Fundamental polytopes of metric trees via parallel connections of matroids

Delucchi, Emanuele; Hoessly, Linard

doi:10.1016/j.ejc.2020.103098

Cited by 13 publications

(15 citation statements)

References 22 publications

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“…The structure of fundamental polytopes of tree-like metric spaces were studied via associated hyperplane arrangements and corresponding decompositions of the matroid in (Delucchi & Hoessly, 2020), enabling explicit formulas for face numbers of tree-like finite metric spaces. Values of the -vectors as well as concrete values for -vectors for "generic" 11 metrics were given in (Gordon & Petrov, 2017).…”

Section: Lipschitz Polytopementioning

confidence: 99%

“…Values of the -vectors as well as concrete values for -vectors for "generic" 11 metrics were given in (Gordon & Petrov, 2017). For more on connections, terminology, history and further context around fundamental polytopes we refer to, e.g., (Ostrovska & Ostrovskii, 2019, x 1.6) or (Delucchi & Hoessly, 2020), where we further remark that direct applications to phylogenetics are still outstanding.…”

Section: Lipschitz Polytopementioning

confidence: 99%

“…Next we concentrate on the case of tree-like metric spaces and their Lipschitz polytopes as in (Delucchi & Hoessly, 2020). Let be a finite set and consider a split = | of , where | | = .…”

Section: Definition 35 the Lipschitz Polytope Of ( ) Is Given As An Intersection Of Halfspaces Bymentioning

confidence: 99%

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Politopos en filogenética

Hoessly¹

2021

teq

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Section: Lipschitz Polytopementioning

confidence: 99%

Section: Lipschitz Polytopementioning

confidence: 99%

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Politopos en filogenética

Hoessly¹

2021

teq

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“…16,19 Similar constructions have also appeared in other contexts. [20][21][22] The adjacency polytope bound is a simplification and relaxation of the Bernshtein-Kushnirenko-Khovanskii bound. 23 Recently, the author proved that the this bound remains a sharp upper bound for the total number of complex synchronization configurations for certain graphs.…”

Section: Adjacency Polytope and Facet Networkmentioning

confidence: 99%

Directed acyclic decomposition of Kuramoto equations

Chen

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The Kuramoto model is one of the most widely studied models for describing synchronization behaviors in a network of coupled oscillators, and it has found a wide range of applications. Finding all possible frequency synchronization configurations in a general non-uniform, heterogeneous, and sparse network is important yet challenging due to complicated nonlinear interactions. From the view point of homotopy deformation, we develop a general framework for decomposing a Kuramoto network into smaller directed acyclic subnetworks, which lays the foundation for a divideand-conquer approach to studying the configurations of frequency synchronization of large Kuramoto networks.The spontaneous synchronization of a network of oscillators is an emergent phenomenon that naturally appears in many seemingly independent complex systems including mechanical, chemical, biological, and even social systems. The Kuramoto model is one of the most widely studied and successful mathematical models for analyzing synchronization behaviors. While much is known about the macro-scale question of whether or not a Kuramoto network can be synchronized, detailed analysis of the possible configurations of the oscillator once it has reached synchronization remains difficult for large networks partly due to the nonlinear interactions involved. In this work, we demonstrate that by dividing the link between two oscillators into two one-way interactions, complex networks can indeed be decomposed into much simpler subnetwork. This is a crucial step toward fully understanding synchronization configurations in large networks. arXiv:1903.04492v2 [math.CO]

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“…In the context of Kuramoto models [15], the geometric structure of adjacency polytopes has been instrumental in solving the root counting problem for algebraic Kuramoto equations [5,6,15]. In the broader context, the adjacency polytope of G is equivalent to the symmetric edge polytope, which has been studied by number theorists, combinatorialists, and discrete geometers motivated by several seemingly-independent problems [11,12,13,16,17,18,19]. These different viewpoints are consolidated in recent work by D'Alì, Delucchi, and Micha lek [10] which, among other contributions, sheds new light on the structure of adjacency polytopes of bipartite graphs, cycles, wheels, and graphs consisting of two subgraphs sharing a single edge.…”

Section: Introductionmentioning

confidence: 99%

Facets and facet subgraphs of symmetric edge polytopes

Chen¹,

Davis²,

Korchevskaia³

2021

Preprint

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Adjacency polytopes, a.k.a. symmetric edge polytopes, associated with undirected graphs have been defined and studied in several seemingly-independent areas including number theory, discrete geometry, and dynamical systems. In particular, the authors are motivated by the study of the algebraic Kuramoto equations of unmixed form whose Newton polytopes are the adjacency polytopes.The interplay between the geometric structure of adjacency polytopes and the topological structure of the underlying graphs is a recurring theme in recent studies. In particular, "facet/face subgraphs" emerged as one of the central concepts in describing this symmetry. Continuing along this line of inquiry we provide a complete description of the correspondence between facets/faces of an adjacency polytope and maximal bipartite subgraphs of the underlying connected graph.

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Fundamental polytopes of metric trees via parallel connections of matroids

Cited by 13 publications

References 22 publications

Politopos en filogenética

Politopos en filogenética

Directed acyclic decomposition of Kuramoto equations

Facets and facet subgraphs of symmetric edge polytopes

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