A novel way of computing dissimilarities between nodes of a graph, with application to collaborative filtering and subspace projection of the graph nodes

This work presents a new perspective on characterizing the similarity be-tween elements of a database or, more generally, nodes of a weighted, undi-rected, graph. It is based on a Markov-chain model of random walk throughthe database. More precisely, we compute quantities (the average commutetime, the pseudoinverse of the Laplacian matrix of the graph, etc) thatprovide similarities between any pair of nodes, having the nice property of in-creasing when the number of paths connecting those elements increases andwhen the “length” of paths decreases. It turns out that the square root of theaverage commute time is a Euclidean distance and that the pseudoinverse ofthe Laplacian matrix is a kernel (it contains inner-products closely related tocommute times). A procedure for computing the subspace projection of thenode vectors of the graph that preserves as much variance as possible in termsof the commute-time distance – a principal components analysis (PCA) of thegraph – is also introduced. This graph PCA provides a nice interpretation tothe “Fiedler vector”, widely used for graph partitioning. The model is evalu-ated on a collaborative-recommendation task where suggestions are made aboutwhich movies people should watch based upon what they watched in the past.Experimental results on the MovieLens database show that the Laplacian-basedsimilarities perform well in comparison with other methods. The model, whichnicely fits into the so-called “statistical relational learning” framework, couldalso be used to compute document or word similarities, and, more generally,could be applied to machine-learning and pattern-recognition tasks involving adatabase.François Fouss, Alain Pirotte and Marco Saerens are with the Information Systems ResearchUnit (ISYS), IAG, Université catholique de Louvain, Place des Doyens 1, B-1348 Louvain-la-Neuve,Belgium. Email: {saerens, pirotte, fouss}@isys.ucl.ac.be.Jean-Michel Renders is with the Xerox Research Center Europe, Chemin de Maupertuis 6, 38240Meylan (Grenoble), France. Email: [email protected].