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A measure of betweenness centrality based on random walks. Social Netw (2005)

by M NEWMAN
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Random walks for image segmentation

by Leo Grady - IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE , 2006
"... A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with user-defined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the ..."
Abstract - Cited by 387 (21 self) - Add to MetaCart
A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with user-defined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels. By assigning each pixel to the label for which the greatest probability is calculated, a high-quality image segmentation may be obtained. Theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits. This algorithm is formulated in discrete space (i.e., on a graph) using combinatorial analogues of standard operators and principles from continuous potential theory, allowing it to be applied in arbitrary dimension on arbitrary graphs.
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...weenness” on the nodes on a graph by considering a node’s “betweenness” measure to be equal to how often a random walk starting at any pair of nodes passes through the node, averaged across all pairs =-=[43]-=-. Such a measure is shown to offer more intuitive behavior over other methods of “betweenness” computation at the cost of an expensive matrix inversion. 3 EXPOSITION OF THE ALGORITHM Although the rand...

Social Network Analysis for Routing in Disconnected Delay-tolerant MANETs

by Elizabeth Daly, Mads Haahr , 2007
"... Message delivery in sparse Mobile Ad hoc Networks (MANETs) is difficult due to the fact that the network graph is rarely (if ever) connected. A key challenge is to find a route that can provide good delivery performance and low end-to-end delay in a disconnected network graph where nodes may move fr ..."
Abstract - Cited by 276 (1 self) - Add to MetaCart
Message delivery in sparse Mobile Ad hoc Networks (MANETs) is difficult due to the fact that the network graph is rarely (if ever) connected. A key challenge is to find a route that can provide good delivery performance and low end-to-end delay in a disconnected network graph where nodes may move freely. This paper presents a multidisciplinary solution based on the consideration of the socalled small world dynamics which have been proposed for economy and social studies and have recently revealed to be a successful approach to be exploited for characterising information propagation in wireless networks. To this purpose, some bridge nodes are identified based on their centrality characteristics, i.e., on their capability to broker information exchange among otherwise disconnected nodes. Due to the complexity of the centrality metrics in populated networks the concept of ego networks is exploited where nodes are not required to exchange information about the entire network topology, but only locally available information is considered. Then SimBet Routing is proposed which exploits the exchange of pre-estimated ‘betweenness’ centrality metrics and locally determined social ‘similarity’ to the destination node. We present simulations using real trace data to demonstrate that SimBet Routing results in delivery performance close to Epidemic Routing but with significantly reduced overhead. Additionally, we show that Sim-Bet Routing outperforms PRoPHET Routing, particularly when the sending and receiving nodes have low connectivity.

Random-walk computation of similarities between nodes of a graph, with application to collaborative recommendation

by Francois Fouss , Alain Pirotte , Jean-Michel Renders , Marco Saerens - 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 Markov-chain model of random walk through the database. More precisely, we compute quantities (the average commu ..."
Abstract - Cited by 194 (19 self) - Add to MetaCart
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 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 (it contains inner-products 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 commute-time 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 collaborative-recommendation 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, could be applied to machine-learning and pattern-recognition tasks involving a database. * François Fouss, Alain Pirotte and Marco Saerens are with the
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... which is closely related to the graph PCA introduced in Section 5. This model could be used in order to compute similarities between nodes, just as any other graph kernel [61]. More recently, Newman =-=[48]-=- suggested a random-walk model to compute a “betweenness centrality” of a given node in a graph. This counts how often a node is traversed during a random walk between two other nodes. Then, this quan...

The slashdot zoo: Mining a social network with negative edges

by Jérôme Kunegis, Andreas Lommatzsch, Christian Bauckhage, Deutsche Telekom - In WWW , 2009
"... christian.bauckhage ..."
Abstract - Cited by 85 (6 self) - Add to MetaCart
christian.bauckhage
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...s with negative edges are described in [17, 10], where they are used in an undirected setting for collaborative filtering. Centrality and trust measures based on the graph Laplacians are described in =-=[21]-=- for graphs with only positive edge weights. 3. THE SLASHDOT ZOO The Slashdot Zoo corpus we consider in this paper contains 77,985 users and 510,157 links. Each link consists of a user who gave an end...

