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2018•ESAIM Control Optimisation and Calculus of VariationsRequires access

Geodesics of minimal length in the set of probability measures on graphs

Wilfrid Gangbo, Wuchen Li, Chenchen Mou

Open publisher page 21 citations

Abstract

We endow the set of probability measures on a weighted graph with a Monge–Kantorovich metric induced by a function defined on the set of edges. The graph is assumed to havenvertices and so the boundary of the probability simplex is an affine (n− 2)-chain. Characterizing the geodesics of minimal length which may intersect the boundary is a challenge we overcome even when the endpoints of the geodesics do not share the same connected components. It is our hope that this work will be a preamble to the theory of mean field games on graphs.

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What this paper is about

We endow the set of probability measures on a weighted graph with a Monge–Kantorovich metric induced by a function defined on the set of edges. The graph is assumed to havenvertices and so the boundary of the probability simplex is an affine (n− 2)-chain. Characterizing the geodesics of minimal length which may intersect the boundary is a challenge we overcome even when the endpoints of the geodesics do not share the same connected components. It is our hope that this work will be a preamble to the theory of mean field games on graphs.

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Available abstract

We endow the set of probability measures on a weighted graph with a Monge–Kantorovich metric induced by a function defined on the set of edges. The graph is assumed to havenvertices and so the boundary of the probability simplex is an affine (n− 2)-chain. Characterizing the geodesics of minimal length which may intersect the boundary is a challenge we overcome even when the endpoints of the geodesics do not share the same connected components. It is our hope that this work will be a preamble to the theory of mean field games on graphs.

Key concepts: Geodesic, Mathematics, Simplex, Combinatorics, Affine transformation, Boundary (topology), Discrete mathematics, Metric (unit)

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