1,509 research outputs found
Flows on Graphs with Random Capacities
We investigate flows on graphs whose links have random capacities. For binary
trees we derive the probability distribution for the maximal flow from the root
to a leaf, and show that for infinite trees it vanishes beyond a certain
threshold that depends on the distribution of capacities. We then examine the
maximal total flux from the root to the leaves. Our methods generalize to
simple graphs with loops, e.g., to hierarchical lattices and to complete
graphs.Comment: 8 pages, 6 figure
Entropy of capacities on lattices and set systems
We propose a definition for the entropy of capacities defined on lattices.
Classical capacities are monotone set functions and can be seen as a
generalization of probability measures. Capacities on lattices address the
general case where the family of subsets is not necessarily the Boolean lattice
of all subsets. Our definition encompasses the classical definition of Shannon
for probability measures, as well as the entropy of Marichal defined for
classical capacities. Some properties and examples are given
Capacities and Games on Lattices: A Survey of Result
We provide a survey of recent developments about capacities (or fuzzy measures) and ccoperative games in characteristic form, when they are defined on more general structures than the usual power set of the universal set, namely lattices. In a first part, we give various possible interpretations and applications of these general concepts, and then we elaborate about the possible definitions of usual tools in these theories, such as the Choquet integral, the Möbius transform, and the Shapley value.capacity, fuzzy measure, game, lattice, Choquet integral,Shapley value
Bipolarization of posets and natural interpolation
The Choquet integral w.r.t. a capacity can be seen in the finite case as a parsimonious linear interpolator between vertices of . We take this basic fact as a starting point to define the Choquet integral in a very general way, using the geometric realization of lattices and their natural triangulation, as in the work of Koshevoy. A second aim of the paper is to define a general mechanism for the bipolarization of ordered structures. Bisets (or signed sets), as well as bisubmodular functions, bicapacities, bicooperative games, as well as the Choquet integral defined for them can be seen as particular instances of this scheme. Lastly, an application to multicriteria aggregation with multiple reference levels illustrates all the results presented in the paper.Interpolation; Choquet integral; Lattice; Bipolar structure
The core of bicapacities and bipolar games
Bicooperative games generalize classical cooperative games in the sense that a player is allowed to play in favor or against some aim, besides non participation. Bicapacities are monotonic bicooperative games, they are useful in decision making where underlying scales are of bipolar nature, i.e., they distinguish between good/satisfactory values and bad/unsatisfactory ones. We propose here a more general framework to represent such situations, called bipolar game. We study the problem of finding the core of such games, i.e., theset of additive dominating games.fuzzy measure, bicapacity, cooperative game, bipolar scale,core
On the decomposition of Generalized Additive Independence models
The GAI (Generalized Additive Independence) model proposed by Fishburn is a
generalization of the additive utility model, which need not satisfy mutual
preferential independence. Its great generality makes however its application
and study difficult. We consider a significant subclass of GAI models, namely
the discrete 2-additive GAI models, and provide for this class a decomposition
into nonnegative monotone terms. This decomposition allows a reduction from
exponential to quadratic complexity in any optimization problem involving
discrete 2-additive models, making them usable in practice
Bipolarization of posets and natural interpolation
The Choquet integral w.r.t. a capacity can be seen in the finite case as a
parsimonious linear interpolator between vertices of . We take this
basic fact as a starting point to define the Choquet integral in a very general
way, using the geometric realization of lattices and their natural
triangulation, as in the work of Koshevoy. A second aim of the paper is to
define a general mechanism for the bipolarization of ordered structures. Bisets
(or signed sets), as well as bisubmodular functions, bicapacities,
bicooperative games, as well as the Choquet integral defined for them can be
seen as particular instances of this scheme. Lastly, an application to
multicriteria aggregation with multiple reference levels illustrates all the
results presented in the paper
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