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Spectral evolution in dynamic networks

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Abstract

We introduce and study the spectral evolution model, which characterizes the growth of large networks in terms of the eigenvalue decomposition of their adjacency matrices: In large networks, changes over time result in a change of a graph’s spectrum, leaving the eigenvectors unchanged. We validate this hypothesis for several large social, collaboration, rating, citation, and communication networks. Following these observations, we introduce two link prediction algorithms based on the learning of the changes to a network’s spectrum. These new link prediction methods generalize several common graph kernels that can be expressed as spectral transformations. The first method is based on reducing the link prediction problem to a one-dimensional curve-fitting problem which can be solved efficiently. The second algorithm extrapolates a network’s spectrum to predict links. Both algorithms are evaluated on fifteen network datasets for which edge creation times are known.

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Notes

  1. This can be seen by noting that a single edge has the adjacency matrix \([0, 1; 1, 0]\), which is indefinite, and in fact only by adding a positive-semidefinite component to a matrix can it be guaranteed that eigenvalues do not shrink; this follows from the interlacing theorem given in [62, p. 97].

References

  1. Adamic L, Adar E (2001) Friends and neighbors on the web. Soc Netw 25:211–230

    Article  Google Scholar 

  2. Barabási A-L, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512

    Article  MathSciNet  Google Scholar 

  3. Blei D, Ng A, Jordan M, Lafferty J (2003) Latent dirichlet allocation. J Mach Learn Res 3:993–1022

    MATH  Google Scholar 

  4. Bradley AP (1997) The use of the area under the ROC curve in the evaluation of machine learning algorithms. Pattern Recogn 30:1145–1159

    Article  Google Scholar 

  5. Breese JS, Heckerman D, Kadie C (1998) Empirical analysis of predictive algorithms for collaborative filtering. In: Proceedings of conference on uncertainty in artificial intelligence, pp 43–52

  6. Brin S, Page L (1998) The anatomy of a large-scale hypertextual Web search engine. Comput Netw ISDN Syst 30(1–7):107–117

    Article  Google Scholar 

  7. Chaintreau A, Hui P, Crowcroft J, Diot C, Gass R, Scott J (2007) Impact of human mobility on opportunistic forwarding algorithms. IEEE Trans Mobile Comput 6(6):606–620

    Article  Google Scholar 

  8. Chapelle O, Weston J, Schölkopf B (2003) Cluster kernels for semi-supervised learning. Adv Neural Inform Process Syst 15:585–592

    Google Scholar 

  9. Chebotarev P, Shamis E (1997) The matrix-forest theorem and measuring relations in small social groups. Autom Remote Control 58(9):1505–1514

    MathSciNet  MATH  Google Scholar 

  10. Choudhury MD, Lin Y-R, Sundaram H, Candan KS, Xie L, Kelliher A (2010) How does the data sampling strategy impact the discovery of information diffusion in social media? In: Proceedings of conference on weblogs and social media, pp 34–41

  11. Choudhury MD, Sundaram H, John A, Seligmann DD (2009) Social synchrony: predicting mimicry of user actions in online social media. In: Proceedings of the international conference on computational science and engineering, pp 151–158

  12. Chung F (1997) Spectral graph theory. American Mathematical Society, Providence

    MATH  Google Scholar 

  13. Doyle PG, Snell JL (1984) Random walks and electric networks. Mathematical Association of America, Washington, DC

    MATH  Google Scholar 

  14. Fouss F, Pirotte A, Renders J-M, Saerens M (2004) A novel way of computing dissimilarities between nodes of a graph, with application to collaborative filtering and subspace projection of the graph nodes. In: Proceedings of European conference on machine learning, pp 26–37

  15. Fouss F, Yen L, Pirotte A, Saerens M (2006) An experimental investigation of graph kernels on a collaborative recommendation task. In: Proceedings of the international conference on data mining, pp 863–868

  16. Goldenberg A, Kubica J, Komarek P (2003) A comparison of statistical and machine learning algorithms on the task of link completion. In Proceedings workshop on link analysis for detecting complex behavior

  17. Guha R, Kumar R, Raghavan P, Tomkins A (2004) Propagation of trust and distrust. In: Proceedings of the international world wide web conference, pp 403–412

  18. Guimerà R, Sales-Pardo M, Amaral LAN (2004) Modularity from fluctuations in random graphs and complex networks. Phys Rev E 70(2):025101

    Article  Google Scholar 

  19. Huang Z, Chung W, Chen H (2004) A graph model for e-commerce recommender systems. Am Soc Inf Sci Technol 55(3):259–274

    Article  Google Scholar 

  20. Huang Z, Zeng D, Chen H (2007) A comparison of collaborative-filtering recommendation algorithms for e-commerce. IEEE Intell Syst 22(5):68–78

