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第1行： − 此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。 − {{Redirect|Complex networks|the company|Complex Networks}}+ − {{short description|Network with non-trivial topological features}}+ + − {{Network Science}}+ − In the context of [[network theory]], a '''complex network''' is a [[Graph (discrete mathematics)|graph]] (network) with non-trivial [[topological]] features—features that do not occur in simple networks such as [[lattice graph|lattices]] or [[random graph]]s but often occur in graphs modelling of real systems. The study of complex networks is a young and active area of scientific research<ref>{{cite journal| author = R. Albert and A.-L. Barabási|year = 2002| title = Statistical mechanics of complex networks| journal=Reviews of Modern Physics|volume = 74| issue = 1| pages = 47-49|doi=10.1103/RevModPhys.74.47| arxiv = cond-mat/0106096}}</ref><ref>{{cite journal| author = Mark Newman| year = 2010| title = Networks: An Introduction | journal = Oxford University Press|isbn=978-0-19-920665-0}}</ref><ref>{{cite journal| author = Reuven Cohen and Shlomo Havlin| year = 2010| title = Complex Networks: Structure, Robustness and Function| journal = Cambridge University Press|isbn=978-0-521-84156-6}}</ref> (since 2000) inspired largely by the empirical study of real-world networks such as [[computer network]]s, [[biological network]]s, technological networks, [[Connectome|brain networks]], [[climate networks]] and [[social network]]s.+ + + + + + + − In the context of network theory, a complex network is a graph (network) with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in graphs modelling of real systems. The study of complex networks is a young and active area of scientific research (since 2000) inspired largely by the empirical study of real-world networks such as computer networks, biological networks, technological networks, brain networks, climate networks and social networks.+ + − 在网络理论的背景下，复杂网络是一个具有非平凡拓扑特征的图形(网络)——这些特征不会出现在简单的网络中，如格子或随机图，而是经常出现在真实系统的图形建模中。复杂网络研究是一个年轻而活跃的科学研究领域(自2000年以来) ，其灵感主要来自对现实世界网络的实证研究，如计算机网络、生物网络、技术网络、大脑网络、气候网络和社会网络。+ + + + + + + + − + − Most [[social network|social]], [[biological network|biological]], and [[Computer network|technological network]]s display substantial non-trivial topological features, with patterns of connection between their elements that are neither purely regular nor purely random. Such features include a heavy tail in the [[degree distribution]], a high [[clustering coefficient]], [[assortativity]] or disassortativity among vertices, [[community structure]], and [[hierarchy|hierarchical structure]]. In the case of directed networks these features also include [[Reciprocity in network|reciprocity]], triad significance profile and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as [[lattice graph|lattices]] and [[random graph]]s, do not show these features. The most complex structures can be realized by networks with a medium number of interactions.<ref>{{cite journal|last=T. Wilhelm|first=J. Kim|title=What is a complex graph?|journal=Physica A|year=2008|volume=387|issue=11|pages=2637–2652|doi=10.1016/j.physa.2008.01.015|bibcode = 2008PhyA..387.2637K }}</ref> This corresponds to the fact that the maximum information content ([[entropy (information theory)|entropy]]) is obtained for medium probabilities.+ + − Most social, biological, and technological networks display substantial non-trivial topological features, with patterns of connection between their elements that are neither purely regular nor purely random. Such features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure, and hierarchical structure. In the case of directed networks these features also include reciprocity, triad significance profile and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features. The most complex structures can be realized by networks with a medium number of interactions. This corresponds to the fact that the maximum information content (entropy) is obtained for medium probabilities.+ − 大多数社会、生物和技术网络表现出大量非平凡的拓扑特征，它们的元素之间的连接模式既不是纯正规则的也不是纯粹随机的。这些特征包括程度分布的重尾，高集聚系数，顶点之间的协调性或不协调性，社区结构和等级结构。在有向网络的情况下，这些特征还包括互惠性、三元显著性特征和其他特征。相比之下，许多过去研究过的网络数学模型，如格和随机图，并没有表现出这些特征。最复杂的结构可以通过具有中等数量相互作用的网络来实现。这相当于这样一个事实，即对于中等概率而言，最大信息量(熵)是获得的。+ + + + + + + + + + + + + − Two well-known and much studied classes of complex networks are [[scale-free networks]]<ref name = "frst">{{cite journal|last=A. Barabasi|first=E. Bonabeau|title=Scale-Free Networks|journal=Scientific American|date= 2003|pages=50–59}}</ref> and [[small-world networks]],<ref name = "sec">{{cite journal|last=S. H. Strogatz|first=D. J. Watts|title=Collective dynamics of 'small-world' networks|journal=Nature|year=1998|volume=393|pages=440–442|doi=10.1038/30918|pmid=9623998|issue=6684|bibcode = 1998Natur.393..440W }}</ref><ref>{{cite journal|last=H.E. Stanley|first=L.A.N. Amaral, A. Scala, M. Barthelemy|title=Classes of small-world networks|journal=PNAS|year=2000|volume=97|issue=21|pages=11149–52|doi= 10.1073/pnas.200327197 |arxiv = cond-mat/0001458 |bibcode = 2000PNAS...9711149A|pmid=11005838|pmc=17168}}</ref> whose discovery and definition are canonical case-studies in the field. Both are characterized by specific structural features—[[power-law]] [[degree distribution]]s for the former and short path lengths and high [[clustering coefficient|clustering]] for the latter. However, as the study of complex networks has continued to grow in importance and popularity, many other aspects of network structure have attracted attention as well.+ + + − Two well-known and much studied classes of complex networks are scale-free networks and small-world networks, whose discovery and definition are canonical case-studies in the field. Both are characterized by specific structural features—power-law degree distributions for the former and short path lengths and high clustering for the latter. However, as the study of complex networks has continued to grow in importance and popularity, many other aspects of network structure have attracted attention as well.+ + + + + − 无标度网络和小世界网络是复杂网络研究的热点，它们的发现和定义是该领域的典型案例。两者都是拥有属性特有的结构特征---- 前者的幂律度分布和后者的短路径长度和高聚集性。然而，随着复杂网络研究的重要性和普及性不断提高，网络结构的许多其他方面也引起了人们的关注。+ + + + + + + + + + + − Recently, the study of complex networks has been expanded to networks of networks.<ref name="BuldyrevParshani2010">{{cite journal|last1=Buldyrev|first1=Sergey V.