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In computer and network science, network theory is the study of graphs as a representation of either symmetric relations or, more generally, of asymmetric relations between discrete objects. Network theory is a part of graph theory.

It has applications in many disciplines including statistical physics, particle physics, computer science, electrical engineering, biology, economics, operations research, and sociology. Applications of network theory include logistical networks, the World Wide Web, Internet, gene regulatory networks, metabolic networks, social networks, epistemological networks, etc; see List of network theory topics for more examples.

Euler's solution of the Seven Bridges of Königsberg problem is considered to be the first true proof in the theory of networks.[1]

Network optimization

Network problems that involve finding an optimal way of doing something are studied under the name 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).
Network analysis
Social network analysis
Visualization of social network analysis.[2]

Social network analysis examines the structure of relationships between social entities.[3] 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.[4] 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[citation needed] 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.
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 context is 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. Similarly, activity motifs are patterns in the attributes of nodes and edges in the network that are over-represented given the network structure. The analysis of biological networks with respect to diseases has led to the development of the field of network medicine.[5]
Narrative network analysis
Narrative network of US Elections 2012[6]

The automatic parsing of textual corpora has enabled the extraction of actors and their relational networks on a vast scale. The resulting networks, which can contain thousands of nodes, are then analysed by using tools from Network theory to identify the key actors, the key communities or parties, and general properties such as robustness or structural stability of the overall network, or centrality of certain nodes.[7] This automates the approach introduced by Quantitative Narrative Analysis,[8] whereby subject-verb-object triplets are identified with pairs of actors linked by an action, or pairs formed by actor-object.[9]

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.[10] When a critical fraction of nodes (or links) is removed the network becomes fragmented into small disconnected clusters. This phenomenon is called percolation,[11] and it represents an order-disorder type of phase transition with critical exponents.

Several Web search ranking algorithms use link-based centrality metrics, including 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. Another use is for classifying pages according to their mention in other pages.[12]
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. For example, eigenvector centrality uses the eigenvectors of the adjacency matrix corresponding to a network, to determine nodes that tend to be frequently visited. Formally established measures of centrality are degree centrality, closeness centrality, betweenness centrality, eigenvector centrality, subgraph centrality and Katz centrality. The purpose or objective of analysis generally determines the type of centrality measure to be used. For example, if one is interested in dynamics on networks or the robustness of a network to node/link removal, often the dynamical importance[13] of a node is the most relevant centrality measure.
Assortative and disassortative mixing

These concepts were made because of the nature of hubs in a network. Hubs are nodes which have lots of links. If we see one link in the hub, there is no difference between the hubs, however, some differences are exited between those nodes; some hubs tend to link to the other nodes and other hubs avoid connecting to the other nodes. We say a hub is assortative when it tends to connect to the other hubs. A dissortative hub avoids connecting to other hubs. If some nodes have some connections with the expected random probabilities, the hubs are neutral. There are three methods to quantify degree correlations.

Content in a complex network can spread via two major methods: conserved spread and non-conserved spread.[14] 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, neural excitation, information and rumors, etc.
Interdependent networks

Interdependent networks 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.[15][16]
Implementations

igraph, an open source C library for the analysis of large-scale complex networks, with interfaces to R, Python and Ruby.
Graph-tool and NetworkX, free and efficient Python modules for manipulation and statistical analysis of networks.[17][18]
Orange, a free data mining software suite, module orngNetwork
Pajek, program for (large) network analysis and visualization.
Tulip, a free data mining and visualization software dedicated to the analysis and visualization of relational data.[19]
SEMOSS, an RDF-based open source context-aware analytics tool written in Java leveraging the SPARQL query language.

