Graph-theoretic interpretation of the generalized row sum method

Shamis, Elena

doi:10.1016/0165-4896(93)00741-c

Cited by 12 publications

(12 citation statements)

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“…It turns out that ridge estimates are more adequate in some cases than the least squares estimates of the object effects in paired comparisons [18]. These ridge estimates can be represented via relative forest accessibilities [19].…”

Section: Observe That the Transformationmentioning

confidence: 99%

The Forest Metrics for Graph Vertices

Chebotarev

Shamis

2002

Electronic Notes in Discrete Mathematics

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We propose a new graph metric and study its properties. In contrast to the standard distance in connected graphs, it takes into account all paths between vertices. Formally, it is defined as d(i,j)=q_{ii}+q_{jj}-q_{ij}-q_{ji}, where q_{ij} is the (i,j)-entry of the {\em relative forest accessibility matrix} Q(\epsilon)=(I+\epsilon L)^{-1}, L is the Laplacian matrix of the (weighted) (multi)graph, and \epsilon is a positive parameter. By the matrix-forest theorem, the (i,j)-entry of the relative forest accessibility matrix of a graph provides the specific number of spanning rooted forests such that i and j belong to the same tree rooted at i. Extremely simple formulas express the modification of the proposed distance under the basic graph transformations. We give a topological interpretation of d(i,j) in terms of the probability of unsuccessful linking i and j in a model of random links. The properties of this metric are compared with those of some other graph metrics. An application of this metric is related to clustering procedures such as "centered partition." In another procedure, the relative forest accessibility and the corresponding distance serve to choose the centers of the clusters and to assign a cluster to each non-central vertex. The notion of cumulative weight of connections between two vertices is proposed. The reasoning involves a reciprocity principle for weighted multigraphs. Connections between the resistance distance and the forest distance are established.Comment: 14 pages, 19 re

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Section: Observe That the Transformationmentioning

confidence: 99%

The Forest Metrics for Graph Vertices

Chebotarev

Shamis

2002

Electronic Notes in Discrete Mathematics

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“…This procedure [53,54] can be considered as a Bayesian modification of the previous one. It is derived axiomatically and has Markov chain and graph theoretic interpretations [55,56]. The scores satisfy the system of equations…”

Section: The Generalized Row Sum Proceduresmentioning

confidence: 99%

Preference fusion when the number of alternatives exceeds two: indirect scoring procedures

Chebotarev¹,

Shamis²

1999

Journal of the Franklin Institute

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We consider the problem of aggregation of incomplete preferences represented by arbitrary binary relations or incomplete paired comparison matrices. For a number of indirect scoring procedures we examine whether or not they satisfy the axiom of self-consistent monotonicity. The class of win-loss combining scoring procedures is introduced which contains a majority of known scoring procedures. Two main results are established. According to the first one, every win-loss combining scoring procedure breaks self-consistent monotonicity. The second result provides a sufficient condition of satisfying self-consistent monotonicity.

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“…It is noted that the formulation in Theorem 1.1 had been mentioned in 2 and had been also proved in 15. Moreover, Merris in 13 established the relationship between the entry ω ij of Ω( G ) and structure of G .…”

Section: Introductionmentioning

confidence: 86%

Vertex degrees and doubly stochastic graph matrices

Zhang

2010

J. Graph Theory

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Abstract:In this article, the relationship between vertex degrees and entries of the doubly stochastic graph matrix has been investigated. In particular, we present an upper bound for the main diagonal entries of a doubly stochastic graph matrix and investigate the relations between a kind of distance for graph vertices and the vertex degrees. These results are used to answer in negative Merris' question on doubly stochastic graph matrices. These results may also be used to establish relations between graph structure and entries of doubly stochastic graph matrices. ᭧

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Graph-theoretic interpretation of the generalized row sum method

Cited by 12 publications

References 1 publication

The Forest Metrics for Graph Vertices

The Forest Metrics for Graph Vertices

Preference fusion when the number of alternatives exceeds two: indirect scoring procedures

Vertex degrees and doubly stochastic graph matrices

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