Stanford Matrix Considered Harmful

The idea for this note arose during the “Web Information Retrieval and Linear Algebra Algorithms” held at Schloss Dagstuhl in February 2007. Many brilliant people working on either side (numerical analysis and web search) had a chance to meet and talk for one week about mathematical and practical aspects of linear methods for ranking, and in particular (not surprisingly) PageRank and HITS. There were many scientific aspects that were surprising for both sides and that were (finally, one might say) stated clearly at the workshop. First of all, PageRank is not the most important factor in Google’s scoring, or in any search engine scoring. It is part of literally hundreds of features that are somehow combined (e.g., by a standard machine-learning framework), and its importance has decreased in time (people in the so-called “search engine optimisation” industry claim that the importance of PageRank dropped drastically around 2003, but this claim is based just on reverse engineering). Yet papers are published every day starting with claims such as “PageRank is the most important ranking. . . ” and so on. Such statements give a distorted view of reality, but clearly work very well for publishing papers. On the other hand, it is also clear that PageRank (and, more generally, link-based ranking) is useful in other areas, such as deciding which pages to crawl and which not. Moreover, when one attacks the real PageRank computation problem suddenly many sophisticated methods devised by the numerical analysis community are pretty useless because of the very large size of the matrix. This problem actually had the very positive effect of stimulating a lot of new research in numerical analysis methods that could be applied to large, sparse, irregular stochastic matrices of dimension 10 and beyond. Much of the classical work on such large systems involved matrices with considerable structure from ODEs, PDEs, queuing theory, etc. Actually, many of the more promising approaches for PageRank computation use some kind of graph structural analysis (usually related to strongly connected components) that is rather meaningless in the classical problems mentioned above. These considerations bring us to the point of this note. The problem of computing PageRank is interesting from a practical viewpoint only if the size of the matrix is large and if the type of the matrix is a web graph. What do we mean by “large”? Currently, search engines claim to index a number of pages in the order of 10. We cannot expect, as scientists, to replicate exactly