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24
A SketchBased Distance Oracle for WebScale Graphs
"... We study the fundamental problem of computing distances between nodes in large graphs such as the web graph and social networks. Our objective is to be able to answer distance queries between pairs of nodes in real time. Since the standard shortest path algorithms are expensive, our approach moves t ..."
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We study the fundamental problem of computing distances between nodes in large graphs such as the web graph and social networks. Our objective is to be able to answer distance queries between pairs of nodes in real time. Since the standard shortest path algorithms are expensive, our approach moves the timeconsuming shortestpath computation offline, and at query time only looks up precomputed values and performs simple and fast computations on these precomputed values. More specifically, during the offline phase we compute and store a small “sketch ” for each node in the graph, and at querytime we look up the sketches of the source and destination nodes and perform a simple computation using these two sketches to estimate the distance. Categories and Subject Descriptors G.2.2 [Graph Theory]: Graph algorithms, path and circuit problems
An experimental investigation of graph kernels on a collaborative recommendation task
 Proceedings of the 6th International Conference on Data Mining (ICDM 2006
, 2006
"... This paper presents a survey as well as a systematic empirical comparison of seven graph kernels and two related similarity matrices (simply referred to as graph kernels), namely the exponential diffusion kernel, the Laplacian exponential diffusion kernel, the von Neumann diffusion kernel, the regul ..."
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Cited by 27 (7 self)
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This paper presents a survey as well as a systematic empirical comparison of seven graph kernels and two related similarity matrices (simply referred to as graph kernels), namely the exponential diffusion kernel, the Laplacian exponential diffusion kernel, the von Neumann diffusion kernel, the regularized Laplacian kernel, the commutetime kernel, the randomwalkwithrestart similarity matrix, and finally, three graph kernels introduced in this paper: the regularized commutetime kernel, the Markov diffusion kernel, and the crossentropy diffusion matrix. The kernelonagraph approach is simple and intuitive. It is illustrated by applying the nine graph kernels to a collaborativerecommendation task and to a semisupervised classification task, both on several databases. The graph methods compute proximity measures between nodes that help study the structure of the graph. Our comparisons suggest that the regularized commutetime and the Markov diffusion kernels perform best, closely followed by the regularized Laplacian kernel. 1
Graph nodes clustering with the sigmoid commutetime kernel: A . . .
 DATA & KNOWLEDGE ENGINEERING
, 2009
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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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An experimental investigation of kernels on graphs for collaborative . . .
 NEURAL NETWORKS
, 2012
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Phase transition in the family of presistances
"... We study the family of presistances on graphs for p 1. This family generalizes the standard resistance distance. We prove that for any fixed graph, for p =1 the presistance coincides with the shortest path distance, for p =2it coincides with the standard resistance distance, and for p!1it converge ..."
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Cited by 7 (0 self)
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We study the family of presistances on graphs for p 1. This family generalizes the standard resistance distance. We prove that for any fixed graph, for p =1 the presistance coincides with the shortest path distance, for p =2it coincides with the standard resistance distance, and for p!1it converges to the inverse of the minimal stcut in the graph. Secondly, we consider the special case of random geometric graphs (such as knearest neighbor graphs) when the number n of vertices in the graph tends to infinity. We prove that an interesting phase transition takes place. There exist two critical thresholds p ⇤ and p ⇤ ⇤ such that if p<p ⇤ , then the presistance depends on meaningful global properties of the graph, whereas if p>p ⇤ ⇤ , it only depends on trivial local quantities and does not convey any useful information. We can explicitly compute the critical values: p ⇤ = 1 + 1/(d 1) and p ⇤ ⇤ = 1 + 1/(d 2) where d is the dimension of the underlying space (we believe that the fact that there is a small gap between p ⇤ and p ⇤ ⇤ is an artifact of our proofs). We also relate our findings to Laplacian regularization and suggest to use qLaplacians as regularizers, where q satisfies 1/p ⇤ +1/q =1. 1
prediction: the power of maximal entropy random walk
 In: Proceedings of the 20th ACM International Conference on Information and Knowledge Management
"... Link prediction is a fundamental problem in social network analysis. The key technique in unsupervised link prediction is to find an appropriate similarity measure between nodes of a network. A class of wildly used similarity measures are based on random walk on graph. The traditional random walk ( ..."
