Network theory

For the theory regarding the regulation of the adaptive immune system, see Immune network theory. For the sociological theory, see Social network.

In mathematics, computer science and network science, network theory is a part of graph theory. It defines networks as graphs where the nodes or edges possess attributes. Network theory analyses these networks over the symmetric relations or asymmetric relations between their (discrete) components.

A small example network with eight vertices (nodes) and ten edges (links)

Network theory has applications in many disciplines, including statistical physics, particle physics, computer science, electrical engineering,^[1][2] biology,^[3] archaeology,^[4] linguistics,^[5][6][7] economics, finance, operations research, climatology, ecology, public health,^[8][9][10] sociology,^[11] psychology,^[12] and neuroscience.^[13][14][15] 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.

Network optimization

Network problems that involve finding an optimal way of doing something are studied as 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 program evaluation and review technique.

Network analysis

Electric network analysis

The analysis of electric power systems could be conducted using network theory from two main points of view:

1. An abstract perspective (i.e., as a graph consists from nodes and edges), regardless of the electric power aspects (e.g., transmission line impedances). Most of these studies focus only on the abstract structure of the power grid using node degree distribution and betweenness distribution, which introduces substantial insight regarding the vulnerability assessment of the grid. Through these types of studies, the category of the grid structure could be identified from the complex network perspective (e.g., single-scale, scale-free). This classification might help the electric power system engineers in the planning stage or while upgrading the infrastructure (e.g., add a new transmission line) to maintain a proper redundancy level in the transmission system.^[1]
2. Weighted graphs that blend an abstract understanding of complex network theories and electric power systems properties.^[2]

Social network analysis

 

Visualization of social network analysis^[16]

Social network analysis examines the structure of relationships between social entities.^[17] These entities are often persons, but may also be groups, organizations, nation states, web sites, or 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.^[18] Amongst many other applications, social network analysis has been used to understand the diffusion of innovations, news and rumors.^[19] Similarly, it has been used to examine the spread of both diseases and health-related behaviors.^[20] 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.^[21] It has been used to study recruitment into political movements, armed groups, and other social organizations.^[22] It has also been used to conceptualize scientific disagreements^[23] as well as academic prestige.^[24] More recently, network analysis (and its close cousin traffic analysis) has gained a significant use in military intelligence,^[25] for uncovering insurgent networks of both hierarchical and leaderless nature.^{[citation needed]}

Biological network analysis

See also: Metabolic network, proteome, metabolome, and omics

With the recent explosion of publicly available high throughput biological data, the analysis of molecular networks has gained significant interest.^[26] 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. Using networks to analyze patterns in biological systems, such as food-webs, allows us to visualize the nature and strength of interactions between species. The analysis of biological networks with respect to diseases has led to the development of the field of network medicine.^[27] Recent examples of application of network theory in biology include applications to understanding the cell cycle^[28] as well as a quantitative framework for developmental processes.^[29]

Narrative network analysis

 

Narrative network of US Elections 2012^[30]

The automatic parsing of textual corpora has enabled the extraction of actors and their relational networks on a vast scale. The resulting narrative networks, which can contain thousands of nodes, are then analyzed 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.^[31] This automates the approach introduced by Quantitative Narrative Analysis,^[32] whereby subject-verb-object triplets are identified with pairs of actors linked by an action, or pairs formed by actor-object.^[30]

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. Links are also derived from similarity of time behavior in both nodes. Examples include climate networks where the links between two locations (nodes) are determined, for example, by the similarity of the rainfall or temperature fluctuations in both sites.^[33][34]

Web link analysis

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' websites or blogs. Another use is for classifying pages according to their mention in other pages.^[35]

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^[36] of a node is the most relevant centrality measure.

Assortative and disassortative mixing

Further information: Assortative mixing

These concepts are used to characterize the linking preferences of hubs in a network. Hubs are nodes which have a large number of links. Some hubs tend to link to other hubs while others avoid connecting to hubs and prefer to connect to nodes with low connectivity. We say a hub is assortative when it tends to connect to other hubs. A disassortative hub avoids connecting to other hubs. If hubs have connections with the expected random probabilities, they are said to be neutral. There are three methods to quantify degree correlations.^[37]

Recurrence networks

The recurrence matrix of a recurrence plot can be considered as the adjacency matrix of an undirected and unweighted network. This allows for the analysis of time series by network measures. Applications range from detection of regime changes over characterizing dynamics to synchronization analysis.^[38][39][40]

Spatial networks

Many real networks are embedded in space. Examples include, transportation and other infrastructure networks, brain neural networks. Several models for spatial networks have been developed.^[41]

Spread

Content in a complex network can spread via two major methods: conserved spread and non-conserved spread.^[42] 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.

