HITS vs. Non-negative Matrix Factorization

Ranking algorithms have been widely used for web and other networks to infer quality/popularity. Both PageRank and HITS were developed for ranking web pages from a web reference graph. Nevertheless, these algorithms have also been applied extensively for a variety of other applications such as question-answer services, author-paper graphs, and others where a graph can be deduced from the data set. The intuition behind HITS has been explained in terms of hubs and spokes as two values are inferred for each node. HITS has also been used extensively for ranking in other applications although it is not clear whether the same intuition carries over. It would be beneficial if we can understand these algorithms mathematically in a general manner so that the results can be interpreted and understood better for different applications. This paper provides such as understanding for applying HITS algorithm to other applications. In this paper, we generalize the graph semantics in terms two underlying concepts: in-link probability (ILP) and out-link probability (OLP). Using these two, the rank scores of nodes in a graph are computed. We propose the standard non-negative matrix factorization (NMF) approach to calculate ILP and OLP vectors. We also establish a relationship between HITS vectors and ILP/OLP vectors which enables us to better understand HITS vectors associated with any graph in terms of these two probabilities. Finally, we illustrate the versatility of our approach using different graph types (representing different application areas) and validate the results. This work provides an alternative way of understanding HITS algorithm for a variety of applications.