网络科学

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具体内容

Network science is an academic field which studies complex networks such as telecommunication networks, computer networks, biological networks, cognitive and semantic networks, and social networks, considering distinct elements or actors represented by nodes (or vertices) and the connections between the elements or actors as links (or edges). The field draws on theories and methods including graph theory from mathematics, statistical mechanics from physics, data mining and information visualization from computer science, inferential modeling from statistics, and social structure from sociology. The United States National Research Council defines network science as "the study of network representations of physical, biological, and social phenomena leading to predictive models of these phenomena."

Network science is an academic field which studies complex networks such as telecommunication networks, computer networks, biological networks, cognitive and semantic networks, and social networks, considering distinct elements or actors represented by nodes (or vertices) and the connections between the elements or actors as links (or edges). The field draws on theories and methods including graph theory from mathematics, statistical mechanics from physics, data mining and information visualization from computer science, inferential modeling from statistics, and social structure from sociology. The United States National Research Council defines network science as "the study of network representations of physical, biological, and social phenomena leading to predictive models of these phenomena."[1]

网络科学Network Science是一个研究复杂网络的学术领域，如通讯网络，计算机网络，生物网络，认知和语义网络，以及社会网络，主要由节点（顶点）表示不同元素或参与者，由连边表示元素和参与者之间的联系。这个领域的理论和方法包括数学的图理论，物理学的统计力学，计算机科学的数据挖掘和信息可视化，统计学的推理建模，社会学的社会结构。 美国国家研究委员会The United States National Research Council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。” [1]

背景和历史

The study of networks has emerged in diverse disciplines as a means of analyzing complex relational data. The earliest known paper in this field is the famous Seven Bridges of Königsberg written by Leonhard Euler in 1736. Euler's mathematical description of vertices and edges was the foundation of graph theory, a branch of mathematics that studies the properties of pairwise relations in a network structure. The field of graph theory continued to develop and found applications in chemistry (Sylvester, 1878).

The study of networks has emerged in diverse disciplines as a means of analyzing complex relational data. The earliest known paper in this field is the famous Seven Bridges of Königsberg written by Leonhard Euler in 1736. Euler's mathematical description of vertices and edges was the foundation of graph theory, a branch of mathematics that studies the properties of pairwise relations in a network structure. The field of graph theory continued to develop and found applications in chemistry (Sylvester, 1878).

网络研究作为分析复杂关系数据的一种手段已经出现在不同的学科中。 在这个领域已知最早的论文是1736年 欧拉 Leonhard Euler 写的著名的柯尼斯堡七桥问题Seven Bridges of Königsberg。 欧拉对顶点和边的数学描述是图论的基础，这是一个研究网络结构中成对关系的性质的数学分支。 图理论的领域继续发展，并在化学中得到应用(Sylvester，1878)。

Dénes Kőnig, a Hungarian mathematician and professor, wrote the first book in Graph Theory, entitled "Theory of finite and infinite graphs", in 1936

Dénes Kőnig, a Hungarian mathematician and professor, wrote the first book in Graph Theory, entitled "Theory of finite and infinite graphs", in 1936 [2] 匈牙利数学家兼教授Dénes Kőnig在1936年写了图论的第一本书《有限图和无限图理论》Theory of finite and infinite graphs。

Moreno's sociogram of a 1st grade class.

In the 1930s Jacob Moreno, a psychologist in the Gestalt tradition, arrived in the United States. He developed the sociogram and presented it to the public in April 1933 at a convention of medical scholars. Moreno claimed that "before the advent of sociometry no one knew what the interpersonal structure of a group 'precisely' looked like" (Moreno, 1953). The sociogram was a representation of the social structure of a group of elementary school students. The boys were friends of boys and the girls were friends of girls with the exception of one boy who said he liked a single girl. The feeling was not reciprocated. This network representation of social structure was found so intriguing that it was printed in The New York Times (April 3, 1933, page 17). The sociogram has found many applications and has grown into the field of social network analysis.

In the 1930s Jacob Moreno, a psychologist in the Gestalt tradition, arrived in the United States. He developed the sociogram and presented it to the public in April 1933 at a convention of medical scholars. Moreno claimed that "before the advent of sociometry no one knew what the interpersonal structure of a group 'precisely' looked like" (Moreno, 1953). The sociogram was a representation of the social structure of a group of elementary school students. The boys were friends of boys and the girls were friends of girls with the exception of one boy who said he liked a single girl. The feeling was not reciprocated. This network representation of social structure was found so intriguing that it was printed in The New York Times (April 3, 1933, page 17). The sociogram has found many applications and has grown into the field of social network analysis.

20世纪30年代，格式塔传统的心理学家雅各布·莫雷诺Jacob Moreno来到美国。 在1933年4月的一次医学学者大会上，他在大会上展示了他制作的社网图 sociogram。 莫雷诺声称“在社会计量学出现之前，没有人知道一个群体的人际关系结构‘精确地’看起来像什么”(莫雷诺，1953)。 这张社网图展示一群小学生的社会结构。男孩是男孩的朋友，女孩是女孩的朋友，只有一个男孩说他喜欢另一个单身女孩但是没有得到回应。 这个网络展示的社会结构非常有趣，也被刊登在《纽约时报》上(1933年4月3日，第17页)。 社网图已经有许多的应用场景，并且已经发展成为社会网络分析social network analysis的一个子领域。

Probabilistic theory in network science developed as an offshoot of graph theory with Paul Erdős and Alfréd Rényi's eight famous papers on random graphs. For social networks the exponential random graph model or p* is a notational framework used to represent the probability space of a tie occurring in a social network. An alternate approach to network probability structures is the network probability matrix, which models the probability of edges occurring in a network, based on the historic presence or absence of the edge in a sample of networks.

Probabilistic theory in network science developed as an offshoot of graph theory with Paul Erdős and Alfréd Rényi's eight famous papers on random graphs. For social networks the exponential random graph model or p* is a notational framework used to represent the probability space of a tie occurring in a social network. An alternate approach to network probability structures is the network probability matrix, which models the probability of edges occurring in a network, based on the historic presence or absence of the edge in a sample of networks.

20世纪90年代，在Paul Erdős和Alfréd Rényi发表了8篇关于随机图的著名论文之后，网络科学中的概率论 Probabilistic theory 作为图论的一个分支发展起来了。 对于社交网络social network来说，指数随机图模型或p* (是一个记号框架），用来表示在一个社交网络中连边在概率空间发生的概率。 网络概率结构的另一种替代表示方法是网络概率矩阵，它根据网络样本中边的历史信息来计算这条边在网络中出现的概率。

In 1998, David Krackhardt and Kathleen Carley introduced the idea of a meta-network with the PCANS Model. They suggest that "all organizations are structured along these three domains, Individuals, Tasks, and Resources". Their paper introduced the concept that networks occur across multiple domains and that they are interrelated. This field has grown into another sub-discipline of network science called dynamic network analysis.