    Article  Google Scholar 

  21. Ito T, Shimbo M, Kudo T, Matsumoto Y (2005) Application of kernels to link analysis. In: Proceedings of the international conference on knowledge discovery in data mining, pp 586–592

  22. Järvelin K, Kekäläinen J (2002) Cumulated gain-based evaluation of IR techniques. ACM Trans Inf Syst 20(4):422–446

    Article  Google Scholar 

  23. Kandola J, Shawe-Taylor J, Cristianini N (2002) Learning semantic similarity. In Advances in neural information processing systems, pp 657–664

  24. Katz L (1953) A new status index derived from sociometric analysis. Psychometrika 18(1):39–43

    Article  MATH  Google Scholar 

  25. Klein DJ, Randić M (1993) Resistance distance. J Math Chem 12(1):81–95

    Article  MathSciNet  Google Scholar 

  26. Klimt B, Yang Y (2004) The Enron corpus: A new dataset for email classification research. In: Proceedings of European conference on machine learning, pp 217–226

  27. Kondor R, Lafferty J (2002) Diffusion kernels on graphs and other discrete structures. In: Proceedings of the international conference on machine learning, pp 315–322

  28. Kunegis J, Fay D, Bauckhage C (2010) Network growth and the spectral evolution model. In: Proceedings of the international conference on information and knowledge management, pp 739–748

  29. Kunegis J, Lommatzsch A (2009) Learning spectral graph transformations for link prediction. In: Proceedings of the international conference on machine learning, pp 561–568

  30. Kunegis J, Lommatzsch A, Bauckhage C (2009) The slashdot zoo: mining a social network with negative edges. In: Proceedings of the international world wide web conference, pp 741–750

  31. Kunegis J, Schmidt S, Lommatzsch A, Lerner J (2010) Spectral analysis of signed graphs for clustering, prediction and visualization. In: Proceedings of SIAM international conference on data mining, pp 559–570

  32. Lü L, Jin C-H, Zhou T (2009) Similarity index based on local paths for link prediction of complex networks. Phys. Rev. E 80(4):046122

    Article  Google Scholar 

  33. Lax PD (1984) Linear algebra and its applications. Wiley, London

    Google Scholar 

  34. Lee DD, Seung SH (2000) Algorithms for non-negative matrix factorization. In: Advances in neural information processing systems, pp 556–562

  35. Leskovec J, Backstrom L, Kumar R, Tomkins A (2008) Microscopic evolution of social networks. In: Proceedings of the international conference on knowledge discovery and data mining, pp 462–470

  36. Leskovec J, Huttenlocher D, Kleinberg J (2010) Governance in social media: A case study of the Wikipedia promotion process. In: Proceedings of th international conference on weblogs and social media

  37. Leskovec J, Kleinberg J, Faloutsos C (2007) Graph evolution: densification and shrinking diameters. ACM Trans Knowl Discov Data 1(1):1–40

    Article  Google Scholar 

  38. Ley M (2002) The DBLP computer science bibliography: evolution, research issues, perspectives. In: Proceedings of the international symposium on string processing and information retrieval, pp 1–10

  39. Liben-Nowell, D. and Kleinberg, J. (2003), The link prediction problem for social networks. In: Proceedings of the international conference on information and knowledge management, pp 556–559

  40. Liu W, Qian B, Cui J, Liu J (2009) Spectral kernel learning for semi-supervised classification. In: Proceedings of the international joint conference on artificial intelligence, pp 1150–1155

  41. Long B, Zhang Z, Yu PS (2010) A general framework for relation graph clustering. J Knowl Inf Syst 24(3):393–413

    Article  Google Scholar 

  42. Lu Z, Jain P, Dhillon IS (2009) Geometry-aware metric learning In: Proceedings of the international conference on machine learning, pp 673–680

  43. von Luxburg U (2007) A tutorial on spectral clustering. Stat Comput 17(4):395–416

    Article  MathSciNet  Google Scholar 

  44. Manning CD, Raghavan P, Schütze H (2008) Introduction to information retrieval. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  45. Massa P, Avesani P (2005) Controversial users demand local trust metrics: an experimental study on epinions.com community. In: Proceedings of American association for artificial intelligence conference, pp 121–126

  46. Mislove A (2009) Online social networks: measurement, analysis, and applications to distributed information systems. PhD thesis, Rice University

  47. Mislove A, Koppula HS, Gummadi KP, Druschel P, Bhattacharjee B (2008) Growth of the Flickr social network. In: Proceedings of the workshop on online social networks, pp 25–30

  48. Mohar B (1991) The Laplacian spectrum of graphs. Graph Theory Comb Appl 2:871–898

    MathSciNet  Google Scholar 

  49. Najork MA, Zaragoza H, Taylor MJ (2007) HITS on the Web: how does it compare? In: Proceedings of the international conference on research and development in information retrieval, pp 471–478

  50. Newman MEJ (2006) Finding community structure in networks using the eigenvectors of matrices. Phys Rev E 74(3):036104