Network immunization

The question of how to immunize efficiently scale free networks which represent realistic networks such as the Internet and social networks has been studied extensively. One such strategy is to immunize the largest degree nodes, i.e., targeted (intentional) attacks^[43] since for this case ${\displaystyle pc}$  is relatively high and fewer nodes are needed to be immunized. However, in most realistic networks the global structure is not available and the largest degree nodes are unknown.

See also

• Complex network
• Congestion game
• Quantum complex network
• Dual-phase evolution
• Network partition
• 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
• Graph partition

References

1. Borchers A, Pieler T (November 2010). "Programming pluripotent precursor cells derived from Xenopus embryos to generate specific tissues and organs". Genes. 1 (3): 413–426. doi:10.3390/en11061381. PMC 3966229. PMID 24710095.
2. Saleh, Mahmoud; Esa, Yusef; Onuorah, Nwabueze; Mohamed, Ahmed A. (2017). "Optimal microgrids placement in electric distribution systems using complex network framework". Optimal microgrids placement in electric distribution systems using complex network framework - IEEE Conference Publication. ieeexplore.ieee.org. pp. 1036–1040. doi:10.1109/ICRERA.2017.8191215. ISBN 978-1-5386-2095-3. S2CID 44685630. Retrieved 2018-06-07.
3. Habibi I, Emamian ES, Abdi A (August 2014). "Quantitative analysis of intracellular communication and signaling errors in signaling networks". BMC Systems Biology. 8: 89. doi:10.1186/s12918-014-0089-z. PMC 4255782. PMID 25115405.
4. Sindbæk S (2007). Networks and nodal points: the emergence of towns in early Viking Age Scandinavia - Antiquity 81(311). Cambridge University Press. pp. 119–132.
5. Paradowski, M. B.; Jarynowski, A.; Jelińska, M.; Czopek, K. (2021). "Selected poster presentations from the American Association of Applied Linguistics conference, Denver, USA, March 2020: Out-of-class peer interactions matter for second language acquisition during short-term overseas sojourns: The contributions of Social Network Analysis". Language Teaching. 54 (1): 139–143. doi:10.1017/S0261444820000580.
6. Paradowski, M. B.; Jarynowski, A.; Czopek, K.; Jelińska, M.; et al. (2021). "Peer interactions and second language learning: The contributions of Social Network Analysis in Study Abroad vs At-Home environments". In Mitchell, Rosamond; Tyne, Henry (eds.). Language, Mobility and Study Abroad in the Contemporary European Context. New York: Routledge. pp. 99–116. doi:10.1017/S0261444820000580. ISBN 978-10-03087-95-3. S2CID 228863564.
7. Paradowski, M. B.; Cierpich-Kozieł, A.; Chen, C.-C.; Ochab, J. K. (2022). "How output outweighs input and interlocutors matter for study-abroad SLA: Computational Social Network Analysis of learner interactions". The Modern Language Journal. 106 (4): 694–725. doi:10.1111/modl.12811. S2CID 255247273.
8. Harris JK, Luke DA, Zuckerman RB, Shelton SC (June 2009). "Forty years of secondhand smoke research: the gap between discovery and delivery". American Journal of Preventive Medicine. 36 (6): 538–548. doi:10.1016/j.amepre.2009.01.039. OCLC 5899755895. PMID 19372026.
9. Varda DM, Forgette R, Banks D, Contractor N (2009). "Social Network Methodology in the Study of Disasters: Issues and Insights Prompted by Post-Katrina Research". Population Research and Policy Review. 28 (1): 11–29. doi:10.1007/s11113-008-9110-9. ISSN 0167-5923. OCLC 5659930640. S2CID 144130904.
10. Sunkersing D, Martin FC, Sullivan P, Bell D (December 2022). "Care and support networks of community-dwelling frail individuals in North West London: a comparison of patient and healthcare workers' perceptions". BMC Geriatrics. 22 (1): 953. doi:10.1186/s12877-022-03561-y. PMC 9737751. PMID 36494627.
11. della Porta D, Diani M (2010). Social Movements 2e: An Introduction (2nd ed.). Wiley-Blackwell. ISBN 978-1-4051-0282-7.
12. Paradowski, M. B.; Jelińska, M. (2023). "The predictors of L2 grit and their complex interactions in online foreign language learning: Motivation, self-directed learning, autonomy, curiosity, and language mindsets". Computer Assisted Language Learning: 1–38. doi:10.1080/09588221.2023.2192762.
13. Bassett DS, Sporns O (February 2017). "Network neuroscience". Nature Neuroscience. 20 (3): 353–364. doi:10.1038/nn.4502. PMC 5485642. PMID 28230844.
14. Alex Fornito. "An Introduction to Network Neuroscience: How to build, model, and analyse connectomes - 0800-10:00 | OHBM". pathlms.com. Retrieved 2020-03-11.
15. Saberi M, Khosrowabadi R, Khatibi A, Misic B, Jafari G (January 2021). "Topological impact of negative links on the stability of resting-state brain network". Scientific Reports. 11 (1): 2176. Bibcode:2021NatSR..11.2176S. doi:10.1038/s41598-021-81767-7. PMC 7838299. PMID 33500525.
16. Grandjean M (2014). "La connaissance est un réseau". Les Cahiers du Numérique. 10 (3): 37–54. doi:10.3166/lcn.10.3.37-54. Retrieved 2014-10-15.
17. Wasserman, Stanley and Katherine Faust. 1994. Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press. Rainie, Lee and Barry Wellman, Networked: The New Social Operating System. Cambridge, MA: MIT Press, 2012.
18. Newman, M.E.J. Networks: An Introduction. Oxford University Press. 2010
