网络商：复杂系统的结构商
Network Quotients: Structural Skeletons of Complex Systems
摘要：
许多大型经验网络的一个决定性特征是它们的内在复杂性。然而，许多网络也包含很大程度的结构重复。一个直接的问题出现了：我们可以描述基本的网络复杂性，同时排除结构冗余？在这篇文章中，我们利用固有的网络对称性，从网络中崩溃的所有冗余信息，导致粗粒度，我们进行的基本结构信息的“父”网络。在代数组合学的上下文中，这种粗粒度化被称为\emph{quotient}。我们系统地探讨了网络代理商的理论属性，并总结了各种“现实世界”代理商相对于其父网络的关键统计数据。特别是，我们发现，子节点可以比它们的父网络小得多，但通常保留了各种关键的功能特性，如复杂性（异质性和枢纽顶点）和通信（直径和平均测地线距离），这表明子节点构成了它们的父网络的基本结构骨架。我们总结与讨论的潜在用途，在生物调控网络的分析和方式，其中使用candidents可以降低网络算法的计算复杂性。
英文摘要：
A defining feature of many large empirical networks is their intrinsic complexity. However, many networks also contain a large degree of structural repetition. An immediate question then arises: can we characterize essential network complexity while excluding structural redundancy? In this article we utilize inherent network symmetry to collapse all redundant information from a network, resulting in a coarse-graining which we show to carry the essential structural information of the `parent' network. In the context of algebraic combinatorics, this coarse-graining is known as the \emph{quotient}. We systematically explore the theoretical properties of network quotients and summarize key statistics of a variety of `real-world' quotients with respect to those of their parent networks. In particular, we find that quotients can be substantially smaller than their parent networks yet typically preserve various key functional properties such as complexity (heterogeneity and hubs vertices) and communication (diameter and mean geodesic distance), suggesting that quotients constitute the essential structural skeleton of their parent network. We summarize with a discussion of potential uses of quotients in analysis of biological regulatory networks and ways in which using quotients can reduce the computational complexity of network algorithms.
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