|last2=Parshani|first2=Roni|last3=Paul|first3=Gerald|last4=Stanley|first4=H. Eugene|last5=Havlin|first5=Shlomo|title=Catastrophic cascade of failures in interdependent networks|journal=Nature|volume=464|issue=7291|year=2010|pages=1025–1028|issn=0028-0836|doi=10.1038/nature08932|pmid=20393559|arxiv = 0907.1182 |bibcode = 2010Natur.464.1025B }}</ref> If those networks are [[interdependent networks|interdependent]], they become significantly more vulnerable to random failures and targeted attacks and exhibit cascading failures and first-order percolation transitions.<ref name="ParshaniBuldyrev2010">{{cite journal|last1=Parshani|first1=Roni|last2=Buldyrev|first2=Sergey V.|last3=Havlin|first3=Shlomo|title=Interdependent Networks: Reducing the Coupling Strength Leads to a Change from a First to Second Order Percolation Transition|journal=Physical Review Letters|volume=105|issue=4|year=2010|issn=0031-9007|doi=10.1103/PhysRevLett.105.048701|bibcode=2010PhRvL.105d8701P|arxiv = 1004.3989|pmid=20867893|pages=048701}}</ref>+ + + − Recently, the study of complex networks has been expanded to networks of networks. If those networks are interdependent, they become significantly more vulnerable to random failures and targeted attacks and exhibit cascading failures and first-order percolation transitions.+ + − 近年来，复杂网络的研究已经扩展到网络的网络。如果这些网络是相互依存的，它们极易受到随机故障和有针对性的攻击，并出现级联故障和一级渗透过渡。+ + + + + + + − Furthermore, the collective behavior of a network in the presence of nodes failure and recovery has been studied.<ref name="MajdandzicPodobnik2013">{{cite journal|last1=Majdandzic|first1=Antonio|last2=Podobnik|first2=Boris|last3=Buldyrev|first3=Sergey V.|last4=Kenett|first4=Dror Y.|last5=Havlin|first5=Shlomo|last6=Eugene Stanley|first6=H.|title=Spontaneous recovery in dynamical networks|journal=Nature Physics|volume=10|issue=1|year=2013|pages=34–38|issn=1745-2473|doi=10.1038/nphys2819|bibcode=2014NatPh..10...34M|doi-access=free}}</ref> It was found that such a network can have spontaneous failures and spontaneous recoveries.+ − Furthermore, the collective behavior of a network in the presence of nodes failure and recovery has been studied. It was found that such a network can have spontaneous failures and spontaneous recoveries.+ + + − 此外，还研究了节点失效和恢复情况下网络的集体行为。研究发现，这样的网络可能会自发失败和自发恢复。+ + + + + + + + − The field continues to develop at a brisk pace, and has brought together researchers from many areas including [[mathematics]], [[physics]], electric power systems,<ref name="Saleh 1381">{{Cite journal|last=Saleh|first=Mahmoud|last2=Esa|first2=Yusef|last3=Mohamed|first3=Ahmed|date=2018-05-29|title=Applications of Complex Network Analysis in Electric Power Systems|journal=Energies|language=en|volume=11|issue=6|pages=1381|doi=10.3390/en11061381|doi-access=free}}</ref> [[biology]]<ref>{{cite journal |author=A. Bashan, R.P. Bartsch, J.W. Kantelhardt, S. Havlin, P.C. Ivanov |year=2012 |title=Network physiology reveals relations between network topology and physiological function |journal=Nature Communications|volume=3 |pages=72|doi=10.1038/ncomms1705 |doi-access=free }}</ref>, [[climate]]<ref>{{cite journal |author=J. Fan, J. Meng, X. Chen, Y. Ashkenazy, S. Havlin |year=2017 |title= Network approaches to climate science |journal=Science China: Physics, Mechanics and Astronomy|volume=60 |issue=1|doi=10.1007/s11433-016-0362-2 |doi-access=free }}</ref>, [[computer science]], [[sociology]], [[epidemiology]]<ref>{{cite journal |author=Lucas D Valdez, Lidia A Braunstein, Shlomo Havlin |year=2020 |title= Epidemic spreading on modular networks: The fear to declare a pandemic |journal=Physical Review E|volume=101 |issue=3|pages=032309|doi=10.1103/PhysRevE.101.032309 |arxiv=1909.09695}}</ref>, and others.<ref>{{cite journal|last=A.E. Motter|first=R. Albert|title=Networks in Motion|journal=Physics Today|year=2012|volume=65|issue=4|pages=43–48|url=http://www.physicstoday.org/resource/1/phtoad/v65/i4/p43_s1|archive-url=https://archive.today/20120906061904/http://www.physicstoday.org/resource/1/phtoad/v65/i4/p43_s1|url-status=dead|archive-date=2012-09-06|doi=10.1063/pt.3.1518|arxiv=1206.2369|bibcode=2012PhT....65d..43M}}</ref> Ideas from network science and engineering have been applied to the analysis of metabolic and genetic regulatory networks; the study of ecosystem stability and robustness;<ref name="johnson2014">{{cite journal |author=Johnson S, Domı́nguez-Garcı́a V, Donetti L, Muñoz MA |year=2014 |title=Trophic coherence determines food-web stability |journal=[[Proc Natl Acad Sci USA]] |volume=111 |issue=50 |pages=17923–17928 |doi=10.1073/pnas.1409077111|pmid=25468963 |pmc=4273378 |arxiv=1404.7728 |bibcode=2014PNAS..11117923J }}</ref> clinical science;<ref>{{cite journal|last=S.G.Hofmann|first=J.E.Curtiss|title=A complex network approach to clinical science|journal=European Journal of Clinical Investigation|year=2018|volume=48|issue=8|pages=e12986|doi=10.1111/eci.12986|pmid=29931701|doi-access=free}}</ref> the modeling and design of scalable communication networks such as the generation and visualization of complex wireless networks;<ref>{{cite thesis|last=Mouhamed Abdulla|date=2012-09-22|title=On the Fundamentals of Stochastic Spatial Modeling and Analysis of Wireless Networks and its Impact to Channel Losses|url=http://spectrum.library.concordia.ca/974847|journal=Ph.D. Dissertation, Dept. Of Electrical and Computer Engineering, Concordia Univ., Montréal, Québec, Canada, Sep. 2012.|volume=|issue=|pages=(Ch.4 develops algorithms for complex network generation and visualization)|doi=|pmid=|access-date=|via=|publisher=Concordia University|type=phd}}</ref> the development of vaccination strategies for the control of disease; and a broad range of other practical issues. Research on networks are regularly published in the most visible scientific journals and obtain vigorous funding in many countries. Network theory has been found recently useful to identify bottlenecks in city traffic.<ref name="LiFu2015">{{cite journal|last1=Li|first1=Daqing|last2=Fu|first2=Bowen|last3=Wang|first3=Yunpeng|last4=Lu|first4=Guangquan|last5=Berezin|first5=Yehiel|last6=Stanley|first6=H. Eugene|last7=Havlin|first7=Shlomo|title=Percolation transition in dynamical traffic network with evolving critical bottlenecks|journal=Proceedings of the National Academy of Sciences|volume=112|issue=3|year=2015|pages=669–672|issn=0027-8424|doi=10.1073/pnas.1419185112|pmid=25552558|pmc=4311803|bibcode=2015PNAS..112..669L}}</ref> Network science is the topic of many conferences in a variety of different fields, and has been the subject of numerous books both for the lay person and for the expert.