The prediction of graph evolution is an important and challenging problem in the analysis of networks and of the Web in particular. But while the appearance of new links is part of virtually every model of Web growth, the disappearance of links has received much less attention in the literature. To fill this gap, our approach DecLiNe (an acronym for DECay of LInks in NEtworks) aims to predict link decay in networks, based on structural analysis of corresponding graph models. In analogy to the link prediction problem, we show that analysis of graph structures can help to identify indicators for superfluous links under consideration of common network models. In doing so, we introduce novel metrics that denote the likelihood of certain links in social graphs to remain in the network, and combine them with state-of-the-art machine learning methods for predicting link decay. Our methods are independent of the underlying network type, and can be applied to such diverse networks as the Web, social networks and any other structure representable as a network, and can be easily combined with case-specific content analysis and adopted for a variety of social network mining, filtering and recommendation applications. In systematic evaluations with large-scale datasets of Wikipedia we show the practical feasibility of the proposed structure-based link decay prediction algorithms.
The evolution of networks such as the Web is subject to many models such as preferential attachment @cite_18 or the spectral evolution model @cite_15 , most of which only model the addition of edges over time. The evolution of the Web hyperlink graph in particular has been studied too, for instance in @cite_23 . The evolution of the Wikipedia hyperlink graph has been studied in 2006 @cite_5 . The concrete problem consisting of predicting the appearance of new links in networks is called @cite_24 . Unlike link prediction, the prediction of link disappearance has been investigated only very little.
@cite_18 @cite_24 @cite_23 @cite_5 @cite_15
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In this thesis, I study the spectral characteristics of large dynamic networks and formulate the spectral evolution model. The spectral evolution model applies to networks that evolve over time, and describes their spectral decompositions such as the eigenvalue and singular value decomposition. The spectral evolution model states that over time, the eigenvalues of a network change while its eigenvectors stay approximately constant. I validate the spectral evolution model empirically on over a hundred network datasets, and theoretically by showing that it generalizes arncertain number of known link prediction functions, including graph kernels, path counting methods, rank reduction and triangle closing. The collection of datasets I use contains 118 distinct network datasets. One dataset, the signed social network of the Slashdot Zoo, was specifically extracted during work on this thesis. I also show that the spectral evolution model can be understood as a generalization of the preferential attachment model, if we consider growth in latent dimensions of a network individually. As applications of the spectral evolution model, I introduce two new link prediction algorithms that can be used for recommender systems, search engines, collaborative filtering, rating prediction, link sign prediction and more. The first link prediction algorithm reduces to a one-dimensional curve fitting problem from which a spectral transformation is learned. The second method uses extrapolation of eigenvalues to predict future eigenvalues. As special cases, I show that the spectral evolution model applies to directed, undirected, weighted, unweighted, signed and bipartite networks. For signed graphs, I introduce new applications of the Laplacian matrix for graph drawing, spectral clustering, and describe new Laplacian graph kernels. I also define the algebraic conflict, a measure of the conflict present in a signed graph based on the signed graph Laplacian. I describe the problem of link sign prediction spectrally, and introduce the signed resistance distance. For bipartite and directed graphs, I introduce the hyperbolic sine and odd Neumann kernels, which generalize the exponential and Neumann kernels for undirected unipartite graphs. I show that the problem of directed and bipartite link prediction are related by the fact that both can be solved by considering spectral evolution in the singular value decomposition. The Web is characterized by an extremely dynamic nature, as it is proved by the rapid and significant growth it has experimented in the last decade and by its continuous evolution through creation or deletion of pages and hyperlinks. Consequently, analyzing the temporal evolution of the Web has become a crucial task that can provide search engines with valuable information for refining crawling policies, improving ranking models or detecting spam. Understanding how the Web evolves over time is a delicate challenge that requires to integrate theoretical efforts of modelization and empirical results. Obtaining such findings is very expensive in terms of bandwidth, computation time and human intervention. Robust software is required to gather the data and provide easy access to the collected information. Apart for commercial engines, there have been only a few attempts to perform such task and to make the data available. Several previous works (e.g., ntoulas2004what,toyoda2006what ) analyzed a sequence of crawls, studying degree and frequency of change both in the content of Web pages and in the hyperlink structure to propose measures of their novelty. In this paper we study a temporal dataset BSVLTAG that has recently been made public. It is made of twelve 100M pages snapshots of the .uk domain. The Web graphs of the single snapshots have been merged into a global graph with labels that provide constant time access to temporal information. bordino2008temporal describes the work done to assess the data and to study some aspects of its evolution at the level of Web pages. We now analyze the structure of this huge time-aware graph at the level of interconnection between hosts. We study the , i.e., the graph in which every node corresponds to a site, whereas a directed edge represents the existence of hyperlinks between pages belonging to two different hosts. Understanding the graph structure of the Web at this macroscopic level can provide valuable insights for improving Web site accessibility and navigation, or discovering related hosts. The notion of hostgraph was proposed by bharat2001who . A few studies analyzed hostgraphs ( baeza2005spain,liu2005china,toyoda2008thai ). baeza2007domain presents a comparison among the results of twelve characterization studies of several national domains. A key ingredient of many current models proposed to capture the topological evolution of complex networks is the hypothesis that highly connected nodes increase their connectivity faster than their less connected peers, a phenomenon called preferential attachment. Measurements on four networks, namely the science citation network, Internet, actor collaboration and science coauthorship network indicate that the rate at which nodes acquire links depends on the node's degree, offering direct quantitative support for the presence of preferential attachment. We find that for the first two systems the attachment rate depends linearly on the node degree, while for the last two the dependence follows a sublinear power law. 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, authorship, rating, citation, communication and tagging networks, covering unipartite, bipartite, signed and unsigned graphs. Following these observations, we introduce a link prediction algorithm based on the extrapolation of a network's spectral evolution. This new link prediction method generalizes several common graph kernels that can be expressed as spectral transformations. In contrast to these graph kernels, the spectral extrapolation algorithm does not make assumptions about specific growth patterns beyond the spectral evolution model. We thus show that it performs particularly well for networks with irregular, but spectral, growth patterns.