus, $\backslash textstyle\; \backslash zeta\_G\; (\; \backslash alpha\; )$ tends to the average degree $\backslash textstyle$ for the graph as $\backslash textstyle\; \backslash alpha\; \backslash rightarrow\; \backslash infty$. :$\backslash langle\; k\; \backslash rangle\; =\; \backslash lim\_\; \backslash zeta\_G\; (\; \backslash alpha\; ).$ The need for taking an average over all nodes can be avoided by using the concept of supremum over nodes, which makes the concept much easier to apply for formally infinite graphs. The definition can be expressed as a weighted sum over the node distances. This gives the Dirichlet series relation :$\backslash zeta\_G\; (\; \backslash alpha\; )\; =\; \backslash sum\_r\; S(r)/r^.$ This definition has been used in the shortcut model to study several processes and their dependence on dimension. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Complex network zeta function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.