+ + − The field continues to develop at a brisk pace, and has brought together researchers from many areas including mathematics, physics, electric power systems, biology, climate, computer science, sociology, epidemiology, and others. Ideas from network science and engineering have been applied to the analysis of metabolic and genetic regulatory networks; the study of ecosystem stability and robustness; clinical science; the modeling and design of scalable communication networks such as the generation and visualization of complex wireless networks; the development of vaccination strategies for the control of disease; and a broad range of other practical issues. Research on networks are regularly published in the most visible scientific journals and obtain vigorous funding in many countries. Network theory has been found recently useful to identify bottlenecks in city traffic. Network science is the topic of many conferences in a variety of different fields, and has been the subject of numerous books both for the lay person and for the expert.+ + − 这个领域继续以快速的步伐发展，汇集了来自数学、物理学、电力系统、生物学、气候学、计算机科学、社会学、流行病学等许多领域的研究人员。来自网络科学和工程学的思想已经应用于代谢和基因调控网络的分析; 生态系统稳定性和健壮性的研究; 临床科学; 可扩展通信网络的建模和设计，例如复杂无线网络的生成和可视化; 疫苗接种战略的发展以控制疾病; 以及广泛的其他实际问题。关于网络的研究定期在最引人注目的科学期刊上发表，并在许多国家获得大量资金。近年来，人们发现网络理论有助于识别城市交通中的瓶颈。网络科学是各个领域许多会议的主题，也是众多外行人和专家著作的主题。+ + + − + + + + + + + + − {{Main|Scale-free networks}}+ − [[File:Social Network Analysis Visualization.png|thumb|An example of complex scale-free network.]]A network is said to be scale-free<ref name="frst" /><ref>{{cite journal |author=R. Albert and A.-L. Barabási |year=2002 |title=Statistical mechanics of complex networks |journal=Reviews of Modern Physics|volume=74 |issue=1 |pages=47–97|ISBN=978-3-540-40372-2}}</ref> if its degree distribution, i.e., the probability that a node selected uniformly at random has a certain number of links (degree), follows a mathematical function called a [[power law]]. The power law implies that the degree distribution of these networks has no characteristic scale. In contrast, networks with a single well-defined scale are somewhat similar to a lattice in that every node has (roughly) the same degree. Examples of networks with a single scale include the [[Erdős–Rényi model|Erdős–Rényi (ER) random graph]], [[random regular graphs]], [[regular lattices]], and [[hypercube]]s. Some models of growing networks that produce scale-invariant degree distributions are the [[Barabási–Albert model]] and the [[fitness model (network theory)|fitness model]]. In a network with a scale-free degree distribution, some vertices have a degree that is orders of magnitude larger than the average - these vertices are often called "hubs", although this language is misleading as, by definition, there is no inherent threshold above which a node can be viewed as a hub. If there were such a threshold, the network would not be scale-free. − An example of complex scale-free network.A network is said to be scale-free if its degree distribution, i.e., the probability that a node selected uniformly at random has a certain number of links (degree), follows a mathematical function called a power law. The power law implies that the degree distribution of these networks has no characteristic scale. In contrast, networks with a single well-defined scale are somewhat similar to a lattice in that every node has (roughly) the same degree. Examples of networks with a single scale include the Erdős–Rényi (ER) random graph, random regular graphs, regular lattices, and hypercubes. Some models of growing networks that produce scale-invariant degree distributions are the Barabási–Albert model and the fitness model. In a network with a scale-free degree distribution, some vertices have a degree that is orders of magnitude larger than the average - these vertices are often called "hubs", although this language is misleading as, by definition, there is no inherent threshold above which a node can be viewed as a hub. If there were such a threshold, the network would not be scale-free.+ + − 复杂无标度网络的一个例子。如果一个网络的度分布(即随机选择的节点均匀地具有一定数量的链路(度)的概率)遵循一个称为幂律的数学函数，那么这个网络就是无标度的。指数定律表明，这些网络的度分布没有特征尺度。相比之下，具有单一定义良好的尺度的网络在某种程度上类似于格子，因为每个节点具有(大致)相同的度数。单尺度网络的例子包括 erd s-Rényi (ER)随机图、随机正则图、正则格和超立方体。Barabási-Albert 模型和适应度模型是产生尺度不变度分布的一些成长网络模型。在一个无标度分布的网络中，一些顶点的度大于平均数量级——这些顶点通常被称为“集线器” ，尽管这种语言具有误导性，因为根据定义，没有固有的阈值可以将一个节点视为一个集线器。如果存在这样一个阈值，网络就不会是无标度的。+ − − − − Interest in scale-free networks began in the late 1990s with the reporting of discoveries of power-law degree distributions in real world networks such as the [[World Wide Web]], the network of [[Autonomous system (Internet)|Autonomous systems]] (ASs), some networks of Internet routers, protein interaction networks, email networks, etc. Most of these reported "power laws" fail when challenged with rigorous statistical testing, but the more general idea of heavy-tailed degree distributions -- which many of these networks do genuinely exhibit (before finite-size effects occur) -- are very different from what one would expect if edges existed independently and at random (i.e., if they followed a [[Poisson distribution]]). There are many different ways to build a network with a power-law degree distribution. The [[Yule-Simon distribution|Yule process]] is a canonical generative process for power laws, and has been known since 1925. However, it is known by many other names due to its frequent reinvention, e.g., The Gibrat principle by [[Herbert A. Simon]], the [[Matthew effect (sociology)|Matthew effect]], cumulative advantage and, [[preferential attachment]] by [[Albert-László Barabási|Barabási]] and Albert for power-law degree distributions. Recently, [[Hyperbolic Geometric Graph]]s have been suggested as yet another way of constructing scale-free networks. − − Interest in scale-free networks began in the late 1990s with the reporting of discoveries of power-law degree distributions in real world networks such as the World Wide Web, the network of Autonomous systems (ASs), some networks of Internet routers, protein interaction networks, email networks, etc. Most of these reported "power laws" fail when challenged with rigorous statistical testing, but the more general idea of heavy-tailed degree distributions -- which many of these networks do genuinely exhibit (before finite-size effects occur) -- are very different from what one would expect if edges existed independently and at random (i.e., if they followed a Poisson distribution). There are many different ways to build a network with a power-law degree distribution. The Yule process is a canonical generative process for power laws, and has been known since 1925. However, it is known by many other names due to its frequent reinvention, e.g., The Gibrat principle by Herbert A. Simon, the Matthew effect, cumulative advantage and, preferential attachment by Barabási and Albert for power-law degree distributions. Recently, Hyperbolic Geometric Graphs have been suggested as yet another way of constructing scale-free networks. − − 对无标度网络的兴趣始于1990年代后期，当时报告了在现实世界网络中幂律度分布的发现，如万维网、自治系统网络、一些互联网路由器网络、蛋白质互动网络、电子邮件网络等。