Randomized Algorithms for Matrices and Data

by Michael W. Mahoney , 2010
"... ..."
Abstract - Cited by 60 (2 self) - Add to MetaCart
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Social Network Analysis for Information Flow in Disconnected Delay-Tolerant MANETs

by Elizabeth M. Daly, Mads Haahr
"... Abstract—Message delivery in sparse mobile ad hoc networks (MANETs) is difficult due to the fact that the network graph is rarely (if ever) connected. A key challenge is to find a route that can provide good delivery performance and low end-to-end delay in a disconnected network graph where nodes ma ..."
Abstract - Cited by 58 (0 self) - Add to MetaCart
Abstract—Message delivery in sparse mobile ad hoc networks (MANETs) is difficult due to the fact that the network graph is rarely (if ever) connected. A key challenge is to find a route that can provide good delivery performance and low end-to-end delay in a disconnected network graph where nodes may move freely. We cast this challenge as an information flow problem in a social network. This paper presents social network analysis metrics that may be used to support a novel and practical forwarding solution to provide efficient message delivery in disconnected delay-tolerant MANETs. These metrics are based on social analysis of a node’s past interactions and consists of three locally evaluated components: a node’s “betweenness ” centrality (calculated using ego networks), a node’s social “similarity ” to the destination node, and a node’s tie strength relationship with the destination node. We present simulations using three real trace data sets to demonstrate that by combining these metrics delivery performance may be achieved close to Epidemic Routing but with significantly reduced overhead. Additionally, we show improved performance when compared to PRoPHET Routing. Index Terms—Delay- and disruption-tolerant networks, MANETs, sparse networks, ego networks, social network analysis.
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... between a node pi and all other reachable nodes [14]. Closeness centrality can be regarded as a measure of how long it will take information to spread from a given node to other nodes in the network =-=[43]-=-. Closeness centrality for a given node, where N is the number of reachable nodes in the network, is calculated as N 1 CCðpiÞ P : N k1 dðpi;pkÞ ð2Þ “Betweenness” centrality measures the extent to wh...

Who's Who in Networks. Wanted: The Key Player

by Coralio Ballester, et al. , 2005
"... Finite population non-cooperative games with linear-quadratic utilities, where each player decides how much action she exerts, can be interpreted as a network game with local payoff complementarities, together with a globally uniform payoff substitutability component and an own-concavity effect. For ..."
Abstract - Cited by 58 (3 self) - Add to MetaCart
Finite population non-cooperative games with linear-quadratic utilities, where each player decides how much action she exerts, can be interpreted as a network game with local payoff complementarities, together with a globally uniform payoff substitutability component and an own-concavity effect. For these games, the Nash equilibrium action of each player is proportional to her Bonacich centrality in the network of local complementarities, thus establishing a bridge with the sociology literature on social networks. We then analyze a policy that consists of targeting the key player, that is, the player who, once removed, leads to the optimal change in aggregate activity. We provide a geometric characterization of the key player identified with an inter-centrality measure, which takes into account both a player’s centrality and her contribution to the centrality of the others.

