# 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

## 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

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...

This is a preview of subscription content, log in to check access.

## 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 106:21068–21073
2. Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D-U (2006) Complex networks: structure and dynamics. Phys Rep 424:275–308
3. Bonacich P (1972) Factoring and weighting approaches to status scores and clique identification. J Math Sociol 2:113–120
4. Chauhan S, Girvan M, Ott E (2009) Spectral properties of networks with community structure. Phys Rev E 80:0561104
5. Chung FRK (1997) Spectral graph theory. AMS Publications, Providence
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):298–305
8. Fortunato S (2010) Community detection in graphs. Phys Rep 48:75–174
9. Girvan M, Newman MEJ (2002) Community structure in social and biological networks. Proc Natl Acad Sci U S A 99:7821
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 80:026106
12. Moody J (2001) Race, school integration, and friendship segregation in America. Amer J Sociol 107:679–716
13. Nascimento MCV, Carvalho ACPF d (2011) Spectral methods for graph clustering-a survey. European J Oper Res 211:221–231
14. Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45(2):167–245
15. Newman MEJ (2006a) Finding community structure in networks using the eigenvectors of matrices. Phys Rev E 74:036104
16. Newman MEJ (2006b) Modularity and community structure in networks. Proc Natl Acad Sci U S A 103:8577–8582
17. Newman MEJ (2012) Communities modules and large-scale structure in networks. Nat Phys 8:25–31
18. Ruhnau B (2000) Eigenvector-centrality – a node-centrality? Soc Networks 22:357–365
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 82:056113
23. Wasserman S, Faust K (1994) Social network analysis. Cambridge University Press, Cambridge
24. Weinan E, Li T, Vanden-Eijnden E (2008) Optimal partition and effective dynamics of complex networks. Proc Natl Acad Sci U S A 105:7907–7912
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