大多数报道的“幂定律”在面对严格的统计测试时都会失败，但是重尾度分布这一更为普遍的概念——其中许多网络确实表现出(在有限大小效应发生之前)——与人们所期望的如果边独立和随机存在(例如，如果它们遵循泊松分佈分布)的概念大相径庭。建立具有幂律度分布的网络有许多不同的方法。圣诞过程是幂定律的典型生成过程，自1925年就为人所知。然而，由于其频繁的重新发明，它被许多其他的名字所熟知，例如，赫伯特·西蒙的吉布拉特原则，马太效应，累积优势，巴拉巴西和阿尔伯特的幂律度分布的优先依恋。近年来，双曲几何图被认为是构造无标度网络的又一种方法。 − − − − Some networks with a power-law degree distribution (and specific other types of structure) can be highly resistant to the random deletion of vertices—i.e., the vast majority of vertices remain connected together in a giant component.<ref name="CohenErez2000">{{cite journal|last1=Cohen|first1=Reuven|last2=Erez|first2=Keren|last3=ben-Avraham|first3=Daniel|last4=Havlin|first4=Shlomo|title=Resilience of the Internet to Random Breakdowns|journal=Physical Review Letters|volume=85|issue=21|year=2000|pages=4626–4628|issn=0031-9007|doi=10.1103/PhysRevLett.85.4626|bibcode=2000PhRvL..85.4626C|pmid=11082612|arxiv=cond-mat/0007048}}</ref> Such networks can also be quite sensitive to targeted attacks aimed at fracturing the network quickly. When the graph is uniformly random except for the degree distribution, these critical vertices are the ones with the highest degree, and have thus been implicated in the spread of disease (natural and artificial) in social and communication networks, and in the spread of fads (both of which are modeled by a [[percolation]] or [[branching process]]). While random graphs (ER) have an average distance of order log N<ref name="sec"/> between nodes, where N is the number of nodes, scale free graph can have a distance of log log N. Such graphs are called ultra small world networks.<ref>{{cite journal|last=R. Cohen|first=S. Havlin|title=Scale-free networks are ultrasmall|journal=Phys. Rev. Lett.|year=2003|volume=90|issue=5|pages=058701|url=http://havlin.biu.ac.il/Publications.php?keyword=Scale-free+networks+are+ultrasmall&year=*&match=all|doi=10.1103/physrevlett.90.058701|arxiv = cond-mat/0205476 |bibcode = 2003PhRvL..90e8701C|pmid=12633404}}</ref> − − Some networks with a power-law degree distribution (and specific other types of structure) can be highly resistant to the random deletion of vertices—i.e., the vast majority of vertices remain connected together in a giant component. Such networks can also be quite sensitive to targeted attacks aimed at fracturing the network quickly. When the graph is uniformly random except for the degree distribution, these critical vertices are the ones with the highest degree, and have thus been implicated in the spread of disease (natural and artificial) in social and communication networks, and in the spread of fads (both of which are modeled by a percolation or branching process). While random graphs (ER) have an average distance of order log N − − 一些具有幂律度分布的网络(以及其他特定类型的网络结构)可以很好地抵抗顶点的随机删除ーー也就是说，绝大多数顶点保持连接在一个巨大的组件中。这种网络对于旨在快速破坏网络的有针对性的攻击也非常敏感。当图是均匀随机的，除了度分布，这些临界顶点是最高度的，因此牵涉到疾病的传播(自然和人为的)在社会和通信网络，并在时尚的传播(两者都是模型渗流或分支过程)。而随机图(ER)具有顺序对数 n 的平均距离 − − − − ==Small-world networks== − − {{Main|Small-world network}} − − A network is called a small-world network<ref name="sec" /> by analogy with the [[small-world phenomenon]] (popularly known as [[six degrees of separation]]). The small world hypothesis, which was first described by the Hungarian writer [[Frigyes Karinthy]] in 1929, and tested experimentally by [[Stanley Milgram]] (1967), is the idea that two arbitrary people are connected by only six degrees of separation, i.e. the diameter of the corresponding graph of social connections is not much larger than six. In 1998, [[Duncan J. Watts]] and [[Steven Strogatz]] published the first small-world network model, which through a single parameter smoothly interpolates between a random graph and a lattice.<ref name="sec"/> Their model demonstrated that with the addition of only a small number of long-range links, a regular graph, in which the diameter is proportional to the size of the network, can be transformed into a "small world" in which the average number of edges between any two vertices is very small (mathematically, it should grow as the logarithm of the size of the network), while the clustering coefficient stays large. It is known that a wide variety of abstract graphs exhibit the small-world property, e.g., random graphs and scale-free networks. Further, real world networks such as the [[World Wide Web]] and the metabolic network also exhibit this property. − − A network is called a small-world network by analogy with the small-world phenomenon (popularly known as six degrees of separation). The small world hypothesis, which was first described by the Hungarian writer Frigyes Karinthy in 1929, and tested experimentally by Stanley Milgram (1967), is the idea that two arbitrary people are connected by only six degrees of separation, i.e. the diameter of the corresponding graph of social connections is not much larger than six. In 1998, Duncan J. Watts and Steven Strogatz published the first small-world network model, which through a single parameter smoothly interpolates between a random graph and a lattice. Their model demonstrated that with the addition of only a small number of long-range links, a regular graph, in which the diameter is proportional to the size of the network, can be transformed into a "small world" in which the average number of edges between any two vertices is very small (mathematically, it should grow as the logarithm of the size of the network), while the clustering coefficient stays large. It is known that a wide variety of abstract graphs exhibit the small-world property, e.g., random graphs and scale-free networks. Further, real world networks such as the World Wide Web and the metabolic network also exhibit this property. − − 类比于小世界现象(通常被称为小世界网络六度分隔理论) ，网络被称为网络。