In 1998, David Krackhardt and Kathleen Carley introduced the idea of a meta-network with the PCANS Model. They suggest that "all organizations are structured along these three domains, Individuals, Tasks, and Resources". Their paper introduced the concept that networks occur across multiple domains and that they are interrelated. This field has grown into another sub-discipline of network science called dynamic network analysis.

1998年，David Krackhardt和Kathleen Carley提出了用PCANS模型构建元网络meta-network的想法。 他们建议所有的组织都通过“个人、任务和资源”来构建。 他们的论文介绍了新的网络概念，即网络发生在多个领域的且相互关联。 这个领域已经发展也成为网络科学的另一个子学科，叫做动态网络分析dynamic network analysis。

More recently other network science efforts have focused on mathematically describing different network topologies. Duncan Watts reconciled empirical data on networks with mathematical representation, describing the small-world network. Albert-László Barabási and Reka Albert developed the scale-free network which is a loosely defined network topology that contains hub vertices with many connections, that grow in a way to maintain a constant ratio in the number of the connections versus all other nodes. Although many networks, such as the internet, appear to maintain this aspect, other networks have long tailed distributions of nodes that only approximate scale free ratios.

More recently other network science efforts have focused on mathematically describing different network topologies. Duncan Watts reconciled empirical data on networks with mathematical representation, describing the small-world network. Albert-László Barabási and Reka Albert developed the scale-free network which is a loosely defined network topology that contains hub vertices with many connections, that grow in a way to maintain a constant ratio in the number of the connections versus all other nodes. Although many networks, such as the internet, appear to maintain this aspect, other networks have long tailed distributions of nodes that only approximate scale free ratios.

最近其他网络科学的研究致力于用数学的方式描述不同网络的拓扑结构。邓肯·瓦茨Duncan Watts 将网络上的经验数据与数学表达相结合，描述了小世界网络。 艾伯特-拉斯洛·巴拉巴西 Albert-László Barabási 和 Reka Albert发现了无标度网络，简单说就是包含Hub中心节点（连边数量很多的节点），且连边的数量和节点数量呈现常数比率的增长方式，即一个网络上节点的度（节点所在连边的数量）分布服从幂指数为2和3之间的幂律分布。尽管许多网络，比如互联网，似乎保持了这样的特性，但是其他网络的节点分布表现出长尾特性，仅仅是接近无标度比例。

思无涯咿呀咿呀（讨论）Albert-László Barabási and Reka Albert developed the scale-free network which is a loosely defined network topology that contains hub vertices with many connections, that grow in a way to maintain a constant ratio in the number of the connections versus all other nodes. 思无涯咿呀咿呀（讨论）这一段话怎么理解。

国防部倡议

The U.S. military first became interested in network-centric warfare as an operational concept based on network science in 1996. John A. Parmentola, the U.S. Army Director for Research and Laboratory Management, proposed to the Army’s Board on Science and Technology (BAST) on December 1, 2003 that Network Science become a new Army research area. The BAST, the Division on Engineering and Physical Sciences for the National Research Council (NRC) of the National Academies, serves as a convening authority for the discussion of science and technology issues of importance to the Army and oversees independent Army-related studies conducted by the National Academies. The BAST conducted a study to find out whether identifying and funding a new field of investigation in basic research, Network Science, could help close the gap between what is needed to realize Network-Centric Operations and the current primitive state of fundamental knowledge of networks.

The U.S. military first became interested in network-centric warfare as an operational concept based on network science in 1996. John A. Parmentola, the U.S. Army Director for Research and Laboratory Management, proposed to the Army’s Board on Science and Technology (BAST) on December 1, 2003 that Network Science become a new Army research area. The BAST, the Division on Engineering and Physical Sciences for the National Research Council (NRC) of the National Academies, serves as a convening authority for the discussion of science and technology issues of importance to the Army and oversees independent Army-related studies conducted by the National Academies. The BAST conducted a study to find out whether identifying and funding a new field of investigation in basic research, Network Science, could help close the gap between what is needed to realize Network-Centric Operations and the current primitive state of fundamental knowledge of networks.

1996年，美国军方第一次对网络中心战感兴趣，将其作为一个基于网络科学的军事概念。 2003年12月1日，美国陆军研究和实验室管理主任 John A. Parmentola 向陆军科学和技术委员会提议，网络科学成为陆军的一个新的研究领域。科学和技术委员会是美国国家学院国家研究委员会工程和物理科学司的下属部门，主要讨论对陆军重要的科学和技术问题，并且监督美国国家学院进行的与陆军相关的独立研究。 科学和技术委员会进行了一项研究，他们想通过确定和资助一个新的研究领域的基础研究：网络科学，希望可以帮助找到网络中心战所必须的知识，从而缩小实现网络中心战与当前网络科学理论基础的差距。

As a result, the BAST issued the NRC study in 2005 titled Network Science (referenced above) that defined a new field of basic research in Network Science for the Army. Based on the findings and recommendations of that study and the subsequent 2007 NRC report titled Strategy for an Army Center for Network Science, Technology, and Experimentation, Army basic research resources were redirected to initiate a new basic research program in Network Science. To build a new theoretical foundation for complex networks, some of the key Network Science research efforts now ongoing in Army laboratories address:

因此，科学和技术委员在2005年发布了NRC的研究报告，题为《网络科学定义了美国陆军网络科学基础研究新领域》。 基于这项研究的发现和建议，以及随后2007年 NRC 题为基于网络科学、技术和实战的战略研究报告，陆军基础研究资源被重新投入在网络科学的一个新的基础研究项目。为了建立复杂网络的全新的理论基础，美国陆军实验室正在进行的一些关键的网络科学研究工作致力于:

• Mathematical models of network behavior to predict performance with network size, complexity, and environment
• Optimized human performance required for network-enabled warfare
• Networking within ecosystems and at the molecular level in cells.

• 网络行为的数学模型，以预测网络规模、复杂度和环境的表现
• 网络化战争network-enabled warfare所需的最佳人类表现
• 生态系统中基于细胞分子层面的网络建模

As initiated in 2004 by Frederick I. Moxley with support he solicited from David S. Alberts, the Department of Defense helped to establish the first Network Science Center in conjunction with the U.S. Army at the United States Military Academy (USMA). Under the tutelage of Dr. Moxley and the faculty of the USMA, the first interdisciplinary undergraduate courses in Network Science were taught to cadets at West Point. In order to better instill the tenets of network science among its cadre of future leaders, the USMA has also instituted a five-course undergraduate minor in Network Science.

