Encyclopedia of Social Network Analysis and Mining

2018 Edition
| Editors: Reda Alhajj, Jon Rokne

Spectral Analysis

Reference work entry
DOI: https://doi.org/10.1007/978-1-4939-7131-2_168

Synonyms

Glossary

Network (graph)

A network G is a triple consisting of a node set V (G), a link set E(G), and a relation that associates each link with two nodes

Adjacency matrix

Let G = (V (G), E(G)) be a network with V (G) = {v1, ···, vn}.

The adjacency matrix A(G) = (aij) of G is n × n matrix with aij = 1 if vi is adjacent to vj, and 0 otherwise

Eigenvalues of a graph

All eigenvalues of the adjacency matrix A(G) of a graph G are called eigenvalues of G and denoted by

λ1 ≥ λ2 ≥ … ≥ λn

Degree diagonal matrix

The degree diagonal matrix D(G) of a network G is the diagonal matrix whose diagonal entries are degrees of the corresponding nodes

Laplacian matrix

The Laplacian matrix L(G) is defined be L(G) = D(G) − A(G), where D(G) is the degree diagonal matrix and A(G) is the adjacency matrix

Laplacian eigenvalues of a graph

All eigenvalues of the Laplacian matrix L(G) of a graph Gare called the Laplacian...

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Notes

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Nos. 11531001 and 11271256), the Joint NSFC-ISF Research Program (jointly funded by the National Natural Science Foundation of China and the Israel Science Foundation (No. 11561141001), Innovation Program of Shanghai Municipal Education Commission (No. 14ZZ016), and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130073110075).

References

  1. Bickel PJ, Chen A (2009) A nonparametric view of network models and Newman-Girvan and other modularities. Proc Natl Acad Sci U S A 106:21068–21073MATHCrossRefGoogle Scholar
  2. Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D-U (2006) Complex networks: structure and dynamics. Phys Rep 424:275–308MathSciNetMATHCrossRefGoogle Scholar
  3. Bonacich P (1972) Factoring and weighting approaches to status scores and clique identification. J Math Sociol 2:113–120CrossRefGoogle Scholar
  4. Chauhan S, Girvan M, Ott E (2009) Spectral properties of networks with community structure. Phys Rev E 80:0561104CrossRefGoogle Scholar
  5. Chung FRK (1997) Spectral graph theory. AMS Publications, ProvidenceMATHGoogle Scholar
  6. Cvetkovi’c D, Doob M, Sachs H (1980) Spectra of graphs-theory and applications. Academic Press, New Work. Third edition, 1995Google Scholar
  7. Fiedler M (1973) Algebra connectivity of graphs. Czechoslovake Mathematical Journal 23(98):298–305MathSciNetMATHGoogle Scholar
  8. Fortunato S (2010) Community detection in graphs. Phys Rep 48:75–174MathSciNetCrossRefGoogle Scholar
  9. Girvan M, Newman MEJ (2002) Community structure in social and biological networks. Proc Natl Acad Sci U S A 99:7821MathSciNetMATHCrossRefGoogle Scholar
  10. Gkantsidis C, Mihail M, Zegura E (2003) Spectral analysis of internet topologies. In: IEEE INFOCOM. San Francisco, CA, USAGoogle Scholar
  11. Li T, Liu J, Weinan E (2009) Probabilistic framework for network partition. Phys Rev E 80:026106CrossRefGoogle Scholar
  12. Moody J (2001) Race, school integration, and friendship segregation in America. Amer J Sociol 107:679–716CrossRefGoogle Scholar
  13. Nascimento MCV, Carvalho ACPF d (2011) Spectral methods for graph clustering-a survey. European J Oper Res 211:221–231MathSciNetMATHCrossRefGoogle Scholar
  14. Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45(2):167–245MathSciNetMATHCrossRefGoogle Scholar
  15. Newman MEJ (2006a) Finding community structure in networks using the eigenvectors of matrices. Phys Rev E 74:036104MathSciNetCrossRefGoogle Scholar
  16. Newman MEJ (2006b) Modularity and community structure in networks. Proc Natl Acad Sci U S A 103:8577–8582CrossRefGoogle Scholar
  17. Newman MEJ (2012) Communities modules and large-scale structure in networks. Nat Phys 8:25–31CrossRefGoogle Scholar
  18. Ruhnau B (2000) Eigenvector-centrality – a node-centrality? Soc Networks 22:357–365CrossRefGoogle Scholar
  19. Scott J (2000) Social network analysis: a handbook. Sage Publications, LondonGoogle Scholar
  20. Seary AJ, Richards WD (2005) Spectral methods for analyzing and visualizing networks: an introduction. In: Breiger R, Carley KM, Pattison P (eds) Dynamic social network Modeling and analysis. National Academies Press, Washington, DC, pp 209–228Google Scholar
  21. Servedio VDP, Colaiori F, Capocci A, Caldarelli G (2004) Community structure from spectral properties in complex network. In: Mendes JFF, Dorogovtsev SN, Abreu FV, Oliveira JG (eds) Science of complex networks: from biology to the internet and WWW; CNRT, pp 277–286Google Scholar
  22. Van Mieghem P, Ge X, Schumm P, Trajanovski S, Wang H (2010) Spectral graph analysis of modularity and assortativity. Phys Rev E 82:056113CrossRefGoogle Scholar
  23. Wasserman S, Faust K (1994) Social network analysis. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  24. Weinan E, Li T, Vanden-Eijnden E (2008) Optimal partition and effective dynamics of complex networks. Proc Natl Acad Sci U S A 105:7907–7912MathSciNetMATHCrossRefGoogle Scholar
  25. Wu L, Ying X, Wu X, Zhou Z.-H (2011) Line orthogonality in adjacency eigenspace with application to community partition. In: Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI11), Barcelona, July 16–22Google Scholar

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© Springer Science+Business Media LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical Science, Ministry of Education Key Laboratory of Scientific and Engineering ComputingShanghai Jiao Tong UniversityShanghaiChina

Section editors and affiliations

  • Suheil Khoury
    • 1
  1. 1.American University of SharjahSharjahUnited Arab Emirates