1929年，匈牙利作家 Frigyes Karinthy 首次提出了小世界假说，并由 Stanley Milgram (1967)进行了实验验证。该假说认为，两个任意的人之间只有一个六度分隔理论的联系。相应的社会关系图的直径不会大于6。1998年，Duncan j. Watts 和 Steven Strogatz 发表了第一个小世界网络模型，该模型通过一个单参数平滑地插值在一个随机图形和一个格子之间。他们的模型证明，只要增加少量的长程链接，一个正则图，其直径与网络大小成正比，就可以转换成一个“小世界” ，其中任意两个顶点之间的平均边数都非常小(从数学上讲，它应该增长为网络大小的对数) ，而集聚系数则保持大。众所周知，各种各样的抽象图都具有小世界特性，例如，随机图和无标度网络。此外，真实世界的网络，如万维网和代谢网络也展示了这一特性。 − − − − In the scientific literature on networks, there is some ambiguity associated with the term "small world". In addition to referring to the size of the diameter of the network, it can also refer to the co-occurrence of a small diameter and a high [[clustering coefficient]]. The clustering coefficient is a metric that represents the density of triangles in the network. For instance, sparse random graphs have a vanishingly small clustering coefficient while real world networks often have a coefficient significantly larger. Scientists point to this difference as suggesting that edges are correlated in real world networks. − − In the scientific literature on networks, there is some ambiguity associated with the term "small world". In addition to referring to the size of the diameter of the network, it can also refer to the co-occurrence of a small diameter and a high clustering coefficient. The clustering coefficient is a metric that represents the density of triangles in the network. For instance, sparse random graphs have a vanishingly small clustering coefficient while real world networks often have a coefficient significantly larger. Scientists point to this difference as suggesting that edges are correlated in real world networks. − − 在关于网络的科学文献中，”小世界”一词有一些含糊不清之处。除了指网络的直径大小之外，它还可以指小直径和高集聚系数的同现现象。集聚系数是一个表示网络中三角形密度的度量单位。例如，稀疏随机图的集聚系数极小，而现实世界网络的系数通常大得多。科学家指出，这种差异表明，在现实世界的网络中，边缘是相关的。 − − − − ==See also== − − {{Div col}} − − * [[Community structure]] − − * [[Complex adaptive system]] − − * [[Complex systems]] − − * [[Dual-phase evolution]] − − * [[Dynamic network analysis]] − − * [[Interdependent networks]] − − * [[Network theory]] − − * [[Network science]] − − * [[Percolation theory]] − − * [[Random graph]] − − * [[Scale-free networks]] − − * [[Small world networks]] − − * [[Spatial network]] − − * [[Trophic coherence]] − − {{div col end}} − − − − ==Books== − − *B. S. Manoj, Abhishek Chakraborty, and Rahul Singh, ''Complex Networks: A Networking and Signal Processing Perspective'', Pearson, New York, USA, February 2018. {{ISBN|978-0134786995}} − − *S.N. Dorogovtsev and J.F.F. Mendes, ''Evolution of Networks: From biological networks to the Internet and WWW'', Oxford University Press, 2003, {{ISBN|0-19-851590-1}} − − *Duncan J. Watts, ''Six Degrees: The Science of a Connected Age'', W. W. Norton & Company, 2003, {{ISBN|0-393-04142-5}} − − *Duncan J. Watts, ''Small Worlds: The Dynamics of Networks between Order and Randomness'', Princeton University Press, 2003, {{ISBN|0-691-11704-7}} − − *Albert-László Barabási, ''Linked: How Everything is Connected to Everything Else'', 2004, {{ISBN|0-452-28439-2}} − − *Alain Barrat, Marc Barthelemy, Alessandro Vespignani, ''Dynamical processes on complex networks'', Cambridge University Press, 2008, {{ISBN|978-0-521-87950-7}} − − *Stefan Bornholdt (Editor) and Heinz Georg Schuster (Editor), ''Handbook of Graphs and Networks: From the Genome to the Internet'', 2003, {{ISBN|3-527-40336-1}} − − *Guido Caldarelli, ''Scale-Free Networks'', Oxford University Press, 2007, {{ISBN|978-0-19-921151-7}} − − *Guido Caldarelli, Michele Catanzaro, ''Networks: A Very Short Introduction'' Oxford University Press, 2012, {{ISBN|978-0-19-958807-7}} − − *E. Estrada, "The Structure of Complex Networks: Theory and Applications", Oxford University Press, 2011, {{ISBN|978-0-199-59175-6}} − − *Reuven Cohen and Shlomo Havlin, ''Complex Networks: Structure, Robustness and Function'', Cambridge University Press, 2010, {{ISBN|978-0-521-84156-6}} − − *Mark Newman, ''Networks: An Introduction'', Oxford University Press, 2010, {{ISBN|978-0-19-920665-0}} − − *Mark Newman, Albert-László Barabási, and Duncan J. Watts, ''The Structure and Dynamics of Networks'', Princeton University Press, Princeton, 2006, {{ISBN|978-0-691-11357-9}} − − *R. Pastor-Satorras and A. Vespignani, ''Evolution and Structure of the Internet: A statistical physics approach'', Cambridge University Press, 2004, {{ISBN|0-521-82698-5}} − − * T. Lewis, Network Science, Wiley 2009, − − *Niloy Ganguly (Editor), Andreas Deutsch (Editor) and Animesh Mukherjee (Editor), ''Dynamics On and Of Complex Networks Applications to Biology, Computer Science, and the Social Sciences'', 2009, {{ISBN|978-0-8176-4750-6}} − − *Vito Latora, Vincenzo Nicosia, Giovanni Russo, ''Complex Networks: Principles, Methods and Applications'', Cambridge University Press, 2017, {{ISBN|978-1107103184}} − − − − ==References== − − {{More footnotes|date=August 2008}} − − {{reflist}} − − *{{cite journal|author=D. J. Watts and S. H. Strogatz|title=Collective dynamics of 'small-world' networks|journal=Nature|volume=393|year=1998|pages=440–442|doi=10.1038/30918|pmid=9623998|issue=6684|bibcode = 1998Natur.393..440W }} − − *{{cite journal|authorlink=Steven Strogatz|author=S. H. Strogatz|title=Exploring Complex Networks|journal=Nature|volume=410|year=2001|pages=268–276|doi=10.1038/35065725|pmid=11258382|issue=6825|bibcode = 2001Natur.410..268S |doi-access=free}} − − *{{cite journal|author=R. Albert and A.-L. Barabási|title=Statistical mechanics of complex networks|journal=Reviews of Modern Physics |volume=74|issue=1|year=2002|pages=47–97|doi=10.1103/RevModPhys.74.47|arxiv=cond-mat/0106096|bibcode=2002RvMP...74...47A}} − − *{{cite journal|author=S. N. Dorogovtsev and J.F.F. Mendes|arxiv=cond-mat/0106144|title=Evolution of Networks|journal=Adv. Phys.|volume=51|issue=4|pages=1079–1187|year=2002|doi=10.1080/00018730110112519|bibcode = 2002AdPhy..51.1079D }} − − *M. E. J. Newman, [https://arxiv.org/abs/cond-mat/0303516 The structure and function of complex networks], SIAM Review 45, 167-256 (2003) − − *S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, ''[https://arxiv.org/abs/0705.0010 Critical phenomena in complex networks]'', Rev. Mod. Phys. 80, 1275, (2008) − − *G. Caldarelli, R. Marchetti, L. Pietronero, The Fractals Properties of Internet, Europhysics Letters 52, 386 (2000). https://arxiv.org/abs/cond-mat/0009178. DOI: 10.1209/epl/i2000-00450-8 − − *R. Cohen, K. Erez, D. ben-Avraham, S. Havlin, "[http://havlin.biu.ac.il/Publications.php?keyword=Resilience+of+the+Internet+to+random+breakdown&year=*&match=all Resilience of the Internet to random breakdown]" ''Phys. Rev. Lett.'' 85, 4626 (2000). https://arxiv.org/abs/1004.3989 − − *R. Cohen, K. Erez, D. ben-Avraham, S. Havlin, "[http://havlin.biu.ac.il/Publications.php?keyword=Breakdown+of+the+Internet+under+intentional+attack&year=*&match=all Breakdown of the Internet under intentional attack]" ''Phys. Rev. Lett.'' 86, 3682 (2001) − − *R. Cohen, S. Havlin, "[http://havlin.biu.ac.il/Publications.php?keyword=Scale-free+networks+are+ultrasmall&year=*&match=all Scale-free networks are ultrasmall]" ''Phys. Rev. Lett.'' 90, 058701 (2003) − − *{{cite journal|author=A. E. Motter|title=Cascade control and defense in complex networks|journal=Phys. Rev. Lett.