Cones of Weighted and Partial Metrics

Deza, Michel; Deza, Elena; Vidali, Janoš

doi:10.1142/9789814366311_0012

Cited by 4 publications

(7 citation statements)

References 15 publications

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“…These equalities are the case k = 3 of k-cyclic symmetry considered in [DD10]. They were derived by another way in [AM11].…”

Section: Cut and Ocut Vector Set-functionsmentioning

confidence: 96%

“…If metrics have weights that satisfy the inequalities (19) and (18), then these metrics are called upweighted metrics. See details in [DD10], [DDV11].…”

Section: Generalizations Of Metricsmentioning

confidence: 99%

“…The cones of weighted quasi-metrics, oriented cuts and other related generalizations of metrics are studied in [DD10] and [DDV11]. The polytope of oriented cuts was considered in [AM11].…”

Section: Introductionmentioning

confidence: 99%

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Cones of weighted quasi-metrics, weighted quasi-hypermetrics and of oriented cuts

Deza¹,

Grishukhin²,

Deza³

2012

Preprint

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We show that the cone of weighted n-point quasi-metrics W QM et n , the cone of weighted quasi-hypermetrics W Hyp n and the cone of oriented cuts OCut n are projections along an extreme ray of the metric cone M et n+1 , of the hypermetric cone Hyp n+1 and of the cut cone Cut n+1 , respectively. This projection is such that if one knows all faces of an original cone then one knows all faces of the projected cone.

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“…These equalities are the case k = 3 of k-cyclic symmetry considered in [DD10]. They were derived by another way in [AM11].…”

Section: Cut and Ocut Vector Set-functionsmentioning

confidence: 96%

“…If metrics have weights that satisfy the inequalities (19) and (18), then these metrics are called upweighted metrics. See details in [DD10], [DDV11].…”

Section: Generalizations Of Metricsmentioning

confidence: 99%

See 1 more Smart Citation

Cones of weighted quasi-metrics, weighted quasi-hypermetrics and of oriented cuts

Deza¹,

Grishukhin²,

Deza³

2012

Preprint

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“…Построение и исследование соответствующих конусов и многогранников на малом числе точек можно провести по той же схеме (см. [3,4]).…”

Section: примеры метрик полуметрик и их ориентированных и многомерных аналоговunclassified

К Вопросу Об Обобщенных Дискретных Метрических Структурах

Deza¹

2020

Итоги науки и техники. Серия «Современная математика и ее прило

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В статье рассмотрены вопросы, связанные с построением и исследованием конусов полуметрик, квазиполуметрик (которые являются ориентированными аналогами симметричного случая полуметрик), и $m$-полуметрик (которые являются многомерными аналогами двумерного случая полуметрик).

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“…Remark 2. The values q j = q (n−1) jj and f ij that satisfy (12) can be calculated by means of elementary matrix algebra, namely, by the following recurrent procedure [9, Proposition 4], which has a polynomial complexity. For k = 0, 1, .…”

Section: A Forest Expression For the Hitting Timesmentioning

confidence: 99%

Hitting time quasi-metric and its forest representation

Chebotarev

Deza

2018

Optim Lett

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Let m ij be the hitting (mean first passage) time from state i to state j in an n-state ergodic homogeneous Markov chain with transition matrix T . Let Γ be the weighted digraph whose vertex set coincides with the set of states of the Markov chain and arc weights are equal to the corresponding transition probabilities. It holds thatwhere f ij is the total weight of 2-tree spanning converging forests in Γ that have one tree containing i and the other tree converging to j, q j is the total weight of spanning trees converging to j in Γ, and q = n j=1 q j is the total weight of all spanning trees in Γ. Moreover, f ij and q j can be calculated by an algebraic recurrent procedure. A forest expression for Kemeny's constant is an immediate consequence of this result. Further, we discuss the properties of the hitting time quasi-metric m on the set of vertices of Γ: m(i, j) = m ij , i = j, and m(i, i) = 0. We also consider a number of other metric structures on the set of graph vertices related to the hitting time quasi-metric m-along with various connections between them. The notions and relationships under study are illustrated by two examples.

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Cones of Weighted and Partial Metrics

Cited by 4 publications

References 15 publications

Cones of weighted quasi-metrics, weighted quasi-hypermetrics and of oriented cuts

Cones of weighted quasi-metrics, weighted quasi-hypermetrics and of oriented cuts

К Вопросу Об Обобщенных Дискретных Метрических Структурах

Hitting time quasi-metric and its forest representation

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