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...
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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
Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D-U (2006) Complex networks: structure and dynamics. Phys Rep 424:275–308
Bonacich P (1972) Factoring and weighting approaches to status scores and clique identification. J Math Sociol 2:113–120
Chauhan S, Girvan M, Ott E (2009) Spectral properties of networks with community structure. Phys Rev E 80:0561104
Chung FRK (1997) Spectral graph theory. AMS Publications, Providence
Cvetkovi’c D, Doob M, Sachs H (1980) Spectra of graphs-theory and applications. Academic Press, New Work. Third edition, 1995
Fiedler M (1973) Algebra connectivity of graphs. Czechoslovake Mathematical Journal 23(98):298–305
Fortunato S (2010) Community detection in graphs. Phys Rep 48:75–174
Girvan M, Newman MEJ (2002) Community structure in social and biological networks. Proc Natl Acad Sci U S A 99:7821
Gkantsidis C, Mihail M, Zegura E (2003) Spectral analysis of internet topologies. In: IEEE INFOCOM. San Francisco, CA, USA
Li T, Liu J, Weinan E (2009) Probabilistic framework for network partition. Phys Rev E 80:026106
Moody J (2001) Race, school integration, and friendship segregation in America. Amer J Sociol 107:679–716
Nascimento MCV, Carvalho ACPF d (2011) Spectral methods for graph clustering-a survey. European J Oper Res 211:221–231
Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45(2):167–245
Newman MEJ (2006a) Finding community structure in networks using the eigenvectors of matrices. Phys Rev E 74:036104
Newman MEJ (2006b) Modularity and community structure in networks. Proc Natl Acad Sci U S A 103:8577–8582
Newman MEJ (2012) Communities modules and large-scale structure in networks. Nat Phys 8:25–31
Ruhnau B (2000) Eigenvector-centrality – a node-centrality? Soc Networks 22:357–365
Scott J (2000) Social network analysis: a handbook. Sage Publications, London
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 209–228
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 277–286
Van Mieghem P, Ge X, Schumm P, Trajanovski S, Wang H (2010) Spectral graph analysis of modularity and assortativity. Phys Rev E 82:056113
Wasserman S, Faust K (1994) Social network analysis. Cambridge University Press, Cambridge
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
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 16–22
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).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2018 Springer Science+Business Media LLC, part of Springer Nature
About this entry
Cite this entry
Zhang, XD. (2018). Spectral Analysis. In: Alhajj, R., Rokne, J. (eds) Encyclopedia of Social Network Analysis and Mining. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7131-2_168
Download citation
DOI: https://doi.org/10.1007/978-1-4939-7131-2_168
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-7130-5
Online ISBN: 978-1-4939-7131-2
eBook Packages: Computer ScienceReference Module Computer Science and Engineering