    Article  MathSciNet  Google Scholar 

  51. Peng W, Li T (2011) Temporal relation co-clustering on directional social network and author-topic evolution. J Knowl Inf Syst 26(3):467–486

    Article  Google Scholar 

  52. Radl A, von Luxburg U, Hein M (2009) The resistance distance is meaningless for large random geometric graphs. In: Proceedings of the workshop on analyzing networks and learning with graphs

  53. Sarwar B, Karypis G, Konstan J, Riedl J (2000) Application of dimensionality reduction in recommender systems-a case study. In: Proceedings of ACM WebKDD Workshop

  54. Smola A, Kondor R (2003) Kernels and regularization on graphs. In: Proceedings of the conference on learning theory and kernel machines, pp 144–158

  55. Stewart GW (1990) Perturbation theory for the singular value decomposition, Technical report. University of Maryland, College Park

  56. Sun J, Tao D, Faloutsos C (2006) Beyond streams and graphs: dynamic tensor analysis. In: Proceedings of the international conference on knowledge discovery and data mining, pp 374–383

  57. Thurau C, Kersting K, Wahabzada M, Bauckhage C (2011) Convex non-negative matrix factorization for massive datasets. J Knowl Inf Syst 29(2):457–478

    Article  Google Scholar 

  58. Viswanath B, Mislove A, Cha M, Gummadi KP (2009) On the evolution of user interaction in Facebook. FIn: Proceedings of the workshop on online social networks, pp 37–42

  59. Wax M, Sheinvald J (1997) A least-squares approach to joint diagonalization. IEEE Signal Process Lett 4(2):52–53

    Article  Google Scholar 

  60. Wedin P-Å (1972) Perturbation bounds in connection with singular value decomposition. BIT Numer Math 12(1):99–111

    Article  MathSciNet  MATH  Google Scholar 

  61. Weyl H (1912) Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differenzialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math Ann 71(4):441–479

    Article  MathSciNet  MATH  Google Scholar 

  62. Wilkinson JH (1965) The algebraic eigenvalue problem. Oxford University Press, Oxford

    MATH  Google Scholar 

  63. Zhang B, Liu R, Massey D, Zhang L (2005) Collecting the internet AS-level topology. SIGCOMM Comput Commun Rev 35(1):53–61

    Article  Google Scholar 

  64. Zhang D, Mao R (2008) Classifying networked entities with modularity kernels. In Proceedings of the conference on information and knowledge management, pp 113–122

  65. Zhu X, Kandola J, Lafferty J, Ghahramani Z (2006) Semi-supervised learning, MIT Press, chapter graph kernels by Spectral Transforms

Download references

Acknowledgments

The research leading to these results has received funding from the European Community’s Seventh Frame Programme under grant agreement no 257859, ROBUST, and from the German Research Foundation (DFG) under the Multipla project (grant 38457858).

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Authors and Affiliations

  1. Institute for Web Science and Technologies, Universität Koblenz–Landau, Universitätsstraße 1, 56070, Koblenz, Germany

    Jérôme Kunegis

  2. University College Cork, Cork, Ireland

    Damien Fay

  3. Fraunhofer IAIS, Sankt Augustin, Germany

    Christian Bauckhage

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  1. Jérôme Kunegis
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Correspondence to Jérôme Kunegis.

Appendix: list of datasets

Appendix: list of datasets

This is the complete list of network datasets; all are from the Koblenz Network Collection (KONECT, konect.uni-koblenz.de). The Haggle dataset (CO) represents contacts between persons [7]. The Digg datasets (DG) contain message replies between users of the Web site Digg [11]. The English Wikipedia vote datasets (EL) represent administrator votes between users of the English Wikipedia [36]. The Enron dataset (EN) is a network of e-mail messages between employees of Enron [26]. Epinions (EP) is a trust/distrust network from the Web site of the same name [45]. The Flickr network (FL) contains user–user friendship links from the Flickr image sharing Web site [47]. The two Facebook datasets (Ol, Ow) contain the friendship links and wall messages between users of the social network Facebook in New Orleans [58]. The DBLP network (Pc) contains the coauthorship links between authors in the DBLP bibliography [38]. The arXiv hep-ph and hep-th networks (PH, TH) also contain the coauthorship links between authors in the corresponding sections of arXiv [37]. The Internet topology dataset (TO) contains the structural network of the Internet [63]. The Twitter network (Wa) contains the user–user mentions using the “@” syntax in Twitter [10]. The English Wikipedia hyperlink network (WP) contains all links between articles of the English Wikipedia [46]. The YouTube dataset (YT) contains the user–user friendships from video-sharing site YouTube [46] (Table 5).

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Kunegis, J., Fay, D. & Bauckhage, C. Spectral evolution in dynamic networks. Knowl Inf Syst 37, 1–36 (2013). https://doi.org/10.1007/s10115-012-0575-9

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