Abstract-This work presents a new perspective on characterizing the similarity between elements of a database or, more generally, nodes of a weighted and undirected graph. It is based on a Markov-chain model of random walk through the database. More precisely, we compute quantities (the average commute time, the pseudoinverse of the Laplacian matrix of the graph, etc.) that provide similarities between any pair of nodes, having the nice property of increasing when the number of paths connecting those elements increases and when the "length" of paths decreases. It turns out that the square root of the average commute time is a Euclidean distance and that the pseudoinverse of the Laplacian matrix is a kernel matrix (its elements are inner products closely related to commute times). A principal component analysis (PCA) of the graph is introduced for computing the subspace projection of the node vectors in a manner that preserves as much variance as possible in terms of the Euclidean commute-time distance. This graph PCA provides a nice interpretation to the "Fiedler vector," widely used for graph partitioning. The model is evaluated on a collaborativerecommendation task where suggestions are made about which movies people should watch based upon what they watched in the past. Experimental results on the MovieLens database show that the Laplacian-based similarities perform well in comparison with other methods. The model, which nicely fits into the so-called "statistical relational learning" framework, could also be used to compute document or word similarities, and, more generally, it could be applied to machine-learning and pattern-recognition tasks involving a relational database.

It sometimes happens (for instance in case control studies) that a classifier is trained on a data set that does not reflect the true a priori probabilities of the target classes on real-world data. This may have a negative effect on the classification accuracy obtained on the real-world data set, especially when the classifier's decisions are based on the a posteriori probabilities of class membership. Indeed, in this case, the trained classifier provides estimates of the a posteriori probabilities that are not valid for this real-world data set (they rely on the a priori probabilities of the training set). Applying the classifier as is (without correcting its outputs with respect to these new conditions) on this new data set may thus be suboptimal. In this note, we present a simple iterative procedure for adjusting the outputs of the trained classifier with respect to these new a priori probabilities without having to refit the model, even when these probabilities are not known in advance. As a by-product, estimates of the new a priori probabilities are also obtained. This iterative algorithm is a straightforward instance of the expectation-maximization (EM) algorithm and is shown to maximize the likelihood of the new data. Thereafter, we discuss a statistical test that can be applied to decide if the a priori class probabilities have changed from the training set to the real-world data. The procedure is illustrated on different classification problems involving a multilayer neural network, and comparisons with a standard procedure for a priori probability estimation are provided. Our original method, based on the EM algorithm, is shown to be superior to the standard one for a priori probability estimation. Experimental results also indicate that the classifier with adjusted outputs always performs better than the original one in terms of classification accuracy, when the a priori probability conditions differ from the training set to the real-world data. The gain in classification accuracy can be significant.

There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into account the global structure of the graph. In this article, we develop the theory of one family of graph node distances, known as the randomized shortest path dissimilarity, which has its foundation in statistical physics. We show that the randomized shortest path dissimilarity can be easily computed in closed form for all pairs of nodes of a graph. Moreover, we come up with a new definition of a distance measure that we call the free energy distance. The free energy distance can be seen as an upgrade of the randomized shortest path dissimilarity as it defines a metric, in addition to which it satisfies the graph-geodetic property. The derivation and computation of the free energy distance are also straightforward. We then make a comparison between a set of generalized distances that interpolate between the shortest path distance and the commute time, or resistance distance. This comparison focuses on the applicability of the distances in graph node clustering and classification. The comparison, in general, shows that the parametrized distances perform well in the tasks. In particular, we see that the results obtained with the free energy distance are among the best in all the experiments.

This work develops a generic framework, called the bag-of-paths (BoP), for link and network data analysis. The central idea is to assign a probability distribution on the set of all paths in a network. More precisely, a Gibbs-Boltzmann distribution is defined over a bag of paths in a network, that is, on a representation that considers all paths independently. We show that, under this distribution, the probability of drawing a path connecting two nodes can easily be computed in closed form by simple matrix inversion. This probability captures a notion of relatedness between nodes of the graph: two nodes are considered as highly related when they are connected by many, preferably low-cost, paths. As an application, two families of distances between nodes are derived from the BoP probabilities. Interestingly, the second distance family interpolates between the shortest path distance and the resistance distance. In addition, it extends the Bellman-Ford formula for computing the shortest path distance in order to integrate sub-optimal paths by simply replacing the minimum operator by the soft minimum operator. Experimental results on semi-supervised classification show that both of the new distance families are competitive with other state-ofthe-art approaches. In addition to the distance measures studied in this paper, the bag-of-paths framework enables straightforward computation of many other relevant network measures.