A novel way of computing similarities 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 similaritybetween elements of a database or, more generally, nodes of a weightedand undirected graph. It is based on a Markov-chain model of randomwalk through the database. More precisely, we compute quantities (theaverage commute time, the pseudoinverse of the Laplacian ma-trix of the graph, etc) that provide similarities between any pair of nodes,having the nice property of increasing when the number of paths connect-ing those elements increases and when the “length” of paths decreases.It turns out that the square root of the average commute time is a Eu-clidean distance and that the pseudoinverse of the Laplacian matrix is akernel (its elements are inner products closely related to commute times).A procedure for computing the subspace projection of the node vectorsof the graph that preserves as much variance as possible in terms of thecommute-time distance – a principal component analysis (PCA) of thegraph – is also introduced. This graph PCA provides a nice interpreta-tion to the “Fiedler vector”, widely used for graph partitioning. Themodel is evaluated on a collaborative-recommendation task where sug-gestions are made about which movies people should watch based uponwhat they watched in the past. Experimental results on the MovieLensdatabase show that the Laplacian-based similarities (the pseudoinverse ofthe Laplacian matrix and the “random-forest matrix”) perform well incomparison with other methods. The model, which nicely fits into theso-called “statistical relational learning” framework, could also be usedto compute document or word similarities, and, more generally, it couldbe applied to machine-learning and pattern-recognition tasks involving adatabase.François Fouss, Alain Pirotte and Marco Saerens are with the Information Systems Re-search Unit (ISYS), IAG, Université catholique de Louvain, Place des Doyens 1, B-1348Louvain-la-Neuve, Belgium. Email: {saerens, pirotte, fouss}@isys.ucl.ac.be.Jean-Michel Renders is with the Xerox Research Center Europe, Chemin de Maupertuis6, 38240 Meylan (Grenoble), France. Email: [email protected].