正如2004年 David S. Alberts支持Frederick I. Moxley那样，国防部帮助美国陆军一起在美国军事学院(USMA)建立了第一个网络科学中心。 在Moxley博士和美国军事学院(USMA)研究员的指导下，西点军校开设了首个网络科学本科生课程。为了更好地向其未来领导干部灌输网络科学的信条，美国军事学院还在网络科学下设置了一个5节课本科生辅修课程。

In 2006, the U.S. Army and the United Kingdom (UK) formed the Network and Information Science International Technology Alliance, a collaborative partnership among the Army Research Laboratory, UK Ministry of Defense and a consortium of industries and universities in the U.S. and UK. The goal of the alliance is to perform basic research in support of Network- Centric Operations across the needs of both nations.

2006年，美国陆军和英国成立了网络和信息科学国际技术联盟，这是美国陆军研究实验室、英国国防部以及美国和英国的工业和大学联盟之间的合作伙伴关系。 该联盟的目标是进行基础研究，以支持两国需要的网络中心作战。

In 2009, the U.S. Army formed the Network Science CTA, a collaborative research alliance among the Army Research Laboratory, CERDEC, and a consortium of about 30 industrial R&D labs and universities in the U.S. The goal of the alliance is to develop a deep understanding of the underlying commonalities among intertwined social/cognitive, information, and communications networks, and as a result improve our ability to analyze, predict, design, and influence complex systems interweaving many kinds of networks.

Subsequently, as a result of these efforts, the U.S. Department of Defense has sponsored numerous research projects that support Network Science. 2009年，美国陆军成立了网络科学 CTA，这是一个由陆军研究实验室、 CERDEC 和美国大约30个工业研发实验室和大学组成的联盟之间的合作研究联盟。 这个联盟的目标是深入理解交织在一起的社会 / 认知、信息和通信网络之间的潜在共性，从而提高我们分析、预测、设计和影响交织在多种网络中的复杂系统的能力。

随后，作为这些努力的结果，美国国防部赞助了许多支持网络科学的研究项目。

Network properties

Often, networks have certain attributes that can be calculated to analyze the properties & characteristics of the network. The behavior of these network properties often define network models and can be used to analyze how certain models contrast to each other. Many of the definitions for other terms used in network science can be found in Glossary of graph theory.

Size

The size of a network can refer to the number of nodes [math]\displaystyle{ N }[/math] or, less commonly, the number of edges [math]\displaystyle{ E }[/math] which (for connected graphs with no multi-edges) can range from [math]\displaystyle{ N-1 }[/math] (a tree) to [math]\displaystyle{ E_{\max} }[/math] (a complete graph). In the case of a simple graph (a network in which at most one (undirected) edge exists between each pair of vertices, and in which no vertices connect to themselves), we have [math]\displaystyle{ E_{\max}=\tbinom N2=N(N-1)/2 }[/math]; for directed graphs (with no self-connected nodes), [math]\displaystyle{ E_{\max}=N(N-1) }[/math]; for directed graphs with self-connections allowed, [math]\displaystyle{ E_{\max}=N^2 }[/math]. In the circumstance of a graph within which multiple edges may exist between a pair of vertices, [math]\displaystyle{ E_{\max}=\infty }[/math].

规模

网络的规模可以由节点的个数[math]\displaystyle{ N }[/math]，或者，少数情况下，连边的数量[math]\displaystyle{ E }[/math]（对于没有重边的连通图），连边的个数E的范围一般是从[math]\displaystyle{ N-1 }[/math] （看做是一个树）到[math]\displaystyle{ E_{\max} }[/math] （看做是一个完全图）。在简单图的例子中（网络中在每对节点之间至多存在一条(无向)边，并且没有节点连向自己），可以计算[math]\displaystyle{ E_{\max}=\tbinom N2=N(N-1)/2 }[/math]；对于有向图（没有自环self-connected的节点），[math]\displaystyle{ E_{\max}=N(N-1) }[/math]；对于有向图且允许存在自环的节点，[math]\displaystyle{ E_{\max}=N^2 }[/math].还有另外一种特殊情况就是一对节点之间存在重边， [math]\displaystyle{ E_{\max}=\infty }[/math].

Density

The density [math]\displaystyle{ D }[/math] of a network is defined as a ratio of the number of edges [math]\displaystyle{ E }[/math] to the number of possible edges in a network with [math]\displaystyle{ N }[/math] nodes, given (in the case of simple graphs) by the binomial coefficient [math]\displaystyle{ \tbinom N2 }[/math], giving [math]\displaystyle{ D =\frac{E-(N-1)}{Emax - (N-1)} = \frac{2(E-N+1)}{N(N-3)+2} }[/math] Another possible equation is [math]\displaystyle{ D = \frac{T-2N+2}{N(N-3)+2}, }[/math] whereas the ties [math]\displaystyle{ T }[/math] are unidirectional (Wasserman & Faust 1994). This gives a better overview over the network density, because unidirectional relationships can be measured.

Planar Network Density

The density [math]\displaystyle{ D }[/math] of a network, where there is no intersection between edges, is defined as a ratio of the number of edges [math]\displaystyle{ E }[/math] to the number of possible edges in a network with [math]\displaystyle{ N }[/math] nodes, given by a graph with no intersecting edges [math]\displaystyle{ (E_{\max}=3N-6) }[/math], giving [math]\displaystyle{ D = \frac{E-N+1}{2N-5}. }[/math]

Average degree

The degree [math]\displaystyle{ k }[/math] of a node is the number of edges connected to it. Closely related to the density of a network is the average degree, [math]\displaystyle{ \langle k\rangle = \tfrac{2E}{N} }[/math] (or, in the case of directed graphs, [math]\displaystyle{ \langle k\rangle = \tfrac{E}{N} }[/math], the former factor of 2 arising from each edge in an undirected graph contributing to the degree of two distinct vertices). In the ER random graph model ([math]\displaystyle{ G(N,p) }[/math]) we can compute the expected value of [math]\displaystyle{ \langle k \rangle }[/math] (equal to the expected value of [math]\displaystyle{ k }[/math] of an arbitrary vertex): a random vertex has [math]\displaystyle{ N-1 }[/math] other vertices in the network available, and with probability [math]\displaystyle{ p }[/math], connects to each. Thus, [math]\displaystyle{ \mathbb{E}[\langle k \rangle]=\mathbb{E}[k]= p(N-1) }[/math].