|volume=93|issue=9|pages=098701|doi=10.1103/PhysRevLett.93.098701|year=2004|arxiv=cond-mat/0401074|bibcode=2004PhRvL..93i8701M}} − − *J. Lehnert, Controlling Synchronization Patterns in Complex Networks, springer 2016 − − *{{cite |last1=Dolev|first1=Shlomi|last2=Elovici|first2=Yuval|last3=Puzis|first3=Rami|title=Routing betweenness centrality|journal=J. ACM|date=2010|volume=57|issue=4|pages=25:1–25:27|doi=10.1145/1734213.1734219}} − − − − [[Category:Network theory]] − − Category:Network theory − − 范畴: 网络理论 − − <noinclude> − − <small>This page was moved from [[wikipedia:en:Complex network]]. Its edit history can be viewed at [[复杂网络/edithistory]]</small></noinclude> − − [[Category:待整理页面]]
无编辑摘要
复杂网络是一种理解现实世界复杂系统的抽象模型。它将复杂系统中的实体抽象成节点 ，将实体之间的关系抽象成连线。虽然数学中的图论也在研究网络， 但是现实中的网络会有更多的随机特性。因此，复杂网络 一般更加关注网络的统计特征。
==定义==
在[http://wiki.swarma.net/index.php/网络理论 网络理论]的研究中，复杂网络是由数量巨大的节点和节点之间错综复杂的关系共同构成的网络结构。用数学的语言来说，就是一个有着足够复杂的拓扑结构特征的图。复杂网络具有简单网络，如晶格网络、随机图等结构所不具备的特性，而这些特性往往出现在真实世界的网络结构中。复杂网络的研究是现今科学研究中的一个热点，与现实中各类高复杂性系统，如的互联网网络、神经网络和社会网络的研究有密切关系。
==复杂网络研究简史==
===欧拉与格尼斯堡七桥问题===
格尼斯堡的七桥是一个历史上著名的数学问题。1736年[https://en.wikipedia.org/wiki/Leonhard_Euler 欧拉·莱昂哈德]的负分辨率奠定了[https://en.wikipedia.org/wiki/Graph_theory 图论]的基础，预示了[https://en.wikipedia.org/wiki/Topology 拓扑学]的概念。
[https://en.wikipedia.org/wiki/Kingdom_of_Prussia 普鲁士]的[https://en.wikipedia.org/wiki/Königsberg 格尼斯堡]（现在是[https://en.wikipedia.org/wiki/Russia 俄罗斯]的[https://en.wikipedia.org/wiki/Kaliningrad 加里宁格勒]）坐落在[https://en.wikipedia.org/wiki/Pregolya_River 普瑞格尔河]的两侧，包括两个大岛——[https://en.wikipedia.org/wiki/Kneiphof 克内夫]和[https://en.wikipedia.org/wiki/Oktyabrsky_Island 洛姆斯]，它们通过7座桥互相连接在一起，且与城市的两个大陆部分相连。问题是设计出一种穿越城市的方式，每座桥都要经过有且只有一次。
通过明确制定逻辑人物的方式，解决方案包括其中任何一个：
1.通过桥梁或其他途径到达岛屿或大陆彼岸
2.不经过另一端而经过桥（但，这是绝对不可能的）
欧拉证明了这个问题没有解决方案。他所面临的困难是开发出一种合适的分析技术，以及后面的测试，这些测试需要用精确的数学方法证明这一论断。
===ER随机网===
在数学领域的[https://en.wikipedia.org/wiki/Graph_theory 图论]中，ER模型与两种用来生成[https://en.wikipedia.org/wiki/Random_graph 随机图像]的模型密切相关。它们是以两个于1959年首次介绍其中一个模型的数学家[https://en.wikipedia.org/wiki/Paul_Erdős 艾迪胥·保罗]和[https://en.wikipedia.org/wiki/Alfréd_Rényi 瑞尼·阿尔弗莱德]命名的，而[https://en.wikipedia.org/wiki/Edgar_Gilbert 吉尔伯特·埃德加]介绍了另一个，同时也[https://en.wikipedia.org/wiki/Independence_(probability_theory) 独立]介绍了艾迪胥和瑞尼介绍的模型。在艾迪胥和瑞尼模型中，所有图形在一个固定的顶点及固定数量的边缘上是等可能的；在吉尔伯特介绍的模型中，每条边都有固定的出现或不出现的概率，与其他边无关。这些模型可用于[https://en.wikipedia.org/wiki/Probabilistic_method 概率方法]，以证明满足各种性质的图像的存在，或提供一个它对[https://en.wikipedia.org/wiki/Almost_all 几乎所有]图形的属性的意义的严格的定义。
===小世界网络===
通过类比[https://en.wikipedia.org/wiki/Small-world_experiment 小世界现象]（通常称作[https://en.wikipedia.org/wiki/Six_degrees_of_separation 六度分离]），网络可以称为小世界网路。由匈牙利作家[https://en.wikipedia.org/wiki/Frigyes_Karinthy 卡林西·弗里杰什]于1929首次提出，[https://en.wikipedia.org/wiki/Stanley_Milgram 米尔格拉姆·斯坦利]于1967年通过实验检验小世界的假设，阐述为任意两个人之间所间隔得人不会超过六个，即相应的社会关系图的直径不大于6。[https://en.wikipedia.org/wiki/Duncan_J._Watts 瓦特·詹姆斯·邓肯]和[https://en.wikipedia.org/wiki/Steven_Strogatz 斯托加茨·斯蒂芬]于1988年发表了第一个小世界网络通过单个参数在随机图像和格子之间的平滑插值体现的模型。该模型证明了只有一小部分远程链接，正则图，在网络的直径大小成正比。该结论可以转换成一个“小世界”中任意两个顶点之间的边的平均数量是非常小的（精确地说，网络的尺寸呈对数增长），而聚类系数大。众所周知，各种各样的抽象图像都具有小世界的特性，例如：随机图像和无标度网络。此外，真实世界的网络，如[https://en.wikipedia.org/wiki/World_Wide_Web 万维网]和代谢网络也显示出这种特性。
在有关网络的科学文献中，“小世界”一词有一些含糊不清的地方。除了表示网络直径的大小，还可以表示小直径和[https://en.wikipedia.org/wiki/Clustering_coefficient 聚类系数]的同时出现。聚类系数时表示网络中三角形密度的指标。例如，系数随机图的聚类系数可以忽略不计，而现实网络的聚类系数往往要大得多。科学家指出，这种差异表明，在现实世界的网络中，边缘是相关的。
===无标度网络===
无标度网络是一个符合[https://en.wikipedia.org/wiki/Power_law 幂律]的[https://en.wikipedia.org/wiki/Degree_distribution 度分布][https://en.wikipedia.org/wiki/Complex_network 网络]，至少是渐近的。也就是说，网络中有k个节点的节点分式P(k)对于k为较大的值，如<ref>[https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f4884468c8f31235f968aed34bab4e769cce4f]</ref>，同时，γ参数的值通常是在2到3之间，虽然有时也可能在这些范围之外。
一些报告表明，许多网络是无标度的，尽管统计分析驳斥了其中许多的说法，并对其他说法提出了严重质疑。他们提出了[https://en.wikipedia.org/wiki/Preferential_attachment 优先依附]和[https://en.wikipedia.org/wiki/Fitness_model_(network_theory) 适应度模型]作为解释真实网络中推测的幂律分布的机制。
===网络科学===
网络科学是研究[https://en.wikipedia.org/wiki/Telecommunications_network 电信网络]、[https://en.wikipedia.org/wiki/Computer_network 计算机网络]、[https://en.wikipedia.org/wiki/Biological_network 生物网络]、认知[https://en.wikipedia.org/wiki/Semantic_network 语义网络]、[https://en.wikipedia.org/wiki/Social_network 社会网络]等[https://en.wikipedia.org/wiki/Complex_network 复杂网络]的学术领域。该领域考虑节点（或顶点）所代表的不同元素或参与者，以及元素或参与者之间为链接（或边）的连接。该领域采用的理论和方法包括数学的[https://en.wikipedia.org/wiki/Graph_theory 图论]、物理的[https://en.wikipedia.org/wiki/Statistical_mechanics 统计力学]、计算机科学的[https://en.wikipedia.org/wiki/Data_mining 数据挖掘]和[https://en.wikipedia.org/wiki/Information_visualization 信息可视化]、统计学的[https://en.wikipedia.org/wiki/Statistical_inference 推理建模]和社会学的[https://en.wikipedia.org/wiki/Social_structure 社会结构]。[https://en.wikipedia.org/wiki/National_Academies_of_Sciences,_Engineering,_and_Medicine 美国国家研究委员会] 将网络科学定义为“研究物理、生物和社会现象的网络表征，从而得出这些现象的预测模型”。
==Definition==
==表示与分类==
===邻接矩阵===
在[https://en.wikipedia.org/wiki/Graph_theory 图论]和[https://en.wikipedia.org/wiki/Computer_science 计算机科学]中，邻接矩阵是用于表示[https://en.wikipedia.org/wiki/Graph_(discrete_mathematics) 有限图]的[https://en.wikipedia.org/wiki/Square_matrix 方阵]。矩阵的元素表明图中顶点对是否相邻。在有限[https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)#Simple_graph 简图]的特殊情况下，邻接矩阵是一个（0，1）对角线[https://en.wikipedia.org/wiki/Logical_matrix 逻辑矩阵]上为0的矩阵。如果图是无向的，则邻接矩阵实对称的[https://en.wikipedia.org/wiki/Symmetric_matrix 对称图像]。[https://en.wikipedia.org/wiki/Spectral_graph_theory 谱图理论]研究了图与其邻接矩阵的[https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors 特征值]和[https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors 特征向量]之间的关系。邻接矩阵应与图的[https://en.wikipedia.org/wiki/Incidence_matrix 关联矩阵]分开，邻接矩阵是一种不同的矩阵表示，其元素表示顶点——对边是否有关联，[https://en.wikipedia.org/wiki/Degree_matrix 度矩阵]包含了每个[https://en.wikipedia.org/wiki/Vertex_(graph_theory) 顶点（图论）]的[https://en.wikipedia.org/wiki/Degree_(graph_theory) 度（图论）]的信息。