Centrality estimation in large networks

by Ulrik Brandes, Christian Pich - INTL. JOURNAL OF BIFURCATION AND CHAOS, SPECIAL ISSUE ON COMPLEX NETWORKS’ STRUCTURE AND DYNAMICS , 2007
"... Centrality indices are an essential concept in network analysis. For those based on shortest-path distances the computation is at least quadratic in the number of nodes, since it usually involves solving the single-source shortest-paths (SSSP) problem from every node. Therefore, exact computation is ..."
Abstract - Cited by 55 (0 self) - Add to MetaCart
Centrality indices are an essential concept in network analysis. For those based on shortest-path distances the computation is at least quadratic in the number of nodes, since it usually involves solving the single-source shortest-paths (SSSP) problem from every node. Therefore, exact computation is infeasible for many large networks of interest today. Centrality scores can be estimated, however, from a limited number of SSSP computations. We present results from an experimental study of the quality of such estimates under various selection strategies for the source vertices.
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...r use maximum network flow instead of shortest paths [Freeman et al. 1991]. Natural variants of closeness and betweenness are also obtained by replacing spread along shortest paths with current flow [=-=Newman 2005-=-, Brandes and Fleischer 2005]. A different class of measures is based on feedback, i.e. the centrality of a vertex directly influences that of its neighbors. Well-known members of this class are eigen...

Fast Approximation of Matrix Coherence and Statistical Leverage

by Petros Drineas, Malik Magdon-ismail, Michael W. Mahoney, David P. Woodruff, Mehryar Mohri
"... The statistical leverage scores of a matrix A are the squared row-norms of the matrix containing its (top) left singular vectors and the coherence is the largest leverage score. These quantities are of interest in recently-popular problems such as matrix completion and Nyström-based low-rank matrix ..."
Abstract - Cited by 53 (11 self) - Add to MetaCart
The statistical leverage scores of a matrix A are the squared row-norms of the matrix containing its (top) left singular vectors and the coherence is the largest leverage score. These quantities are of interest in recently-popular problems such as matrix completion and Nyström-based low-rank matrix approximation as well as in large-scale statistical data analysis applications more generally; moreover, they are of interest since they define the key structural nonuniformity that must be dealt with in developing fast randomized matrix algorithms. Our main result is a randomized algorithm that takes as input an arbitrary n×d matrix A, with n ≫ d, and that returns as output relative-error approximations to all n of the statistical leverage scores. The proposed algorithm runs (under assumptions on the precise values of n and d) in O(nd logn) time, as opposed to the O(nd 2) time required by the naïve algorithm that involves computing an orthogonal basis for the range of A. Our analysis may be viewed in terms of computing a relative-error approximation to an underconstrained least-squares approximation problem, or, relatedly, it may be viewed as an application of Johnson-Lindenstrauss type ideas. Several practically-important extensions of our basic result are also described, including the approximation of so-called cross-leverage scores, the extension of these ideas to matrices with n≈d, and the extension to streaming environments.
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...es in data graphs, very small clusters in noisy data, coherence of information, articulation points between clusters, the value of a customer in a network, space localization in sensor networks, etc. =-=[9, 37, 34, 27, 29]-=-. In genetics, 4 dense matrices of size thousands by hundreds of thousands (a size scale at which even traditional deterministic QR algorithms fail to run) constructed from DNA Single Nucleotide Polym...

A regularization framework for learning from graph data

by Dengyong Zhou, Bernhard Schölkopf - ICML Workshop on Statistical Relational Learning , 2004
"... The data in many real-world problems can be thought of as a graph, such as the web, co-author networks, and biological networks. We propose a general regularization framework on graphs, which is applicable to the classification, ranking, and link prediction problems. We also show that the method can ..."
Abstract - Cited by 51 (1 self) - Add to MetaCart
The data in many real-world problems can be thought of as a graph, such as the web, co-author networks, and biological networks. We propose a general regularization framework on graphs, which is applicable to the classification, ranking, and link prediction problems. We also show that the method can be explained as lazy random walks. We evaluate the method on a number of experiments. 1.
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... = (I − αS) −1 y. Then rank the vertices v according with f(v) (largest ranked first) [8]. We address a toy ranking problem shown in the top panel of Figure 3. This toy network was first suggested by =-=[4]-=- for the problem measuring betweenness centrality in social networks. Assume that A is the query. The task is to rank the remaining vertices with respect to A. According with our intuition, D should b...

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