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.

This letter addresses the problem of designing the transition probabilities of a finite Markov chain (the policy) in order to minimize the expected cost for reaching a destination node from a source node while maintaining a fixed level of entropy spread throughout the network (the exploration). It is motivated by the following scenario. Suppose you have to route agents through a network in some optimal way, for instance by minimizing the total travel cost. Nothing particular up to now -you could use a standard shortest-path algorithm. Suppose, however, that you want to avoid pure deterministic routing policies in order, for instance, to allow some continual exploration of the network, to avoid congestion, or to avoid complete predictability of your routing strategy. In other words, you want to introduce some randomness/unpredictability in the routing policy, i.e., the routing policy is randomized. This problem, which will be called the randomized shortest-path problem (RSP), is investigated in this work. The global level of randomness of the routing policy is quantified by the expected Shannon entropy spread throughout the network, and is provided a priori by the designer. Then, necessary conditions allowing to compute the optimal randomized policy -minimizing the expected routing cost -are derived. Iterating these necessary conditions, reminiscent of Bellman's value iteration equations, allows to compute an optimal policy, that is, a set of transition probabilities in each node. Interestingly and surprisingly enough, this first model, while formulated in a totally different framework, is equivalent to Akamatsu's model (Akamatsu (1996)), appearing in transportation science, for a special choice of the entropy constraint. We therefore revisit Akamatsu's model by recasting it into a sum-over-paths statistical-physics formalism allowing to easily derive all the quantities of interest in an elegant, unified, way. For instance, it is shown that the unique optimal policy can be obtained by solving a simple linear system of equations. This second * Marco Saerens is also a Research Fellow of the IRIDIA Laboratory, Université Libre de Bruxelles. 1 model is therefore more convincing because of its computational efficiency and soundness. Finally, simulation results obtained on simple, illustrative, examples show that the models behave as expected.

Abstract. This work presents a novel procedure for computing (1) distances between nodes of a weighted, undirected, graph, called the Euclidean Commute Time Distance (ECTD), and (2) a subspace projection of the nodes of the graph that preserves as much variance as possible, in terms of the ECTD -a principal components analysis of the graph. It is based on a Markov-chain model of random walk through the graph. The model assigns transition probabilities to the links between nodes, so that a random walker can jump from node to node. A quantity, called the average commute time, computes the average time taken by a random walker for reaching node j for the first time when starting from node i, and coming back to node i. The square root of this quantity, the ECTD, is a distance measure between any two nodes, and has the nice property of decreasing when the number of paths connecting two nodes increases and when the "length" of any path decreases. The ECTD can be computed from the pseudoinverse of the Laplacian matrix of the graph, which is a kernel. We finally define the Principal Components Analysis (PCA) of a graph as the subspace projection that preserves as much variance as possible, in terms of the ECTD. This graph PCA has some interesting links with spectral graph theory, in particular spectral clustering.

Network data are produced automatically by everyday interactions-social networks, power grids, and citations between documents are a few examples. Such data capture social and economic behavior in a form that can be analyzed using powerful computational tools. This book is a guide to both basic and advanced techniques and algorithms for extracting useful information from network data. The content is organized around "tasks," grouping the algorithms needed to gather specific types of information and thus answer specific types of questions. Examples include similarity between nodes in a network, prestige or centrality of individual nodes, and dense regions or communities in a network. Algorithms are derived in detail and summarized in pseudo-code. The book is intended primarily for computer scientists, engineers, statisticians, and physicists, but is accessible to network scientists based in the social sciences. Matlab/Octave code illustrating some of the algorithms will

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