Average shortest path length (or characteristic path length)

The average shortest path length is calculated by finding the shortest path between all pairs of nodes, and taking the average over all paths of the length thereof (the length being the number of intermediate edges contained in the path, i.e., the distance [math]\displaystyle{ d_{u,v} }[/math] between the two vertices [math]\displaystyle{ u,v }[/math] within the graph). This shows us, on average, the number of steps it takes to get from one member of the network to another. The behavior of the expected average shortest path length (that is, the ensemble average of the average shortest path length) as a function of the number of vertices [math]\displaystyle{ N }[/math] of a random network model defines whether that model exhibits the small-world effect; if it scales as [math]\displaystyle{ O(\ln N) }[/math], the model generates small-world nets. For faster-than-logarithmic growth, the model does not produce small worlds. The special case of [math]\displaystyle{ O(\ln\ln N) }[/math] is known as ultra-small world effect.

Diameter of a network

As another means of measuring network graphs, we can define the diameter of a network as the longest of all the calculated shortest paths in a network. It is the shortest distance between the two most distant nodes in the network. In other words, once the shortest path length from every node to all other nodes is calculated, the diameter is the longest of all the calculated path lengths. The diameter is representative of the linear size of a network. If node A-B-C-D are connected, going from A->D this would be the diameter of 3 (3-hops, 3-links).模板:Fact

Clustering coefficient

The clustering coefficient is a measure of an "all-my-friends-know-each-other" property. This is sometimes described as the friends of my friends are my friends. More precisely, the clustering coefficient of a node is the ratio of existing links connecting a node's neighbors to each other to the maximum possible number of such links. The clustering coefficient for the entire network is the average of the clustering coefficients of all the nodes. A high clustering coefficient for a network is another indication of a small world.

The clustering coefficient of the [math]\displaystyle{ i }[/math]'th node is

[math]\displaystyle{ C_i = {2e_i\over k_i{(k_i - 1)}}\,, }[/math]

where [math]\displaystyle{ k_i }[/math] is the number of neighbours of the [math]\displaystyle{ i }[/math]'th node, and [math]\displaystyle{ e_i }[/math] is the number of connections between these neighbours. The maximum possible number of connections between neighbors is, then,

[math]\displaystyle{ {\binom {k}{2}} = {{k(k-1)}\over 2}\,. }[/math]

From a probabalistic standpoint, the expected local clustering coefficient is the likelihood of a link existing between two arbitrary neighbors of the same node.

Connectedness

The way in which a network is connected plays a large part into how networks are analyzed and interpreted. Networks are classified in four different categories:

• Clique/Complete Graph: a completely connected network, where all nodes are connected to every other node. These networks are symmetric in that all nodes have in-links and out-links from all others.
• Giant Component: A single connected component which contains most of the nodes in the network.
• Weakly Connected Component: A collection of nodes in which there exists a path from any node to any other, ignoring directionality of the edges.
• Strongly Connected Component: A collection of nodes in which there exists a directed path from any node to any other.

Node centrality

Main article: Centrality

Centrality indices produce rankings which seek to identify the most important nodes in a network model. Different centrality indices encode different contexts for the word "importance." The betweenness centrality, for example, considers a node highly important if it form bridges between many other nodes. The eigenvalue centrality, in contrast, considers a node highly important if many other highly important nodes link to it. Hundreds of such measures have been proposed in the literature.

Centrality indices are only accurate for identifying the most central nodes. The measures are seldom, if ever, meaningful for the remainder of network nodes. Also, their indications are only accurate within their assumed context for importance, and tend to "get it wrong" for other contexts. For example, imagine two separate communities whose only link is an edge between the most junior member of each community. Since any transfer from one community to the other must go over this link, the two junior members will have high betweenness centrality. But, since they are junior, (presumably) they have few connections to the "important" nodes in their community, meaning their eigenvalue centrality would be quite low.

The concept of centrality in the context of static networks was extended, based on empirical and theoretical research, to dynamic centrality in the context of time-dependent and temporal networks.

Node influence

Main article: Node influence metric

Limitations to centrality measures have led to the development of more general measures. Two examples are the accessibility, which uses the diversity of random walks to measure how accessible the rest of the network is from a given start node, and the expected force, derived from the expected value of the force of infection generated by a node. Both of these measures can be meaningfully computed from the structure of the network alone.

Network models

Network models serve as a foundation to understanding interactions within empirical complex networks. Various random graph generation models produce network structures that may be used in comparison to real-world complex networks.

Erdős–Rényi random graph model

文件:ER model.svg

This Erdős–Rényi model is generated with N = 4 nodes. For each edge in the complete graph formed by all N nodes, a random number is generated and compared to a given probability. If the random number is less than p, an edge is formed on the model.

The Erdős–Rényi model, named for Paul Erdős and Alfréd Rényi, is used for generating random graphs in which edges are set between nodes with equal probabilities. It can be used in the probabilistic method to prove the existence of graphs satisfying various properties, or to provide a rigorous definition of what it means for a property to hold for almost all graphs.

To generate an Erdős–Rényi model [math]\displaystyle{ G(n, p) }[/math] two parameters must be specified: the total number of nodes n and the probability p that a random pair of nodes has an edge.

Because the model is generated without bias to particular nodes, the degree distribution is binomial: for a randomly chosen vertex [math]\displaystyle{ v }[/math],

[math]\displaystyle{ P(\deg(v) = k) = {n-1\choose k} p^k (1-p)^{n-1-k}. }[/math]

In this model the clustering coefficient is 0 a.s. The behavior of [math]\displaystyle{ G(n, p) }[/math] can be broken into three regions.

Subcritical [math]\displaystyle{ n p \lt 1 }[/math]: All components are simple and very small, the largest component has size [math]\displaystyle{ |C_1| = O(\log n) }[/math];

Critical [math]\displaystyle{ n p = 1 }[/math]: [math]\displaystyle{ |C_1| = O(n^\frac{2}{3}) }[/math];

Supercritical [math]\displaystyle{ n p \gt 1 }[/math]:[math]\displaystyle{ |C_1| \approx yn }[/math] where [math]\displaystyle{ y = y(n p) }[/math] is the positive solution to the equation [math]\displaystyle{ e^{-p n y }=1-y }[/math].

The largest connected component has high complexity. All other components are simple and small [math]\displaystyle{ |C_2| = O(\log n) }[/math].