===链表表示===
===网络分类===
====有向网====
====加权网====
加权网是直接点之间的连结具有分配给它们权值的[https://en.wikipedia.org/wiki/Flow_network 网络]。网络是一个元素以某种方式连接在一起的系统（沃瑟曼和浮士德，于1994年）。系统的元素表示为节点（也称为参与者或顶点），交互元素之间的连接被称为连结、边、弧或链接。节点可以是神经元、个人、团体、组织、航站楼，甚至是国家，而连结可以是友谊、交流、合作、联盟、流动或贸易等形式。
在许多现实世界的网络中，并非网络中的所有连结都具有相同的容量。事实上，连结往往与权数有关，以他们的力度、强度、能力 （巴拉特.et al.,于2004年；霍瓦特，于2011年）区分。一方面，[https://en.wikipedia.org/wiki/Mark_Granovetter 格兰诺维·马克]（于1973年）认为，[https://en.wikipedia.org/wiki/Social_network 社交网络]中社交关系的强度是其持续时间、情感强度、亲密度和服务交换的函数。另一方面，非社会网络，权重通常指的是执行的函数关系，例如，在[https://en.wikipedia.org/wiki/Food_web 食物网]中，物种之间的碳流量（毫克/m²/天）（拉士克维施.et al.，于2003年），神经结的数量和缝隙连接神经网络（沃茨和斯特里伽茨，于1998年），或在交通网络中的流动的流量（奥普萨尔 et al.,于2008年）。
通过记录连结强度，可以创建一个加权网络（也称价值网络）。下面是这样一个网络的例子（权重也可以通过赋予变换不同的宽度来可视化）：
[[https://upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Weighted_network.svg/430px-Weighted_network.svg.png]]
加权网络也广泛应用于基因组和[https://en.wikipedia.org/wiki/Systems_biology 系统生物学]应用。（霍瓦特，于2011年）。例如，加权基因共表达网络分析（WGCNA）常用于构建基于基因表达（如[https://en.wikipedia.org/wiki/Microarray 微阵列]）数据的基因（或基因产物）之间的加权网络（张某和霍瓦特，于2005年）。一般而言，[https://en.wikipedia.org/wiki/Weighted_correlation_network_analysis 加权相关网络]可以用软阈值法定义变量之间的两两相关关系（如基因检测）。
====流网络====
在[https://en.wikipedia.org/wiki/Graph_theory 图论]中，流网络（也称运输网络）是一个[https://en.wikipedia.org/wiki/Directed_graph 有向图]，其中每条边都有一个容量，每条边接受一个流量。边缘上的流量不能超过边缘的流量。通常来说，在运筹学[https://en.wikipedia.org/wiki/Operations_research运筹学]中，有向图成为网络，那些顶点被称为节点，这些边称为弧。流量必须满足流入节点的流量等于流出节点的流量的限制，除非它是一个只有流出流或汇聚即只有流入流的源。一个网络可以用来模拟道路系统中的交通，需求环流，管道中的流体，电路中的电流，或任何类似的东西通过一个节点网络时的情况。
====空间网====
空间网络(有时也称为[https://en.wikipedia.org/wiki/Geometric_graph_theory 几何图形])是指[https://en.wikipedia.org/wiki/Vertex_(graph_theory) 顶点]或[https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms#edge 边]是与[https://en.wikipedia.org/wiki/Geometry 几何]对象相关联的空间元素的[https://en.wikipedia.org/wiki/Graph_(discrete_mathematics) 图像]，即节点位于具有一定[https://en.wikipedia.org/wiki/Metric_(mathematics) 度量]的空间中。最简单的数学实现是[https://en.wikipedia.org/wiki/Lattice_graph 格子]或随机[https://en.wikipedia.org/wiki/Random_geometric_graph 几何图形]，它们的节点在二维平面上均匀分布；如果[https://en.wikipedia.org/wiki/Euclidean_distance 欧几里得距离]小于给定的邻域半径，则连接一对节点。[https://en.wikipedia.org/wiki/Transport_network 交通和移动网络]、[https://en.wikipedia.org/wiki/Internet 互联网]、[https://en.wikipedia.org/wiki/Cellular_network 移动电话网络]、[https://en.wikipedia.org/wiki/Electrical_grid 电网]、[https://en.wikipedia.org/wiki/Social_network 社交和联系网络]以及[https://en.wikipedia.org/wiki/Neural_network 神经网络]都是底层空间相关的例子，而图的[https://en.wikipedia.org/wiki/Topology拓扑]本身并不包含所有信息。描述和理解空间网络的结构、弹性和演化对从城市主义到流行病学的许多不同领域都至关重要。
====二分网====
二分网是一类特殊的[https://en.wikipedia.org/wiki/Complex_network 复杂网络]，其节点被划分为两个集合X和Y，只允许不同集合中两个节点之间的连接。为了方便直接显示特定一组节点之间的关系结构，通常采用单模投影的方式压缩二分网。这意味着接下来的网络只包含两个集合中的任意一个节点，并且只有当两个X(或者Y)节点至少有一个公共相邻的Y(或者X)节点时，它们才会连接。
最简单的方法涉及到双方的网络投射到一个未加权的网络，但不考虑网络的拓扑结构或共享一个连接的频率对立的元素集。由于在这种情况下，双方的网络很大程度上与不同的结构可以有完全相同的一种模式表示,一个清晰的插图的原始网络拓扑通常需要使用一些加权法。[[https://upload.wikimedia.org/wikipedia/commons/f/f9/Bipartite_network_projection.png]]
==统计指标==
在统计和[https://en.wikipedia.org/wiki/Research_design 研究设计]中，指标是一种[https://en.wikipedia.org/wiki/Composite_measure 复合统计]——衡量一组具有代表性的单个数据点的变化，或者换句话说，综合多个[https://en.wikipedia.org/wiki/Indicator_(statistics) 指标]的复合度量。指标总结和排列特定的观察。
社会科学领域的许多数据都用各种指数表示，如[https://en.wikipedia.org/wiki/Global_Gender_Gap_Report 性别差距指数]、[https://en.wikipedia.org/wiki/Human_Development_Index 人类发展指数]或道琼斯工业[https://en.wikipedia.org/wiki/Dow_Jones_Industrial_Average 平均指数]。
各项指标通常权重相等，除非有一些理由违反(例如，如果两个项本质上反映了变量的相同方面，那么它们的权重可能为0.5)。
构建项目涉及四个步骤。首先，项目的选择应该基于它们的面部效度[https://en.wikipedia.org/wiki/Face_validity面部效度]、单维性、[https://en.wikipedia.org/wiki/Sensitivity_and_specificity 测量维度]的特异性程度以及它们的[https://en.wikipedia.org/wiki/Variance 方差]。项目之间应该有经验上的联系，这引向了检验它们的多元关系的第二步。第三，设计索引得分，这涉及到确定它们的得分范围和项目的权重。最后，需要对指标进行验证，这涉及到是否能够预测与构建中未使用的测量变量相关的指标。
===度===
在[https://en.wikipedia.org/wiki/Graph_theory 图论]中，[https://en.wikipedia.org/wiki/Graph_(discrete_mathematics) 图]的[https://en.wikipedia.org/wiki/Vertex_(graph_theory) 顶点]的度(或值)是指与该顶点[https://en.wikipedia.org/wiki/Incidence_matrix 关联的]、[https://en.wikipedia.org/wiki/Loop_(graph_theory) 循环数]为2次的[https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms#edge 边]的数量。一个顶点v是表示程度的度(v)或度诉一个图G的最大程度,用Δ(G)和最低程度的一个图表,用δ(G),最大和最小程度的顶点。在右边的图中，最大值是5，最小值是0。在一个[https://en.wikipedia.org/wiki/Regular_graph 普通的图]中，所有的度都是一样的，所以我们可以说这个是图的度。[[https://upload.wikimedia.org/wikipedia/commons/thumb/d/d6/UndirectedDegrees_%28Loop%29.svg/283px-UndirectedDegrees_%28Loop%29.svg.png]]
===平均路径长度===
平均路径长度是[https://en.wikipedia.org/wiki/Network_topology 拓扑网络]中的一个概念，定义为所有可能的网络[https://en.wikipedia.org/wiki/Node_(networking) 节点]对的最短路径上的平均步数。它是网络上信息或大众运输效率的一种度量。
===直径===
在[https://en.wikipedia.org/wiki/Geometry 几何学]中，[https://en.wikipedia.org/wiki/Circle 圆]的直径是任何穿过圆中心、端点在圆上的直[https://en.wikipedia.org/wiki/Line_segment 线段]。它也可以定义为圆的最长[https://en.wikipedia.org/wiki/Chord_(geometry) 弦]。这两个定义对于[https://en.wikipedia.org/wiki/Sphere 球体]的直径也是有效的。
在现代化的用法中，直径的长度也被称为直径。从这个意义上说，我们谈论的是直径（指距离）而不是直径(指的是直线本身)，因为一个圆或球体的所有直径都有相同的长度，这是[https://en.wikipedia.org/wiki/Radius 半径]r的两倍。
d = 2r⇒r = d / 2。
对于平面上的[https://en.wikipedia.org/wiki/Convex_set 凸形]，其直径定义为两条相切于其边界的[https://en.wikipedia.org/wiki/Parallel_(geometry) 平行线]之间可以形成的最大距离，而宽度通常定义为这种距离的最小值。这两个量都可以用[https://en.wikipedia.org/wiki/Rotating_caliper 旋转卡尺]有效地计算出来。对于一个[https://en.wikipedia.org/wiki/Curve_of_constant_width 等宽的曲线]，例如[https://en.wikipedia.org/wiki/Reuleaux_triangle 鲁洛三角形]，其宽度和直径是相同的，因为所有的平行切线的长度相同。
而对于[https://en.wikipedia.org/wiki/Ellipse 椭圆]，标准术语是不同的。椭圆的直径是任何穿过椭圆中心的弦。例如，[https://en.wikipedia.org/wiki/Conjugate_diameters 共轭直径]具有这样的性质:其中一个端点与椭圆的切线平行于另一个端点。最长的直径称为[https://en.wikipedia.org/wiki/Semi-major_and_semi-minor_axes 长轴]。
“直径”一词来源于[https://en.wikipedia.org/wiki/Ancient_Greek 希腊]διάμετρος(diametros)、“一圈直径”,从διά(dia)”,通过“和μέτρον(密特隆),“衡量”。[3]通常缩写DIA,dia,d,或⌀。
===集聚系数===
在[https://en.wikipedia.org/wiki/Graph_theory 图论]中，聚类系数是图中节点聚类的程度的度量。有证据表明，在大多数真实世界的网络中，特别是在[https://en.wikipedia.org/wiki/Social_network 社交网络]中，节点往往以相对高密度的联系为特征，形成紧密的群体;这种可能性往往大于两个节点之间随机建立平局的平均概率(霍兰德和莱因哈特，于1971年; 瓦茨和斯托盖茨，于1998年)。
这种方法有两种版本:全局和局部。全局版本的设计是为了给出网络中集群的总体指示，而局部版本则表示单个节点的嵌入性。
===相关性===
关联性是指一个主题与另一个主题之间的[https://en.wikipedia.org/wiki/Premise 联系]，在考虑第一个主题时考虑第二个主题是有作用的。关联的概念被研究在许多不同的领域，包括认知科学，逻辑，图[https://en.wikipedia.org/wiki/Library_and_information_science 图书馆信息科学]。然而，最根本的是，它属于[https://en.wikipedia.org/wiki/Epistemology 认识学]（也称认识论）的研究范围。不同的知识理论对被认为相关的东西有不同的含义，这些基本观点对所有其他领域也都有影响。
