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Distance in graphs

About: The article was published on 1990-01-01 and is currently open access. It has received 1185 citations till now. The article focuses on the topics: Graph theory & Convexity.
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TL;DR: The model, which nicely fits into the so-called "statistical relational learning" framework, could also be used to compute document or word similarities, and could be applied to machine-learning and pattern-recognition tasks involving a relational database.
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 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, it could be applied to machine-learning and pattern-recognition tasks involving a relational database

1,276 citations


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Journal ArticleDOI
01 May 2001-Ecology
TL;DR: In this paper, a set of analyses using a hypothetical landscape mosaic of habitat patches in a nonhabitat matrix is developed. And the results suggest that a simple graph construct, the minimum spanning tree, can serve as a powerful guide to decisions about the relative importance of individual patches to overall landscape con- nectivity.
Abstract: Ecologists are familiar with two data structures commonly used to represent landscapes. Vector-based maps delineate land cover types as polygons, while raster lattices represent the landscape as a grid. Here we adopt a third lattice data structure, the graph. A graph represents a landscape as a set of nodes (e.g., habitat patches) connected to some degree by edges that join pairs of nodes functionally (e.g., via dispersal). Graph theory is well developed in other fields, including geography (transportation networks, routing ap- plications, siting problems) and computer science (circuitry and network optimization). We present an overview of basic elements of graph theory as it might be applied to issues of connectivity in heterogeneous landscapes, focusing especially on applications of metapo- pulation theory in conservation biology. We develop a general set of analyses using a hypothetical landscape mosaic of habitat patches in a nonhabitat matrix. Our results suggest that a simple graph construct, the minimum spanning tree, can serve as a powerful guide to decisions about the relative importance of individual patches to overall landscape con- nectivity. We then apply this approach to an actual conservation scenario involving the

1,253 citations


Cites background from "Distance in graphs"

  • ...(Buckley and Harary 1990, Thulasiraman and Swamy 1992)....

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Journal ArticleDOI
TL;DR: The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph as discussed by the authors, defined as the distance between all vertices in a graph.
Abstract: The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. The paper outlines the results known for W of trees: methods for computation of W and combinatorial expressions for W for various classes of trees, the isomorphism–discriminating power of W, connections between W and the center and centroid of a tree, as well as between W and the Laplacian eigenvalues, results on the Wiener indices of the line graphs of trees, on trees extremal w.r.t. W, and on integers which cannot be Wiener indices of trees. A few conjectures and open problems are mentioned, as well as the applications of W in chemistry, communication theory and elsewhere.

1,015 citations


Cites background from "Distance in graphs"

  • ...In the mathematical literature W seems to be first studied only in 1976 [32]; for a long time mathematicians were unaware of the (earlier) work on W done in chemistry (cf. the book [ 6 ])....

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  • ...The Doyle–Graver formula holds not only for trees, but for all geodetic graphs [ 6 ]....

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Journal ArticleDOI
TL;DR: For simple random walk on aN-vertex graph, the mean time to cover all vertices is at leastcN log(N), wherec>0 is an absolute constant, deduced from a more general result about stationary finite-state reversible Markov chains.
Abstract: For simple random walk on aN-vertex graph, the mean time to cover all vertices is at leastcN log(N), wherec>0 is an absolute constant. This is deduced from a more general result about stationary finite-state reversible Markov chains. Under weak conditions, the covering time for such processes is at leastc times the covering time for the corresponding i.i.d. process.

942 citations


Cites background from "Distance in graphs"

  • ...This is one of several notions of “centrality” of vertices and edges which arise in our discussion—see Buckley and Harary [81] for a treatment of centrality in the general graph context, and for the standard graph-theoretic terminology....

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