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THE ELECTRICAL RESISTANCE OF A GRAPH CAPTURES ITS COMMUTE AND COVER TIMES
"... View an nvertex, medge undirected graph as an electrical network with unit resistors as edges. We extend known relations between random walks and electrical networks by showing that resistance in this network is intimately connected with the lengths of random walks on the graph. For example, the c ..."
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Cited by 196 (5 self)
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View an nvertex, medge undirected graph as an electrical network with unit resistors as edges. We extend known relations between random walks and electrical networks by showing that resistance in this network is intimately connected with the lengths of random walks on the graph. For example, the commute time between two vertices s and t (the expected length of a random walk from s to t and back) is precisely characterized by the e ective resistance Rst between s and t: commute time = 2mRst. As a corollary, the cover time (the expected length of a random walk visiting all vertices) is characterized by the maximum resistance R in the graph to within a factor of log n: mR cover time O(mR log n). For many graphs, the bounds on cover time obtained in this manner are better than those obtained from previous techniques such as the eigenvalues of the adjacency matrix. In particular, we improve known bounds on cover times for highdegree graphs and expanders, and give new proofs of known results for multidimensional meshes. Moreover, resistance seems to provide an intuitively appealing and tractable approach to these problems.
Randomwalk computation of similarities between nodes of a graph, with application to collaborative recommendation
 IEEE Transactions on Knowledge and Data Engineering
"... ABSTRACT This work presents a new perspective on characterizing the similarity between elements of a database or, more generally, nodes of a weighted, undirected, graph. It is based on a Markovchain model of random walk through the database. More precisely, we compute quantities (the average commu ..."
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Cited by 194 (19 self)
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ABSTRACT This work presents a new perspective on characterizing the similarity between elements of a database or, more generally, nodes of a weighted, undirected, graph. It is based on a Markovchain 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 (it contains innerproducts closely related to commute times). A procedure for computing the subspace projection of the node vectors of the graph that preserves as much variance as possible in terms of the commutetime distance a principal components analysis (PCA) of the graph is also introduced. 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 Laplacianbased similarities perform well in comparison with other methods. The model, which nicely fits into the socalled "statistical relational learning" framework, could also be used to compute document or word similarities, and, more generally, could be applied to machinelearning and patternrecognition tasks involving a database. * François Fouss, Alain Pirotte and Marco Saerens are with the
The Principal Components Analysis of a Graph, and its Relationships to Spectral Clustering
 Proceedings of the 15th European Conference on Machine Learning (ECML 2004). Lecture Notes in Artificial Intelligence
, 2004
"... 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 princi ..."
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Cited by 101 (21 self)
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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 Markovchain 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 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.
A Comprehensive Survey of Neighborhoodbased Recommendation Methods.
 In Recommender Systems Handbook,
, 2011
"... Abstract Among collaborative recommendation approaches, methods based on nearestneighbors still enjoy a huge amount of popularity, due to their simplicity, their efficiency, and their ability to produce accurate and personalized recommendations. This chapter presents a comprehensive survey of neig ..."
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Cited by 69 (0 self)
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Abstract Among collaborative recommendation approaches, methods based on nearestneighbors still enjoy a huge amount of popularity, due to their simplicity, their efficiency, and their ability to produce accurate and personalized recommendations. This chapter presents a comprehensive survey of neighborhoodbased methods for the item recommendation problem. In particular, the main benefits of such methods, as well as their principal characteristics, are described. Furthermore, this document addresses the essential decisions that are required while implementing a neighborhoodbased recommender system, and gives practical information on how to make such decisions. Finally, the problems of sparsity and limited coverage, often observed in large commercial recommender systems, are discussed, and a few solutions to overcome these problems are presented.
Collective entity linking in web text: A graphbased method
 in: Proceedings of the 34th international Conference on Research and Development in Information Retrieval
, 2011
"... Entity Linking (EL) is the task of linking name mentions in Web text with their referent entities in a knowledge base. Traditional EL methods usually link name mentions in a document by assuming them to be independent. However, there is often additional interdependence between different EL decisions ..."
