In many applications, we need to measure similarity between nodes in a large network based on features of their neighborhoods. Although in-network node similarity based on proximity has been well investigated, surprisingly, measuring in-network node similarity based on neighborhoods remains a largely untouched problem in literature. One challenge is that in different applications we may need different measurements that manifest different meanings of similarity. Furthermore, we often want to make trade-offs between specificity of neighborhood matching and efficiency. In this paper, we investigate the problem in a principled and systematic manner. We develop a unified parametric model and a series of four instance measures. Those instance similarity measures not only address a spectrum of various meanings of similarity, but also present a series of trade-offs between computational cost and strictness of matching between neighborhoods of nodes being compared. By extensive experiments and case studies, we demonstrate the effectiveness of the proposed model and its instances.
Node proximity is widely adopted as similarity measures in many data mining and machine learning tasks. Koutra @cite_18 proposed a unified framework of Guilt-by-Association. A wide range of popular proximity measures, such as Personalized PageRank @cite_9 , Random Walk with Restart @cite_10 and Generalized Belief Propagation @cite_28 , are special cases or can be approximated by Guilt-by-Association. The basic assumption behind those proximity-based measures is that nodes in a network influence each other through edges and paths. Thus, nodes within the same community are often more similar than those from different communities. Apparently, proximity-based measures cannot be used to solve the problems demonstrated in Example .
@cite_28 @cite_9 @cite_18 @cite_10
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Important inference problems in statistical physics, computer vision, error-correcting coding theory, and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs. The belief propagation (BP) algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree, but only approximate when the factor graph has cycles. We show that BP fixed points correspond to the stationary points of the Bethe approximation of the free energy for a factor graph. We explain how to obtain region-based free energy approximations that improve the Bethe approximation, and corresponding generalized belief propagation (GBP) algorithms. We emphasize the conditions a free energy approximation must satisfy in order to be a "valid" or "maxent-normal" approximation. We describe the relationship between four different methods that can be used to generate valid approximations: the "Bethe method", the "junction graph method", the "cluster variation method", and the "region graph method". Finally, we explain how to tell whether a region-based approximation, and its corresponding GBP algorithm, is likely to be accurate, and describe empirical results showing that GBP can significantly outperform BP. Recent web search techniques augment traditional text matching with a global notion of "importance" based on the linkage structure of the web, such as in Google's PageRank algorithm. For more refined searches, this global notion of importance can be specialized to create personalized views of importance--for example, importance scores can be biased according to a user-specified set of initially-interesting pages. Computing and storing all possible personalized views in advance is impractical, as is computing personalized views at query time, since the computation of each view requires an iterative computation over the web graph. We present new graph-theoretical results, and a new technique based on these results, that encode personalized views as partial vectors. Partial vectors are shared across multiple personalized views, and their computation and storage costs scale well with the number of views. Our approach enables incremental computation, so that the construction of personalized views from partial vectors is practical at query time. We present efficient dynamic programming algorithms for computing partial vectors, an algorithm for constructing personalized views from partial vectors, and experimental results demonstrating the effectiveness and scalability of our techniques. If several friends of Smith have committed petty thefts, what would you say about Smith? Most people would not be surprised if Smith is a hardened criminal. Guilt-by-association methods combine weak signals to derive stronger ones, and have been extensively used for anomaly detection and classification in numerous settings (e.g., accounting fraud, cyber-security, calling-card fraud). The focus of this paper is to compare and contrast several very successful, guilt-by-association methods: Random Walk with Restarts, Semi-Supervised Learning, and Belief Propagation (BP). Our main contributions are two-fold: (a) theoretically, we prove that all the methods result in a similar matrix inversion problem; (b) for practical applications, we developed FaBP, a fast algorithm that yields 2× speedup, equal or higher accuracy than BP, and is guaranteed to converge. We demonstrate these benefits using synthetic and real datasets, including YahooWeb, one of the largest graphs ever studied with BP. How closely related are two nodes in a graph? How to compute this score quickly, on huge, disk-resident, real graphs? Random walk with restart (RWR) provides a good relevance score between two nodes in a weighted graph, and it has been successfully used in numerous settings, like automatic captioning of images, generalizations to the "connection subgraphs", personalized PageRank, and many more. However, the straightforward implementations of RWR do not scale for large graphs, requiring either quadratic space and cubic pre-computation time, or slow response time on queries. We propose fast solutions to this problem. The heart of our approach is to exploit two important properties shared by many real graphs: (a) linear correlations and (b) block- wise, community-like structure. We exploit the linearity by using low-rank matrix approximation, and the community structure by graph partitioning, followed by the Sherman- Morrison lemma for matrix inversion. Experimental results on the Corel image and the DBLP dabasets demonstrate that our proposed methods achieve significant savings over the straightforward implementations: they can save several orders of magnitude in pre-computation and storage cost, and they achieve up to 150x speed up with 90 + quality preservation.