===模块性===
从广义上讲，模块化是指系统的组件可以分离和重组的程度，通常具有灵活性和多样化的使用优势。模块化的概念主要用于通过将系统分解为不同程度的相互依赖和独立性来降低复杂性，并“将每个部分的复杂性隐藏在抽象和接口之后”。然而，模块化的概念可以扩展到多个学科，每个学科都有自己的概念上的细微差别。尽管有这些细微的差别，关于模块化系统的一致主题还是出现了。
===节点中心性===
在[https://en.wikipedia.org/wiki/Graph_theory 图论]和[https://en.wikipedia.org/wiki/Network_theory 网络分析]中，中心性指标确定图中最重要的[https://en.wikipedia.org/wiki/Vertex_(graph_theory) 顶点]。应用程序包括识别[https://en.wikipedia.org/wiki/Social_network 社交网络]中最有影响力的人、[https://en.wikipedia.org/wiki/Internet 互联网]或[https://en.wikipedia.org/wiki/Transport_network 城市网络]中的关键基础设施节点和疾病的[https://en.wikipedia.org/wiki/Super-spreader 超级传播者]。中心性概念最早出现在[https://en.wikipedia.org/wiki/Social_network_analysis 社会网络分析]中，许多衡量中心性的术语反映了它们的[https://en.wikipedia.org/wiki/Sociology 社会学]渊源。它们不应该与[https://en.wikipedia.org/wiki/Node_influence_metric 节点影响度量]相混淆，节点影响度量试图量化网络中每个节点的影响。[[https://upload.wikimedia.org/wikipedia/commons/1/11/6_centrality_measures.png]]
==实证数据==
实证数据，也被称为感觉经验，是通过[https://en.wikipedia.org/wiki/Sense 感觉]获得的[https://en.wikipedia.org/wiki/Information 信息]，特别是通过[https://en.wikipedia.org/wiki/Observation 观察]和记录通过[https://en.wikipedia.org/wiki/Experiment 实验]的模式和行为。该词来自[https://en.wikipedia.org/wiki/Ancient_Greek 希腊]语的经验,ἐμπειρία(empeiria)。
在[https://en.wikipedia.org/wiki/Immanuel_Kant 康德·伊曼努尔]之后，哲学中通常把获得的知识称为[https://en.wikipedia.org/wiki/A_priori_and_a_posteriori 后验知识](与[https://en.wikipedia.org/wiki/A_priori_and_a_posteriori 先验]知识相反)。
==统计特征==
===可观测性===
统计量是一种可观察的[https://en.wikipedia.org/wiki/Random_variable 随机变量]，它不同于描述[https://en.wikipedia.org/wiki/Statistical_population 统计总体]特性的一般不可观察量的[https://en.wikipedia.org/wiki/Statistical_parameter 参数]，也不同于不可观察的随机变量，例如观察到的测量值和总体平均值之间的差值。如果整个总体可以观察到没有误差，一个参数只能精确地计算；例如，在一个完美的人口普查或[https://en.wikipedia.org/wiki/Standardized_test 标准化考试]的考生群体中。
统计学家经常考虑一个参数化的概率分[https://en.wikipedia.org/wiki/Probability_distribution 概率分布]布族[https://en.wikipedia.org/wiki/Parametric_family 参数族]，其中的任何成员都可能是人口中每个成员的某种可测量方面的分布，从中随机抽取一个样本。例如，这个参数可能是北美25岁男性的平均身高。测量了100个这样的人的样本的身高;这100个数字的平均值是一个统计数据。人口中所有成员的平均身高不是一个统计数据，除非这个数据也以某种方式被确定(例如通过测量人口中的每一个成员)。所有25岁北美男性的平均身高是一个参数，而不是一个统计数据。
===统计特性===
统计的重要潜在特性包括完整性、一致性、充分性、无偏性、最小均方误差、低方差、稳健性和计算便利性。
===信息的统计===
模型参数的统计信息可以用几种方法定义。最常见的是费雪信息，它是在由统计量引起的统计模型上定义的。也可以使用库尔贝克信息度量。
===无标度===
一个网络的度分布被称为无标度，即一个均匀随机选择的节点有一定数量链路度的概率，遵循一个被称为[https://en.wikipedia.org/wiki/Power_law 幂律]的特殊数学函数。幂律表明，这些网络的度分布没有特征尺寸。相比之下，具有单一的、规模定义明确的网络在某种程度上类似于网格，因为每个节点（大致）都在同一个程度。单一的规模网络的例子，包括[https://en.wikipedia.org/wiki/Erdős–Rényi_modelER Erdős-Renyi(ER)]随机图像和[https://en.wikipedia.org/wiki/Hypercube 超立方体]。产生尺度不变的度分布的网络模型有[https://en.wikipedia.org/wiki/Barabási–Albert_model 巴拉巴西-艾伯特模型]和[https://en.wikipedia.org/wiki/Fitness_model_(network_theory) 适应度模型]。在一个具有无标度度分布的网络里，一些顶点有大于平均梯度的阶的等级——这些顶点通常被称为枢纽，尽管这有点误导因为它没有固定的阈值，一个节点可以被视为一个枢纽。如果有这样的阈值，网络就不会是无标度。
人们对无标度网络的兴趣起始于1990年代末的关于幂律度分布在真实世界的网络如[https://en.wikipedia.org/wiki/World_Wide_Web 万维网]、[https://en.wikipedia.org/wiki/Autonomous_system_(Internet) 自制系统]网络（ASs)，一些互联网路由器网络、蛋白质相互作用网络、电子邮件网络等的发现报告。大部分在报告幂律时，在挑战严格的统计测试时，都失败了，但是，对于重尾度分布的较普遍看法——许多这样的网络确实表现出这种分布（在有限尺寸影响发生之前）——与人们所预期的边缘独立存在且随机存在（即假设它们遵循[https://en.wikipedia.org/wiki/Poisson_distribution 泊松分布]）。建立一个幂律度分布的网络有很多种不同的方法。尤尔过程[https://en.wikipedia.org/wiki/Yule–Simon_distribution尤尔过程]是幂律的典型生成过程，自1925年以来就为人所知。然而，由于其频繁的再创造，它的许多其他名字被人们所熟知，例如，[https://en.wikipedia.org/wiki/Herbert_A._Simon 西蒙·亚历山大·赫伯特]的直布陀罗原则、[https://en.wikipedia.org/wiki/Matthew_effect 马太效应]、累积优势，以及[https://en.wikipedia.org/wiki/Albert-László_Barabás i巴拉巴西]和艾伯特的幂律度分布的[https://en.wikipedia.org/wiki/Preferential_attachment 优先链接]。近期，[https://en.wikipedia.org/wiki/Hyperbolic_geometric_graph 双曲几何图形]被认为是另一种构建无标度网络的方法。
一些具有幂律度分布的网络（以及特定的其他类型的结构）可以高度抵抗垂直的随机删除——即绝大多数顶点仍然与一个巨大的指数连接在一起。这样的网络对旨在快速破坏网络的攻击也异常敏感。当图像是除度分布外均匀随机的时候，这些关键的顶点是最高阶的，因此在社会和通信网络涉及疾病和病毒的传播（自然和人工），以及时尚的传播（这两种都是由[https://en.wikipedia.org/wiki/Percolation 渗流]或[https://en.wikipedia.org/wiki/Branching_process 分支流程]建模的）。而随即图像（ER)节点间的平均距离为o(log N),其中N为节点数，无标度图像的距离为log log N。这种图像被称为超小世界网络。
===小世界===
这个小世界实验由[https://en.wikipedia.org/wiki/Stanley_Milgram 米尔格拉姆·斯坦利]和其他研究人员进行的几个实验组成，他们研究了美国人[https://en.wikipedia.org/wiki/Social_network 社交网络]的[https://en.wikipedia.org/wiki/Average_path_length 平均路径长度]。该研究具有开创性，它表明人类社会是一个以短路径长度为特征的[https://en.wikipedia.org/wiki/Small-world_network 小世界网络]。这些实验经常与“[https://en.wikipedia.org/wiki/Six_degrees_of_separation 六度分离]”联系在一起，尽管米尔格拉姆本人并没有使用这个术语。
===Rich Club系数===
Rich-club系数是[https://en.wikipedia.org/wiki/Graph_(discrete_mathematics) 图形]和[https://en.wikipedia.org/wiki/Complex_network 网络]的度量标准，旨在衡量连接良好的节点之间相互连接的程度。具有较高Rich-club系数的网络具有较强的Rich-club效应，且节点间的连接程度较高。这种效应已经在科学协作网络和航空运输网络中得到了测量和注意。[https://en.wikipedia.org/wiki/Biological_network#Protein-protein_interaction_networks 蛋白质相互作用网络]明显缺乏Rich-club系数。
===社区划分===
==基本模型==
===SW模型===
==Scale-free networks==
===BA模型===
艾伯特-巴拉巴西（BA）模型是一种使用[https://en.wikipedia.org/wiki/Preferential_attachment 优先连接]机制生成随机[https://en.wikipedia.org/wiki/Scale-free_network 无标度网络]的算法。一些自然的和人为的系统，包括[https://en.wikipedia.org/wiki/Internet 因特网]、[https://en.wikipedia.org/wiki/World_Wide_Web 万维网]、[https://en.wikipedia.org/wiki/Citation_analysis 引文网络]和一些[https://en.wikipedia.org/wiki/Social_network 社交网络]，被认为是近似无尺度的，并且肯定包含很少的节点(称为集线器)，与网络的其他节点相比具有异常高的程度。BA模型试图解释真实网络中存在这样的节点。该算法以发明者[https://en.wikipedia.org/wiki/Albert-László_Barabási 艾伯特·拉斯洛·巴拉巴西]和[https://en.wikipedia.org/wiki/Réka_Albert 艾伯特·瑞卡]的名字命名，它是一种更通用的模型的特例，称为[https://en.wikipedia.org/wiki/Price%27s_model 普利斯模型]。
===双曲几何模型===
有很多不同的伪球面在很大范围内都有一个恒定的负高斯曲率，其中[https://en.wikipedia.org/wiki/Pseudosphere 伪球面]是最著名的。
但是在其他模型上做双曲几何比较容易。
[[https://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Poincare_disc_hyperbolic_parallel_lines.svg/500px-Poincare_disc_hyperbolic_parallel_lines.svg.png]]
[[https://upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Hyperbolic_tiling_omnitruncated_3-7.png/500px-Hyperbolic_tiling_omnitruncated_3-7.png]]
双曲几何常用的[https://en.wikipedia.org/wiki/Mathematical_model 模型]有四种：[https://en.wikipedia.org/wiki/Klein_model 克莱恩模型]、[https://en.wikipedia.org/wiki/Poincaré_disk_model 庞加莱圆盘模型]、[https://en.wikipedia.org/wiki/Poincaré_half-plane_model 庞加莱半平面模型]和洛伦茨或[https://en.wikipedia.org/wiki/Hyperboloid_model 双曲面模型]。这些模型定义了满足双曲几何公理的双曲平面。尽管它们的名字，上面提到的前三个模型是由[https://en.wikipedia.org/wiki/Eugenio_Beltrami 贝尔特拉米]作为双曲空间模型引入的，而不是由[https://en.wikipedia.org/wiki/Henri_Poincaré 庞加莱]或[https://en.wikipedia.org/wiki/Felix_Klein 克莱恩]提出的。所有这些模型都可以扩展到更多的维度。
==网络动力学==
==参考文献==
<references />
==相关wiki==