19. Al-Taie MZ, Kadry S (2017). "Information Diffusion in Social Networks". Python for Graph and Network Analysis. Advanced Information and Knowledge Processing. pp. 165–184. doi:10.1007/978-3-319-53004-8_8. ISBN 978-3-319-53003-1. PMC 7123536.
20. Luke DA, Harris JK (April 2007). "Network analysis in public health: history, methods, and applications". Annual Review of Public Health. 28 (1): 69–93. doi:10.1146/annurev.publhealth.28.021406.144132. PMID 17222078.
21. Odabaş M, Holt TJ, Breiger RL (October 2017). "Markets as Governance Environments for Organizations at the Edge of Illegality: Insights From Social Network Analysis". American Behavioral Scientist. 61 (11): 1267–1288. doi:10.1177/0002764217734266. hdl:10150/631238. S2CID 158776581.
22. Larson JM (11 May 2021). "Networks of Conflict and Cooperation". Annual Review of Political Science. 24 (1): 89–107. doi:10.1146/annurev-polisci-041719-102523.
23. Leng RI (24 May 2018). "A network analysis of the propagation of evidence regarding the effectiveness of fat-controlled diets in the secondary prevention of coronary heart disease (CHD): Selective citation in reviews". PLOS ONE. 13 (5): e0197716. Bibcode:2018PLoSO..1397716L. doi:10.1371/journal.pone.0197716. PMC 5968408. PMID 29795624.
24. Burris V (April 2004). "The Academic Caste System: Prestige Hierarchies in PhD Exchange Networks". American Sociological Review. 69 (2): 239–264. doi:10.1177/000312240406900205. S2CID 143724478. Retrieved 22 September 2021.
25. Roberts N, Everton SF. "Strategies for Combating Dark Networks" (PDF). Journal of Social Structure. 12. Retrieved 22 September 2021.
26. Habibi I, Emamian ES, Abdi A (2014-10-07). "Advanced fault diagnosis methods in molecular networks". PLOS ONE. 9 (10): e108830. Bibcode:2014PLoSO...9j8830H. doi:10.1371/journal.pone.0108830. PMC 4188586. PMID 25290670.
27. Barabási AL, Gulbahce N, Loscalzo J (January 2011). "Network medicine: a network-based approach to human disease". Nature Reviews. Genetics. 12 (1): 56–68. doi:10.1038/nrg2918. PMC 3140052. PMID 21164525.
28. Jailkhani N, Ravichandran S, Hegde SR, Siddiqui Z, Mande SC, Rao KV (December 2011). "Delineation of key regulatory elements identifies points of vulnerability in the mitogen-activated signaling network". Genome Research. 21 (12): 2067–2081. doi:10.1101/gr.116145.110. PMC 3227097. PMID 21865350.
29. Jackson MD, Duran-Nebreda S, Bassel GW (October 2017). "Network-based approaches to quantify multicellular development". Journal of the Royal Society, Interface. 14 (135): 20170484. doi:10.1098/rsif.2017.0484. PMC 5665831. PMID 29021161.
30. 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
31. Network analysis of narrative content in large corpora; S Sudhahar, G De Fazio, R Franzosi, N Cristianini; Natural Language Engineering, 1–32, 2013
32. Quantitative Narrative Analysis; Roberto Franzosi; Emory University © 2010
33. Tsonis AA, Swanson KL, Roebber PJ (2006). "What Do Networks Have to Do with Climate?". Bulletin of the American Meteorological Society. 87 (5): 585–595. Bibcode:2006BAMS...87..585T. doi:10.1175/BAMS-87-5-585. ISSN 0003-0007.
34. Boers N, Bookhagen B, Barbosa HM, Marwan N, Kurths J, Marengo JA (October 2014). "Prediction of extreme floods in the eastern Central Andes based on a complex networks approach". Nature Communications. 5: 5199. Bibcode:2014NatCo...5.5199B. doi:10.1038/ncomms6199. PMID 25310906. S2CID 3032237.
35. Attardi G, Di Marco S, Salvi D (1998). "Categorization by Context" (PDF). Journal of Universal Computer Science. 4 (9): 719–736.
36. Restrepo JG, Ott E, Hunt BR (September 2006). "Characterizing the dynamical importance of network nodes and links". Physical Review Letters. 97 (9): 094102. arXiv:cond-mat/0606122. Bibcode:2006PhRvL..97i4102R. doi:10.1103/PhysRevLett.97.094102. PMID 17026366. S2CID 18365246.
37. M. E. J. Newman (2003). "Mixing patterns in networks". Physical Review E. 67 (2): 026126. arXiv:cond-mat/0209450. Bibcode:2003PhRvE..67b6126N. doi:10.1103/PhysRevE.67.026126. PMID 12636767. S2CID 15186389.
38. Marwan N, Donges JF, Zou Y, Donner RV, Kurths J (2009). "Complex network approach for recurrence analysis of time series". Physics Letters A. 373 (46): 4246–4254. arXiv:0907.3368. Bibcode:2009PhLA..373.4246M. doi:10.1016/j.physleta.2009.09.042. ISSN 0375-9601. S2CID 7761398.
39. Donner RV, Heitzig J, Donges JF, Zou Y, Marwan N, Kurths J (2011). "The Geometry of Chaotic Dynamics – A Complex Network Perspective". European Physical Journal B. 84 (4): 653–672. arXiv:1102.1853. Bibcode:2011EPJB...84..653D. doi:10.1140/epjb/e2011-10899-1. ISSN 1434-6036. S2CID 18979395.
40. Feldhoff JH, Donner RV, Donges JF, Marwan N, Kurths J (2013). "Geometric signature of complex synchronisation scenarios". Europhysics Letters. 102 (3): 30007. arXiv:1301.0806. Bibcode:2013EL....10230007F. doi:10.1209/0295-5075/102/30007. ISSN 1286-4854. S2CID 119118006.
41. Waxman BM (1988). "Routing of multipoint connections". IEEE Journal on Selected Areas in Communications. 6 (9): 1617–1622. doi:10.1109/49.12889.
42. Newman M, Barabási AL, Watts DJ, eds. (2006). The Structure and Dynamics of Networks. Princeton, N.J.: Princeton University Press.
43. Callaway DS, Newman ME, Strogatz SH, Watts DJ (December 2000). "Network robustness and fragility: percolation on random graphs". Physical Review Letters. 85 (25): 5468–5471. arXiv:cond-mat/0007300. Bibcode:2000PhRvL..85.5468C. doi:10.1103/PhysRevLett.85.5468. PMID 11136023. S2CID 2325768.