Configuration model

The configuration model takes a degree sequence or degree distribution (which subsequently is used to generate a degree sequence) as the input, and produces randomly connected graphs in all respects other than the degree sequence. This means that for a given choice of the degree sequence, the graph is chosen uniformly at random from the set of all graphs that comply with this degree sequence. The degree [math]\displaystyle{ k }[/math] of a randomly chosen vertex is an independent and identically distributed random variable with integer values. When [math]\displaystyle{ \mathbb E [k^2] - 2 \mathbb E [k]\gt 0 }[/math], the configuration graph contains the giant connected component, which has infinite size. The rest of the components have finite sizes, which can be quantified with the notion of the size distribution. The probability [math]\displaystyle{ w(n) }[/math] that a randomly sampled node is connected to a component of size [math]\displaystyle{ n }[/math] is given by convolution powers of the degree distribution:[math]\displaystyle{ w(n)=\begin{cases} \frac{\mathbb E [k]}{n-1} u_1^{*n}(n-2),& n\gt 1, \\ u(0) & n=1, \end{cases} }[/math]where [math]\displaystyle{ u(k) }[/math] denotes the degree distribution and [math]\displaystyle{ u_1(k)=\frac{(k+1)u(k+1)}{\mathbb E[k]} }[/math]. The giant component can be destroyed by randomly removing the critical fraction [math]\displaystyle{ p_c }[/math] of all edges. This process is called percolation on random networks. When the second moment of the degree distribution is finite, [math]\displaystyle{ \mathbb E [k^2]\lt \infty }[/math], this critical edge fraction is given by [math]\displaystyle{ p_c=1-\frac{\mathbb E[k]}{ \mathbb E [k^2] - \mathbb E[k]} }[/math], and the average vertex-vertex distance [math]\displaystyle{ l }[/math] in the giant component scales logarithmically with the total size of the network, [math]\displaystyle{ l = O(\log N) }[/math].

In the directed configuration model, the degree of a node is given by two numbers, in-degree [math]\displaystyle{ k_\text{in} }[/math] and out-degree [math]\displaystyle{ k_\text{out} }[/math], and consequently, the degree distribution is two-variate. The expected number of in-edges and out-edges coincides, so that [math]\displaystyle{ \mathbb E [k_\text{in}]=\mathbb E [k_\text{out}] }[/math]. The directed configuration model contains the giant component iff[math]\displaystyle{ 2\mathbb{E}[k_\text{in}] \mathbb{E}[k_\text{in}k_\text{out}] - \mathbb{E}[k_\text{in}] \mathbb{E}[k_\text{out}^2] - \mathbb{E}[k_\text{in}] \mathbb{E}[k_\text{in}^2] +\mathbb{E}[k_\text{in}^2] \mathbb{E}[k_\text{out}^2] - \mathbb{E}[k_\text{in} k_\text{out}]^2 \gt 0. }[/math]Note that [math]\displaystyle{ \mathbb E [k_\text{in}] }[/math] and [math]\displaystyle{ \mathbb E [k_\text{out}] }[/math] are equal and therefore interchangeable in the latter inequality. The probability that a randomly chosen vertex belongs to a component of size [math]\displaystyle{ n }[/math] is given by:[math]\displaystyle{ h_\text{in}(n)=\frac{\mathbb E[k_{in}]}{n-1} \tilde u_\text{in}^{*n}(n-2), \;n\gt 1, \; \tilde u_\text{in}=\frac{k_\text{in}+1}{\mathbb E[k_\text{in}]}\sum\limits_{k_\text{out}\geq 0}u(k_\text{in}+1,k_\text{out}), }[/math]for in-components, and

[math]\displaystyle{ h_\text{out}(n)=\frac{\mathbb E[k_\text{out}]}{n-1} \tilde u_\text{out}^{*n}(n-2), \;n\gt 1, \;\tilde u_\text{out}=\frac{k_\text{out}+1}{\mathbb E[k_\text{out}]}\sum\limits_{k_\text{in}\geq0}u(k_\text{in},k_\text{out}+1), }[/math]

for out-components.

Watts–Strogatz small world model

文件:Watts-Strogatz-rewire.png

The Watts and Strogatz model uses the concept of rewiring to achieve its structure. The model generator will iterate through each edge in the original lattice structure. An edge may changed its connected vertices according to a given rewiring probability. [math]\displaystyle{ \langle k\rangle = 4 }[/math] in this example.

The Watts and Strogatz model is a random graph generation model that produces graphs with small-world properties.

An initial lattice structure is used to generate a Watts–Strogatz model. Each node in the network is initially linked to its [math]\displaystyle{ \langle k\rangle }[/math] closest neighbors. Another parameter is specified as the rewiring probability. Each edge has a probability [math]\displaystyle{ p }[/math] that it will be rewired to the graph as a random edge. The expected number of rewired links in the model is [math]\displaystyle{ pE = pN\langle k\rangle/2 }[/math].

As the Watts–Strogatz model begins as non-random lattice structure, it has a very high clustering coefficient along with high average path length. Each rewire is likely to create a shortcut between highly connected clusters. As the rewiring probability increases, the clustering coefficient decreases slower than the average path length. In effect, this allows the average path length of the network to decrease significantly with only slightly decreases in clustering coefficient. Higher values of p force more rewired edges, which in effect makes the Watts–Strogatz model a random network.

Barabási–Albert (BA) preferential attachment model

The Barabási–Albert model is a random network model used to demonstrate a preferential attachment or a "rich-get-richer" effect. In this model, an edge is most likely to attach to nodes with higher degrees.The network begins with an initial network of m₀ nodes. m₀ ≥ 2 and the degree of each node in the initial network should be at least 1, otherwise it will always remain disconnected from the rest of the network.

The Barabási–Albert model is a random network model used to demonstrate a preferential attachment or a "rich-get-richer" effect. In this model, an edge is most likely to attach to nodes with higher degrees. The network begins with an initial network of m0 nodes. m0 ≥ 2 and the degree of each node in the initial network should be at least 1, otherwise it will always remain disconnected from the rest of the network.

Barabási–Albert (BA) 优先链接模型

BA模型是一个随机网络模型，用于说明偏好依附效应（优先链接）preferential attachment或“富人越富”效应。 在这个模型中，边最有可能附着在度数较高的节点上。 这个网络从一个 m0节点的初始网络开始。 M0≥2，初始网络中每个节点的度至少为1，否则它将始终与网络的其余部分断开。

In the BA model, new nodes are added to the network one at a time. Each new node is connected to [math]\displaystyle{ m }[/math] existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Formally, the probability p_i that the new node is connected to node i is

[math]\displaystyle{ p_i = \frac{k_i}{\sum_j k_j}, }[/math]

where k_i is the degree of node i. Heavily linked nodes ("hubs") tend to quickly accumulate even more links, while nodes with only a few links are unlikely to be chosen as the destination for a new link. The new nodes have a "preference" to attach themselves to the already heavily linked nodes.

文件:Barabasi-albert model degree distribution.svg

The degree distribution of the BA Model, which follows a power law. In loglog scale the power law function is a straight line.

