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

http://outobox.cs.umn.edu/Random_Walks_Collaborative_Recommendation_Fouss.pdf

Random-Walk Computation of Similarities between Nodes of a Graph with Application to Collaborative Recommendation

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

Landscape connectivity: a graph‐theoretic perspective

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.

Wiener Index of Trees: Theory and Applications

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.

Lower bounds for covering times for reversible Markov chains and random walks on graphs

/pdf/reversible-markov-chains-and-random-walks-on-graphs-38pfceaw4a.pdf

Reversible Markov Chains and Random Walks on Graphs

Distance in graphs

Extremal results on the geodetic number of a graph

Let S be a subset of the vertex set V(G) of a nontrivial connected graph G. The geodetic closure (S) of S is the set of all vertices on geodesics between two vertices in S. The first player A chooses a vertex v1 of G. The second player B then picks v2 ≠ v1 and forms the geodetic closure (S2) = ({v1, v2}). Now A selects v3 e V—S2 and forms (S3) = ({v1, v2, v3}), etc. The player who first selects a vertex vn such that (Sn) = V wins the achievement game, but loses the avoidance game. These geodetic achievement and avoidance games are solved for several families of graphs by determining which player is the winner.

Geodetic games for graphs

A graph G is called invertible if its adjacency matrix A has an inverse which is the adjacency matrix of some graph H. All such graphs were shown by Harary and Minc to have the form nK2. We now introduce signed invertible (or briefly s-invertible) graphs G as those whose inverse H is a signed graph. We identify two infinite classes of s-invertible graphs: the paths P2n of even order, and the corona of any graph with K1. We then characterize s-invertible trees.

On graphs with signed inverses

Following Erdos, Harary, and Tutte, the euclidean dimension of a graphG is the minimumn such thatG can be embedded in euclideann-spaceR

 n
 so that each edge ofG has length 1. We present constructive proofs which give the euclidean dimension of a wheel and of a complete tripartite graph. We also define the generalized wheelW
m,n as the join
 
$$\bar K_m + C_n $$

 and determine the euclidean dimension of all generalized wheels.

Fred Buckley

Papers

Distance in graphs

Extremal results on the geodetic number of a graph

Geodetic games for graphs

On graphs with signed inverses

On the euclidean dimension of a wheel