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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].

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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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