Books

• Dorogovtsev SN, Mendes JR (2003). Evolution of Networks: from biological networks to the Internet and WWW. Oxford University Press. ISBN 978-0-19-851590-6.
• Caldarelli G (2007). Scale-Free Networks. Oxford University Press. ISBN 978-0-19-921151-7.
• Barrat A, Barthelemy M, Vespignani A (2008). Dynamical Processes on Complex Networks. Cambridge University Press. ISBN 978-0-521-87950-7.
• Estrada E (2011). The Structure of Complex Networks: Theory and Applications. Oxford University Press. ISBN 978-0-199-59175-6.
• Soramaki K, Cook S (2016). Network Theory and Financial Risk. Risk Books. ISBN 978-1-78272-219-9.
• Latora V, Nicosia V, Russo G (2017). Complex Networks: Principles, Methods and Applications. Cambridge University Press. ISBN 978-1-107-10318-4.

External links

 

Wikiquote has quotations related to Network theory.

• 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
• Optimization of the Large Network doi:10.13140/RG.2.2.20183.06565/6
• Network analysis of computer networks
• Network analysis of organizational networks
• Link Analysis: An Information Science Approach (book)
• (documentary)
• A short course on complex networks
• The Journal of Network Theory in Finance
• Network theory in Operations Research from the Institute for Operations Research and the Management Sciences (INFORMS)