The degree distribution resulting from the BA model is scale free, in particular, it is a power law of the form:

[math]\displaystyle{ P(k)\sim k^{-3} \, }[/math]

Hubs exhibit high betweenness centrality which allows short paths to exist between nodes. As a result, the BA model tends to have very short average path lengths. The clustering coefficient of this model also tends to 0. While the diameter, D, of many models including the Erdős Rényi random graph model and several small world networks is proportional to log N, the BA model exhibits D~loglogN (ultrasmall world). Note that the average path length scales with N as the diameter.

Mediation-driven attachment (MDA) model

In the mediation-driven attachment (MDA) model in which a new node coming with [math]\displaystyle{ m }[/math] edges picks an existing connected node at random and then connects itself not with that one but with [math]\displaystyle{ m }[/math] of its neighbors chosen also at random. The probability [math]\displaystyle{ \Pi(i) }[/math] that the node [math]\displaystyle{ i }[/math] of the existing node picked is

[math]\displaystyle{ \Pi(i) = \frac{k_i} N \frac{ \sum_{j=1}^{k_i} \frac 1 {k_j} }{k_i}. }[/math]

The factor [math]\displaystyle{ \frac{\sum_{j=1}^{k_i}{\frac{1}{k_j}}}{k_i} }[/math] is the inverse of the harmonic mean (IHM) of degrees of the [math]\displaystyle{ k_i }[/math] neighbors of a node [math]\displaystyle{ i }[/math]. Extensive numerical investigation suggest that for an approximately [math]\displaystyle{ m\gt 14 }[/math] the mean IHM value in the large [math]\displaystyle{ N }[/math] limit becomes a constant which means [math]\displaystyle{ \Pi(i) \propto k_i }[/math]. It implies that the higher the links (degree) a node has, the higher its chance of gaining more links since they can be reached in a larger number of ways through mediators which essentially embodies the intuitive idea of rich get richer mechanism (or the preferential attachment rule of the Barabasi–Albert model). Therefore, the MDA network can be seen to follow the PA rule but in disguise.

However, for [math]\displaystyle{ m=1 }[/math] it describes the winner takes it all mechanism as we find that almost [math]\displaystyle{ 99\% }[/math] of the total nodes have degree one and one is super-rich in degree. As [math]\displaystyle{ m }[/math] value increases the disparity between the super rich and poor decreases and as [math]\displaystyle{ m\gt 14 }[/math] we find a transition from rich get super richer to rich get richer mechanism.

Fitness model

Another model where the key ingredient is the nature of the vertex has been introduced by Caldarelli et al. Here a link is created between two vertices [math]\displaystyle{ i,j }[/math] with a probability given by a linking function [math]\displaystyle{ f(\eta_i,\eta_j) }[/math] of the fitnesses of the vertices involved. The degree of a vertex i is given by

[math]\displaystyle{ k(\eta_i)=N\int_0^\infty f(\eta_i,\eta_j) \rho(\eta_j) \, d\eta_j }[/math]

If [math]\displaystyle{ k(\eta_i) }[/math] is an invertible and increasing function of [math]\displaystyle{ \eta_i }[/math], then the probability distribution [math]\displaystyle{ P(k) }[/math] is given by

[math]\displaystyle{ P(k)=\rho(\eta(k)) \cdot \eta'(k) }[/math]

As a result, if the fitnesses [math]\displaystyle{ \eta }[/math] are distributed as a power law, then also the node degree does.

Less intuitively with a fast decaying probability distribution as [math]\displaystyle{ \rho(\eta)=e^{-\eta} }[/math] together with a linking function of the kind

[math]\displaystyle{ f(\eta_i,\eta_j)=\Theta(\eta_i+\eta_j-Z) }[/math]

with [math]\displaystyle{ Z }[/math] a constant and [math]\displaystyle{ \Theta }[/math] the Heavyside function, we also obtain scale-free networks.

Such model has been successfully applied to describe trade between nations by using GDP as fitness for the various nodes [math]\displaystyle{ i,j }[/math] and a linking function of the kind

[math]\displaystyle{ \frac{\delta \eta_i\eta_j}{1+ \delta \eta_i\eta_j}. }[/math]

Network analysis

Social network analysis

Social network analysis examines the structure of relationships between social entities. These entities are often persons, but may also be groups, organizations, nation states, web sites, scholarly publications.

Since the 1970s, the empirical study of networks has played a central role in social science, and many of the mathematical and statistical tools used for studying networks have been first developed in sociology. Amongst many other applications, social network analysis has been used to understand the diffusion of innovations, news and rumors. Similarly, it has been used to examine the spread of both diseases and health-related behaviors. It has also been applied to the study of markets, where it has been used to examine the role of trust in exchange relationships and of social mechanisms in setting prices. Similarly, it has been used to study recruitment into political movements and social organizations. It has also been used to conceptualize scientific disagreements as well as academic prestige. More recently, network analysis (and its close cousin traffic analysis) has gained a significant use in military intelligence, for uncovering insurgent networks of both hierarchical and leaderless nature.In criminology, it is being used to identify influential actors in criminal gangs, offender movements, co-offending, predict criminal activities and make policies.

Dynamic network analysis

Dynamic network analysis examines the shifting structure of relationships among different classes of entities in complex socio-technical systems effects, and reflects social stability and changes such as the emergence of new groups, topics, and leaders. Dynamic Network Analysis focuses on meta-networks composed of multiple types of nodes (entities) and multiple types of links. These entities can be highly varied. Examples include people, organizations, topics, resources, tasks, events, locations, and beliefs.

Dynamic network techniques are particularly useful for assessing trends and changes in networks over time, identification of emergent leaders, and examining the co-evolution of people and ideas.

Biological network analysis

With the recent explosion of publicly available high throughput biological data, the analysis of molecular networks has gained significant interest. The type of analysis in this content are closely related to social network analysis, but often focusing on local patterns in the network. For example, network motifs are small subgraphs that are over-represented in the network. Activity motifs are similar over-represented patterns in the attributes of nodes and edges in the network that are over represented given the network structure. The analysis of biological networks has led to the development of network medicine, which looks at the effect of diseases in the interactome.

Link analysis

Link analysis is a subset of network analysis, exploring associations between objects. An example may be examining the addresses of suspects and victims, the telephone numbers they have dialed and financial transactions that they have partaken in during a given timeframe, and the familial relationships between these subjects as a part of police investigation. Link analysis here provides the crucial relationships and associations between very many objects of different types that are not apparent from isolated pieces of information. Computer-assisted or fully automatic computer-based link analysis is increasingly employed by banks and insurance agencies in fraud detection, by telecommunication operators in telecommunication network analysis, by medical sector in epidemiology and pharmacology, in law enforcement investigations, by search engines for relevance rating (and conversely by the spammers for spamdexing and by business owners for search engine optimization), and everywhere else where relationships between many objects have to be analyzed.