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Cited by 52 (2 self)
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Entity Linking (EL) is the task of linking name mentions in Web text with their referent entities in a knowledge base. Traditional EL methods usually link name mentions in a document by assuming them to be independent. However, there is often additional interdependence between different EL decisions, i.e., the entities in the same document should be semantically related to each other. In these cases, Collective Entity Linking, in which the name mentions in the same document are linked jointly by exploiting the interdependence between them, can improve the entity linking accuracy. This paper proposes a graphbased collective EL method, which can model and exploit the global interdependence between different EL decisions. Specifically, we first propose a graphbased representation, called Referent Graph, which can model the global interdependence between different EL decisions. Then we propose a collective inference algorithm, which can jointly infer the referent entities of all name mentions by exploiting the interdependence captured in Referent Graph. The key benefit of our method comes from: 1) The global interdependence model of EL decisions; 2) The purely collective nature of the inference algorithm, in which evidence for related EL decisions can be reinforced into highprobability decisions. Experimental results show that our method can achieve significant performance improvement over the traditional EL methods.
Clustering Using a Random Walk Based Distance Measure
, 2005
"... This work proposes a simple way to improve a clustering algorithm. ..."
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Cited by 34 (3 self)
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This work proposes a simple way to improve a clustering algorithm.
A family of dissimilarity measures between nodes generalizing both the shortestpath and the commutetime distances
 in Proceedings of the 14th SIGKDD International Conference on Knowledge Discovery and Data Mining
"... This work introduces a new family of linkbased dissimilarity measures between nodes of a weighted directed graph. This measure, called the randomized shortestpath (RSP) dissimilarity, depends on a parameter θ and has the interesting property of reducing, on one end, to the standard shortestpath d ..."
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Cited by 24 (11 self)
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This work introduces a new family of linkbased dissimilarity measures between nodes of a weighted directed graph. This measure, called the randomized shortestpath (RSP) dissimilarity, depends on a parameter θ and has the interesting property of reducing, on one end, to the standard shortestpath distance when θ is large and, on the other end, to the commutetime (or resistance) distance when θ is small (near zero). Intuitively, it corresponds to the expected cost incurred by a random walker in order to reach a destination node from a starting node while maintaining a constant entropy (related to θ) spread in the graph. The parameter θ is therefore biasing gradually the simple random walk on the graph towards the shortestpath policy. By adopting a statistical physics approach and computing a sum over all the possible paths (discrete path integral), it is shown that the RSP dissimilarity from every node to a particular node of interest can be computed efficiently by solving two linear systems of n equations, where n is the number of nodes. On the other hand, the dissimilarity between every couple of nodes is obtained by inverting an n × n matrix. The proposed measure can be used for various graph mining tasks such as computing betweenness centrality, finding dense communities, etc, as shown in the experimental section.
A Novel Way of Computing Dissimilarities between Nodes of a Graph, with Application to Collaborative Filtering
, 2004
"... This work presents some general procedures for computing dissimilarities between elements of a database or, more generally, nodes of a weighted, undirected, graph. It is based on a Markovchain model of random walk through the database. The model assigns transition probabilities to the links betw ..."
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Cited by 20 (1 self)
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This work presents some general procedures for computing dissimilarities between elements of a database or, more generally, nodes of a weighted, undirected, graph. It is based on a Markovchain model of random walk through the database. The model assigns transition probabilities to the links between elements, so that a random walker can jump from element to element. A quantity, called the average firstpassage cost, computes the average cost incurred by a random walker for reaching element k for the first time when starting from element i.
The sumoverpaths covariance kernel: A novel covariance measure between nodes of a directed graph
 IEEE Transactions on Pattern Analysis and Machine Intelligence
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Commute Times for a Directed Graph using an Asymmetric Laplacian
"... The expected commute times for a strongly connected directed graph are related to an asymmetric Laplacian matrix as a direct extension to similar well known formulas for undirected graphs. We show the close relationships between the asymmetric Laplacian and the socalled Fundamental matrix. We give ..."
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Cited by 15 (6 self)
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The expected commute times for a strongly connected directed graph are related to an asymmetric Laplacian matrix as a direct extension to similar well known formulas for undirected graphs. We show the close relationships between the asymmetric Laplacian and the socalled Fundamental matrix. We give bounds for the commute times in terms of the stationary probabilities for a random walk over the graph together with the asymmetric Laplacian and show how this can be approximated by a symmetrized Laplacian derived from a related weighted undirected graph.