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Cited by 6 (1 self)
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Link prediction is a fundamental problem in social network analysis. The key technique in unsupervised link prediction is to find an appropriate similarity measure between nodes of a network. A class of wildly used similarity measures are based on random walk on graph. The traditional random walk (TRW) considers the link structures by treating all nodes in a network equivalently, and ignores the centrality of nodes of a network. However, in many real networks, nodes of a network not only prefer to link to the similar node, but also prefer to link to the central nodes of the network. To address this issue, we use maximal entropy random walk (MERW) for link prediction, which incorporates the centrality of nodes of the network. First, we study certain important properties of MERW on graph G by constructing an eigenweighted graph G. We show that the transition matrix and stationary distribution of MERW on G are identical to the ones of TRW on G. Based on G, we further give the maximal entropy graph Laplacians, and show how to fast compute the hitting time and commute time of MERW. Second, we propose four new graph kernels and two similarity measures based on MERW for link prediction. Finally, to exhibit the power of MERW in link prediction, we compare 27 various link prediction methods over 3 synthetic and 8 real networks. The results show that our newly proposed MERW based methods outperform the stateoftheart method on most datasets.
A continuousstate version of discrete randomized shortestpaths
 Proceedings of the 50th IEEE International Conference on Decision and Control (IEEE CDC 2011
, 2011
"... Abstract — This work investigates the continuousstate counterpart of the discrete randomized shortestpath framework (RSP, [23]) on a graph. Given a weighted directed graph G, the RSP considers the policy that minimizes the expected cost (exploitation) to reach a destination node from a source node ..."
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Abstract — This work investigates the continuousstate counterpart of the discrete randomized shortestpath framework (RSP, [23]) on a graph. Given a weighted directed graph G, the RSP considers the policy that minimizes the expected cost (exploitation) to reach a destination node from a source node, while maintaining a constant relative entropy spread in the graph (exploration). This results in a Boltzmann probability distribution on the (usually infinite) set of paths connecting the source node and the destination node, depending on an inverse temperature parameter θ. This framework defines a biased random walk on the graph that gradually favors lowcost paths as θ increases. It is shown that the continuousstate counterpart requires the solution of two partial differential equations – providing forward and backward variables – from which all the quantities of interest can be computed. For instance, the best local move is obtained by taking the gradient of the logarithm of one of these solutions, namely the backward variable. These partial differential equations are the socalled steadystate Bloch equations to which the FeynmanKac formula provides a path integral solution. The RSP framework is therefore a discretestate equivalent of the continuous FeynmanKac diffusion process involving the Wiener measure. Finally, it is shown that the continuoustime continuousstate optimal randomized policy is obtained by solving a diffusion equation with an external drift provided by the gradient of the logarithm of the backward variable, playing the role of a potential. I.
Developments in the theory of randomized shortest paths with a comparison of graph node distances
, 2012
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1 The SumoverForests density index: identifying dense regions in a graph
"... Abstract—This work introduces a novel nonparametric density index defined on graphs, the SumoverForests (SoF) density index. It is based on a clear and intuitive idea: highdensity regions in a graph are characterized by the fact that they contain a large amount of lowcost trees with high outdegr ..."
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Abstract—This work introduces a novel nonparametric density index defined on graphs, the SumoverForests (SoF) density index. It is based on a clear and intuitive idea: highdensity regions in a graph are characterized by the fact that they contain a large amount of lowcost trees with high outdegrees while lowdensity regions contain few ones. Therefore, inspired by [1], a Boltzmann probability distribution on the countable set of forests in the graph is defined so that large (highcost) forests occur with a low probability while short (lowcost) forests occur with a high probability. Then, the SoF density index of a node is defined as the expected outdegree of this node in a nontrivial tree of the forest, thus providing a measure of density around that node. Following the matrixforest theorem [2], [3] and a statistical physics framework, it is shown that the SoF density index can be easily computed in closed form through a simple matrix inversion. Experiments on artificial and real data sets show that the proposed index performs well on finding dense regions, for graphs of various origins. Index Terms—Graph mining, density index, dense regions on graphs, matrixforest theorem.