text: The automated analysis of social networks has become an important problem due to the proliferation of social networks, such as LiveJournal, Flickr and Facebook. The scale of these social networks is massive and continues to grow rapidly. An important problem in social network analysis is proximity estimation that infers the closeness of different users. Link prediction, in turn, is an important application of proximity estimation. However, many methods for computing proximity measures have high computational complexity and are thus prohibitive for large-scale link prediction problems. One way to address this problem is to estimate proximity measures via low-rank approximation. However, a single low-rank approximation may not be sufficient to represent the behavior of the entire network. In this paper, we propose Multi-Scale Link Prediction (MSLP), a framework for link prediction, which can handle massive networks. The basis idea of MSLP is to construct low rank approximations of the network at multiple scales in an efficient manner. Based on this approach, MSLP combines predictions at multiple scales to make robust and accurate predictions. Experimental results on real-life datasets with more than a million nodes show the superior performance and scalability of our method.
  @cite_0: Proximity measures quantify the closeness or similarity between nodes in a social network and form the basis of a range of applications in social sciences, business, information technology, computer networks, and cyber security. It is challenging to estimate proximity measures in online social networks due to their massive scale (with millions of users) and dynamic nature (with hundreds of thousands of new nodes and millions of edges added daily). To address this challenge, we develop two novel methods to efficiently and accurately approximate a large family of proximity measures. We also propose a novel incremental update algorithm to enable near real-time proximity estimation in highly dynamic social networks. Evaluation based on a large amount of real data collected in five popular online social networks shows that our methods are accurate and can easily scale to networks with millions of nodes. To demonstrate the practical values of our techniques, we consider a significant application of proximity estimation: link prediction, i.e., predicting which new edges will be added in the near future based on past snapshots of a social network. Our results reveal that (i) the effectiveness of different proximity measures for link prediction varies significantly across different online social networks and depends heavily on the fraction of edges contributed by the highest degree nodes, and (ii) combining multiple proximity measures consistently yields the best link prediction accuracy.
  @cite_16: Abstract We explore methods for approximating the commute time and Katz score between a pair of nodes. These methods are based on the approach of matrices, moments, and quadrature developed in the numerical linear algebra community. They rely on the Lanczos process and provide upper and lower bounds on an estimate of the pairwise scores. We also explore methods to approximate the commute times and Katz scores from a node to all other nodes in the graph. Here, our approach for the commute times is based on a variation of the conjugate gradient algorithm, and it provides an estimate of all the diagonals of the inverse of a matrix. Our technique for the Katz scores is based on exploiting an empirical localization property of the Katz matrix. We adapt algorithms used for personalized PageRank computing to these Katz scores and theoretically show that this approach is convergent. We evaluate these methods on 17 real-world graphs ranging in size from 1000 to 1,000,000 nodes. Our results show that our pairwise c...
  @cite_23: Given a snapshot of a social network, can we infer which new interactions among its members are likely to occur in the near future? We formalize this question as the link prediction problem, and develop approaches to link prediction based on measures the "proximity" of nodes in a network. Experiments on large co-authorship networks suggest that information about future interactions can be extracted from network topology alone, and that fairly subtle measures for detecting node proximity can outperform more direct measures.
  @cite_8: One of the core tasks in social network analysis is to predict the formation of links (i.e. various types of relationships) over time. Previous research has generally represented the social network in the form of a graph and has leveraged topological and semantic measures of similarity between two nodes to evaluate the probability of link formation. Here we introduce a novel local probabilistic graphical model method that can scale to large graphs to estimate the joint co-occurrence probability of two nodes. Such a probability measure captures information that is not captured by either topological measures or measures of semantic similarity, which are the dominant measures used for link prediction. We demonstrate the effectiveness of the co-occurrence probability feature by using it both in isolation and in combination with other topological and semantic features for predicting co-authorship collaborations on real datasets.
summary: Many popular proximity measures that are used for link prediction have high computational complexity and do not scale well to large-scale networks. A great deal of recent work has been devoted to speedup the computation. For example, @cite_8 truncates the series expansion of Katz and only considers paths of length up to some threshold. In @cite_23 @cite_0 , dimensionality reduction methods, such as the eigen-decomposition, are used to construct low rank approximations of a graph, which are then used to compute approximated proximity measures. The more recent work in @cite_16 applies the Lanczos Stieltjes procedure to iteratively compute upper and lower bounds of a single Katz value and shows that these eventually converge to the real Katz value.