Network robustness

The structural robustness of networks is studied using percolation theory. When a critical fraction of nodes is removed the network becomes fragmented into small clusters. This phenomenon is called percolation and it represents an order-disorder type of phase transition with critical exponents.

Pandemic analysis

The SIR model is one of the most well known algorithms on predicting the spread of global pandemics within an infectious population.

Susceptible to infected

[math]\displaystyle{ S = \beta\left(\frac 1 N \right) }[/math]

The formula above describes the "force" of infection for each susceptible unit in an infectious population, where β is equivalent to the transmission rate of said disease.

To track the change of those susceptible in an infectious population:

[math]\displaystyle{ \Delta S = \beta \times S {1\over N} \, \Delta t }[/math]

Infected to recovered

[math]\displaystyle{ \Delta I = \mu I \, \Delta t }[/math]

Over time, the number of those infected fluctuates by: the specified rate of recovery, represented by [math]\displaystyle{ \mu }[/math] but deducted to one over the average infectious period [math]\displaystyle{ {1\over \tau} }[/math], the numbered of infectious individuals, [math]\displaystyle{ I }[/math], and the change in time, [math]\displaystyle{ \Delta t }[/math].

Infectious period

Whether a population will be overcome by a pandemic, with regards to the SIR model, is dependent on the value of [math]\displaystyle{ R_0 }[/math] or the "average people infected by an infected individual."

[math]\displaystyle{ R_0 = \beta\tau = {\beta\over\mu} }[/math]

Web link analysis

Several Web search ranking algorithms use link-based centrality metrics, including (in order of appearance) Marchiori's Hyper Search, Google's PageRank, Kleinberg's HITS algorithm, the CheiRank and TrustRank algorithms. Link analysis is also conducted in information science and communication science in order to understand and extract information from the structure of collections of web pages. For example, the analysis might be of the interlinking between politicians' web sites or blogs.

PageRank

PageRank works by randomly picking "nodes" or websites and then with a certain probability, "randomly jumping" to other nodes. By randomly jumping to these other nodes, it helps PageRank completely traverse the network as some webpages exist on the periphery and would not as readily be assessed.

Each node, [math]\displaystyle{ x_i }[/math], has a PageRank as defined by the sum of pages [math]\displaystyle{ j }[/math] that link to [math]\displaystyle{ i }[/math] times one over the outlinks or "out-degree" of [math]\displaystyle{ j }[/math] times the "importance" or PageRank of [math]\displaystyle{ j }[/math].

[math]\displaystyle{ x_i = \sum_{j\rightarrow i}{1\over N_j}x_j^{(k)} }[/math]

Random jumping

As explained above, PageRank enlists random jumps in attempts to assign PageRank to every website on the internet. These random jumps find websites that might not be found during the normal search methodologies such as Breadth-First Search and Depth-First Search.

In an improvement over the aforementioned formula for determining PageRank includes adding these random jump components. Without the random jumps, some pages would receive a PageRank of 0 which would not be good.

The first is [math]\displaystyle{ \alpha }[/math], or the probability that a random jump will occur. Contrasting is the "damping factor", or [math]\displaystyle{ 1 - \alpha }[/math].

[math]\displaystyle{ R{(p)} = {\alpha\over N} + (1 - \alpha) \sum_{j\rightarrow i} {1\over N_j} x_j^{(k)} }[/math]

Another way of looking at it:

[math]\displaystyle{ R(A) = \sum {R_B\over B_\text{(outlinks)}} + \cdots + {R_n \over n_\text{(outlinks)}} }[/math]

Centrality measures

Information about the relative importance of nodes and edges in a graph can be obtained through centrality measures, widely used in disciplines like sociology. Centrality measures are essential when a network analysis has to answer questions such as: "Which nodes in the network should be targeted to ensure that a message or information spreads to all or most nodes in the network?" or conversely, "Which nodes should be targeted to curtail the spread of a disease?". Formally established measures of centrality are degree centrality, closeness centrality, betweenness centrality, eigenvector centrality, and katz centrality. The objective of network analysis generally determines the type of centrality measure(s) to be used.

• Degree centrality of a node in a network is the number of links (vertices) incident on the node.
• Closeness centrality determines how "close" a node is to other nodes in a network by measuring the sum of the shortest distances (geodesic paths) between that node and all other nodes in the network.
• Betweenness centrality determines the relative importance of a node by measuring the amount of traffic flowing through that node to other nodes in the network. This is done by measuring the fraction of paths connecting all pairs of nodes and containing the node of interest. Group Betweenness centrality measures the amount of traffic flowing through a group of nodes.
• Eigenvector centrality is a more sophisticated version of degree centrality where the centrality of a node not only depends on the number of links incident on the node but also the quality of those links. This quality factor is determined by the eigenvectors of the adjacency matrix of the network.
• Katz centrality of a node is measured by summing the geodesic paths between that node and all (reachable) nodes in the network. These paths are weighted, paths connecting the node with its immediate neighbors carry higher weights than those which connect with nodes farther away from the immediate neighbors.

Spread of content in networks

Content in a complex network can spread via two major methods: conserved spread and non-conserved spread. In conserved spread, the total amount of content that enters a complex network remains constant as it passes through. The model of conserved spread can best be represented by a pitcher containing a fixed amount of water being poured into a series of funnels connected by tubes. Here, the pitcher represents the original source and the water is the content being spread. The funnels and connecting tubing represent the nodes and the connections between nodes, respectively. As the water passes from one funnel into another, the water disappears instantly from the funnel that was previously exposed to the water. In non-conserved spread, the amount of content changes as it enters and passes through a complex network. The model of non-conserved spread can best be represented by a continuously running faucet running through a series of funnels connected by tubes. Here, the amount of water from the original source is infinite. Also, any funnels that have been exposed to the water continue to experience the water even as it passes into successive funnels. The non-conserved model is the most suitable for explaining the transmission of most infectious diseases.

The SIR model

In 1927, W. O. Kermack and A. G. McKendrick created a model in which they considered a fixed population with only three compartments, susceptible: [math]\displaystyle{ S(t) }[/math], infected, [math]\displaystyle{ I(t) }[/math], and recovered, [math]\displaystyle{ R(t) }[/math]. The compartments used for this model consist of three classes:

• [math]\displaystyle{ S(t) }[/math] is used to represent the number of individuals not yet infected with the disease at time t, or those susceptible to the disease
• [math]\displaystyle{ I(t) }[/math] denotes the number of individuals who have been infected with the disease and are capable of spreading the disease to those in the susceptible category
• [math]\displaystyle{ R(t) }[/math] is the compartment used for those individuals who have been infected and then recovered from the disease. Those in this category are not able to be infected again or to transmit the infection to others.

