This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
A Spectrum-Based Framework for Quantifying Randomness of Social Networks
December 2011 (vol. 23 no. 12)
pp. 1842-1856
Xiaowei Ying, University of North Carolina at Charlotte, Charlotte
Leting Wu, University of North Carolina at Charlotte, Charlotte
Xintao Wu, University of North Carolina at Charlotte, Charlotte
Social networks tend to contain some amount of randomness and some amount of nonrandomness. The amount of randomness versus nonrandomness affects the properties of a social network. In this paper, we theoretically analyze graph randomness and present a framework which provides a series of nonrandomness measures at levels of edge, node, subgraph, and the overall graph. We show that graph nonrandomness can be obtained mathematically from the spectra of the adjacency matrix of the network. We derive the upper bound and lower bound of nonrandomness value of the overall graph. We investigate whether other graph spectra (such as Laplacian and normal spectra) could also be used to derive a nonrandomness framework. Our theoretical results showed that they are unlikely, if not impossible, to have a consistent framework to evaluate randomness. We also compare our proposed nonrandomness measures with some traditional measures such as modularity. Our theoretical and empirical studies show our proposed nonrandomness measures can characterize and capture graph randomness.

[1] D. Chakrabarti and C. Faloutsos, "Graph Mining: Laws, Generators, and Algorithms," ACM Computing Surveys, vol. 38, no. 1, p. 2, 2006.
[2] F. Chung and R. Graham, "Sparse Quasi-Random Graphs," Combinatorica, vol. 22, no. 2, pp. 217-244, 2002.
[3] F. Chung, Spectral Graph Theory. Am. Math. Soc., 1997.
[4] D. Cvetkovic and P. Rowlinson, "The Largest Eigenvalue of a Graph: A Survey," Linear and Multilinear Algebra, vol. 28, pp. 3-33, 1990.
[5] D. Cvetkovic, P. Rowlinson, and S. Simic, Eigenspaces of Graphs. Cambridge Univ. Press, 1997.
[6] L. Costa, F. Rodrigues, G. Travieso, and P. Boas, "Characterization of Complex Networks: A Survey of Measurements," Advances in Physics, vol. 56, pp. 167-242, 2007.
[7] C. Ding, X. He, H. Zha, M. Gu, and H. Simon, "A Min-Max Cut Algorithm for Graph Partitioning and Data Clustering," Proc. IEEE Int'l Conf. Data Mining (ICDM '01), pp. 107-114, 2001.
[8] P. Erdos and A. Renyi, "On Random Graphs i," Publicationes Mathematicae, vol. 6, pp. 290-297, 1959.
[9] I. Farkas, I. Derenyi, A. Barabasi, and T. Vicsek, "Spectra of "Real-World" Graphs: Beyond the Semi-Circle Law," Physical Rev. E, vol. 64, article no. 026704, 2001.
[10] Z. Furedi and J. Komlos, "The Eigenvalues of Random Symmetric Matrices," Combinatorica, vol. 1, no. 3, pp. 233-41, 1981.
[11] G. Golub and C. Van Loan, Matrix Computations. Johns Hopkins Univ. Press, 1996.
[12] J. Kleinberg, "Authoritative Sources in a Hyperlinked Environment," J. ACM, vol. 46, no. 5, pp. 604-632, 1999.
[13] V. Krebs, http:/www.orgnet.com/. 2006.
[14] M. Newman, "The Structure and Function of Complex Networks," SIAM Rev., vol. 45, p. 167, 2003.
[15] M. Newman, "Detecting Community Structure in Networks," The European Physical J. B—Condensed Matter, vol. 38, no. 2, pp. 321-330, Mar. 2004.
[16] M. Newman, "Finding Community Sturcture in Network Using the Eigenvectors of Matrices," Physical Rev. E, vol. 74, p. 036104, 2006.
[17] A. Seary and W. Richards, "Spectral Methods for Analyzing and Visualizing Networks: An Introduction," Proc. Nat'l Research Council, Dynamic Social Network Modelling and Analysis: Workshop Summary and Papers, pp. 209-228, 2003.
[18] J. Shetty and J. Adibi, "The Enron Email Dataset Database Schema and Brief Statistical Report," technical report, Information Sciences Inst., Univ. of Southern California, 2004.
[19] J. Shi and J. Malik, "Normalized Cuts and Image Segmentation," Proc. Conf. Computer Vision and Pattern Recognition (CVPR '97), p. 731, 1997.
[20] N. Shrivastava, A. Majumder, and R. Rastogi, "Mining (Social) Network Graphs to Detect Random Link Attacks," Proc. IEEE 24th Int'l Conf. Data Eng. (ICDE), 2008.
[21] G. Stewart and J. Sun, Matrix Perturbation Theory. Academic Press, 1990.
[22] S. Strogatz, "Exploring Complex Networks," Nature, vol. 410, pp. 268-276, 2001.
[23] Y. Wang, D. Chakrabarti, C. Wang, and C. Faloutsos, "Epidemic Spreading in Real Networks: An Eigenvalue Viewpoint," Proc. 22nd Int'l Symp. Reliable Distributed Systems, 2003.
[24] Y. Weiss, "Segmentation Using Eigenvectors: A Unifying View," Proc. IEEE Int'l Conf. Computer Vision, pp. 975-982, 1999.
[25] X. Ying and X. Wu, "On Randomness Measures for Social Networks," Proc. Ninth SIAM Conf. Data Mining, pp. 709-720, 2009.

Index Terms:
Randomness measures, graph spectra, social networks.
Citation:
Xiaowei Ying, Leting Wu, Xintao Wu, "A Spectrum-Based Framework for Quantifying Randomness of Social Networks," IEEE Transactions on Knowledge and Data Engineering, vol. 23, no. 12, pp. 1842-1856, Dec. 2011, doi:10.1109/TKDE.2010.218
Usage of this product signifies your acceptance of the Terms of Use.