Complex network
Quantum complex network
Constructal law[20]
Dual-phase evolution
Percolation
Network science
Network theory in risk assessment
Network topology
Network analyzer
Seven Bridges of Königsberg
Small-world networks
Social network
Scale-free networks
Network dynamics
Sequential dynamical systems
Pathfinder networks
Human disease network
Biological network
Network medicine

References

Newman, M. E. J. "The structure and function of complex networks" (PDF). Department of Physics, University of Michigan.
Grandjean, Martin (2014). "La connaissance est un réseau". Les Cahiers du Numérique 10 (3): 37–54. Retrieved 2014-10-15.
Wasserman, Stanley and Katherine Faust. 1994. Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press.
Newman, M.E.J. Networks: An Introduction. Oxford University Press. 2010
Barabási, A. L., Gulbahce, N., & Loscalzo, J. (2011). Network medicine: a network-based approach to human disease. Nature Reviews Genetics, 12(1), 56–68.
Automated analysis of the US presidential elections using Big Data and network analysis; S Sudhahar, GA Veltri, N Cristianini; Big Data & Society 2 (1), 1–28, 2015
Network analysis of narrative content in large corpora; S Sudhahar, G De Fazio, R Franzosi, N Cristianini; Natural Language Engineering, 1–32, 2013
Quantitative Narrative Analysis; Roberto Franzosi; Emory University © 2010
Automated analysis of the US presidential elections using Big Data and network analysis; S Sudhahar, GA Veltri, N Cristianini; Big Data & Society 2 (1), 1–28, 2015
R. Cohen, S. Havlin (2010). Complex Networks: Structure, Robustness and Function. Cambridge University Press.
A. Bunde, S. Havlin (1996). Fractals and Disordered Systems. Springer.
Attardi, G.; S. Di Marco; D. Salvi (1998). "Categorization by Context" (PDF). Journal of Universal Computer Science 4 (9): 719–736.
Restrepo, Juan; E. Ott; B. R. Hunt (2006). "Characterizing the Dynamical Importance of Network Nodes and Links". Phys. Rev. Lett 97 (9): 094102. doi:10.1103/PhysRevLett.97.094102. PMID 17026366.
Newman, M., Barabási, A.-L., Watts, D.J. [eds.] (2006) The Structure and Dynamics of Networks. Princeton, N.J.: Princeton University Press.
S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, S. Havlin (2010). "Catastrophic cascade of failures in interdependent networks". Nature 464 (7291): 1025–28. doi:10.1038/nature08932. PMID 20393559.
Jianxi Gao; Sergey V. Buldyrev; Shlomo Havlin; H. Eugene Stanley (2011). "Robustness of a Network of Networks". Phys. Rev. Lett 107 (19): 195701. doi:10.1103/PhysRevLett.107.195701. PMID 22181627.
http://graph-tool.skewed.de/
http://networkx.lanl.gov/
http://tulip.labri.fr/

[1] Bejan A., Lorente S., The Constructal Law of Design and Evolution in Nature. Philosophical Transactions of the Royal Society B, Biological Science, Vol. 365, 2010, pp. 1335–1347.

Books

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
G. Caldarelli, "Scale-Free Networks", Oxford University Press, 2007, ISBN 978-0-19-921151-7
A. Barrat, M. Barthelemy, A. Vespignani , "Dynamical Processes on Complex Networks", Cambridge University Press, 2008, ISBN 978-0521879507
E. Estrada, "The Structure of Complex Networks: Theory and Applications", Oxford University Press, 2011, ISBN 978-0-199-59175-6

netwiki Scientific wiki dedicated to network theory
New Network Theory International Conference on 'New Network Theory'
Network Workbench: A Large-Scale Network Analysis, Modeling and Visualization Toolkit
Network analysis of computer networks
Network analysis of organizational networks
Network analysis of terrorist networks
Network analysis of a disease outbreak
Link Analysis: An Information Science Approach (book)
Connected: The Power of Six Degrees (documentary)
Influential Spreaders in Networks, M. Kitsak, L. K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H. E. Stanley, H.A. Makse, Nature Physics 6, 888 (2010)
A short course on complex networks
A course on complex network analysis by Albert-László Barabási

Mathematics Encyclopedia