The flow of this model may be considered as follows:

[math]\displaystyle{ \mathcal{S} \rightarrow \mathcal{I} \rightarrow \mathcal{R} }[/math]

Using a fixed population, [math]\displaystyle{ N = S(t) + I(t) + R(t) }[/math], Kermack and McKendrick derived the following equations:

[math]\displaystyle{ \begin{align} \frac{dS}{dt} & = - \beta S I \\[8pt] \frac{dI}{dt} & = \beta S I - \gamma I \\[8pt] \frac{dR}{dt} & = \gamma I \end{align} }[/math]

Several assumptions were made in the formulation of these equations: First, an individual in the population must be considered as having an equal probability as every other individual of contracting the disease with a rate of [math]\displaystyle{ \beta }[/math], which is considered the contact or infection rate of the disease. Therefore, an infected individual makes contact and is able to transmit the disease with [math]\displaystyle{ \beta N }[/math] others per unit time and the fraction of contacts by an infected with a susceptible is [math]\displaystyle{ S/N }[/math]. The number of new infections in unit time per infective then is [math]\displaystyle{ \beta N (S/N) }[/math], giving the rate of new infections (or those leaving the susceptible category) as [math]\displaystyle{ \beta N (S/N)I = \beta SI }[/math] (Brauer & Castillo-Chavez, 2001). For the second and third equations, consider the population leaving the susceptible class as equal to the number entering the infected class. However, infectives are leaving this class per unit time to enter the recovered/removed class at a rate [math]\displaystyle{ \gamma }[/math] per unit time (where [math]\displaystyle{ \gamma }[/math] represents the mean recovery rate, or [math]\displaystyle{ 1/\gamma }[/math] the mean infective period). These processes which occur simultaneously are referred to as the Law of Mass Action, a widely accepted idea that the rate of contact between two groups in a population is proportional to the size of each of the groups concerned (Daley & Gani, 2005). Finally, it is assumed that the rate of infection and recovery is much faster than the time scale of births and deaths and therefore, these factors are ignored in this model.

More can be read on this model on the Epidemic model page.

The master equation approach

A master equation can express the behaviour of an undirected growing network where, at each time step, a new node is added to the network, linked to an old node (randomly chosen and without preference). The initial network is formed by two nodes and two links between them at time [math]\displaystyle{ t = 2 }[/math], this configuration is necessary only to simplify further calculations, so at time [math]\displaystyle{ t = n }[/math] the network have [math]\displaystyle{ n }[/math] nodes and [math]\displaystyle{ n }[/math] links.

The master equation for this network is:

[math]\displaystyle{ p(k,s,t+1) = \frac 1 t p(k-1,s,t) + \left(1 - \frac 1 t \right)p(k,s,t), }[/math]

where [math]\displaystyle{ p(k,s,t) }[/math] is the probability to have the node [math]\displaystyle{ s }[/math] with degree [math]\displaystyle{ k }[/math] at time [math]\displaystyle{ t+1 }[/math], and [math]\displaystyle{ s }[/math] is the time step when this node was added to the network. Note that there are only two ways for an old node [math]\displaystyle{ s }[/math] to have [math]\displaystyle{ k }[/math] links at time [math]\displaystyle{ t+1 }[/math]:

• The node [math]\displaystyle{ s }[/math] have degree [math]\displaystyle{ k-1 }[/math] at time [math]\displaystyle{ t }[/math] and will be linked by the new node with probability [math]\displaystyle{ 1/t }[/math]
• Already has degree [math]\displaystyle{ k }[/math] at time [math]\displaystyle{ t }[/math] and will not be linked by the new node.

After simplifying this model, the degree distribution is [math]\displaystyle{ P(k) = 2^{-k}. }[/math]

Based on this growing network, an epidemic model is developed following a simple rule: Each time the new node is added and after choosing the old node to link, a decision is made: whether or not this new node will be infected. The master equation for this epidemic model is:

[math]\displaystyle{ p_r(k,s,t) = r_t \frac 1 t p_r(k-1,s,t) + \left(1 - \frac 1 t \right) p_r(k,s,t), }[/math]

where [math]\displaystyle{ r_t }[/math] represents the decision to infect ([math]\displaystyle{ r_t = 1 }[/math]) or not ([math]\displaystyle{ r_t = 0 }[/math]). Solving this master equation, the following solution is obtained: [math]\displaystyle{ \tilde{P}_r(k) = \left(\frac r 2 \right)^k. }[/math]

Interdependent networks

Main article: Interdependent networks

An interdependent network is a system of coupled networks where nodes of one or more networks depend on nodes in other networks. Such dependencies are enhanced by the developments in modern technology. Dependencies may lead to cascading failures between the networks and a relatively small failure can lead to a catastrophic breakdown of the system. Blackouts are a fascinating demonstration of the important role played by the dependencies between networks. A recent study developed a framework to study the cascading failures in an interdependent networks system.

Multilayer networks

Main article: Multidimensional network

Multilayer networks are networks with multiple kinds of relations. Attempts to model real-world systems as multidimensional networks have been used in various fields such as social network analysis, economics, history, urban and international transport, ecology, psychology, medicine, biology, commerce, climatology, physics, computational neuroscience, operations management, and finance.

Network optimization

Network problems that involve finding an optimal way of doing something are studied under the name of combinatorial optimization. Examples include network flow, shortest path problem, transport problem, transshipment problem, location problem, matching problem, assignment problem, packing problem, routing problem, Critical Path Analysis and PERT (Program Evaluation & Review Technique).

See also

• Climate as complex networks
• Collaborative innovation network
• Communicative ecology
• Complex network
• Core-periphery structures in networks
• Dual-phase evolution
• Erdős–Rényi model
• Glossary of graph theory
• Higher category theory
• Immune network theory
• Irregular warfare
• Network analyzer
• Network dynamics
• Network formation
• Network theory in risk assessment
• Network topology
• Networks in labor economics
• Non-linear preferential attachment
• Percolation
• Policy network analysis
• Polytely
• Quantum complex network
• Random networks
• Rumor spread in social network
• Scale-free networks
• Sequential dynamical system
• Service network
• Small-world networks
• Structural cut-off
• Systems theory

Further reading

• "Connected: The Power of Six Degrees," https://web.archive.org/web/20111006191031/http://ivl.slis.indiana.edu/km/movies/2008-talas-connected.mov
• Cohen, R.; Erez, K.; Havlin, S. (2000). "Resilience of the Internet to random breakdown". Phys. Rev. Lett. 85 (21): 4626–4628. arXiv:cond-mat/0007048. Bibcode:2000PhRvL..85.4626C. CiteSeerX 10.1.1.242.6797. doi:10.1103/physrevlett.85.4626. PMID 11082612.
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