A Discrete Phase–Metric Coupling Model on Graphs: Small–Coupling Regime and a Fully Worked Two–Node Prototype Article Swipe
We study a finite–dimensional gradient system on a finite graph, where a discrete “metric” g on edges and a scalar phase field \psi on vertices evolve together under the gradient flow of a coupled energy. The free energy splits as F = F_0 + \varepsilon C, where F_0 is strictly convex and the coupling term C is bounded but has an unbounded Hessian. In a first step, we prove that, for sufficiently small coupling \varepsilon, the full energy F is bounded from below and coercive. This yields global well–posedness of the gradient flow and strict energy dissipation. Next, we show that a naive perturbative argument for global strong convexity fails, because the Hessian of C is not uniformly bounded on the configuration space. Finally, we analyse in full detail the simplest nontrivial graph with two nodes and one edge. In this “two–node prototype” we identify an explicit small–coupling threshold, prove existence and uniqueness of a global nondegenerate minimiser, and establish global exponential convergence of every trajectory of the gradient flow towards this minimiser. This discrete model is intended as a fully worked, calculable reference block within a broader research programme on phase–metric flows (PMF) and emergent geometry.
Related Topics
Concepts
Bounded function
Balanced flow
Convexity
Mathematics
Uniqueness
Hessian matrix
Coupling (piping)
Mathematical analysis
Regular polygon
Scalar field
Flow (mathematics)
Convergence (economics)
Graph
Exponential function
Energy (signal processing)
Scalar (mathematics)
Domain (mathematical analysis)
Term (time)
Boundary (topology)
Boundary value problem
Convex function
Trajectory
Pure mathematics
Energy functional
Vector field
Energy flow
Field (mathematics)
Discrete time and continuous time
Relaxation (psychology)
Metadata
- Type
- article
- Language
- enc
- Landing Page
- https://doi.org/10.5281/zenodo.17816726
- OA Status
- green
- OpenAlex ID
- https://openalex.org/W7108641209
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
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https://openalex.org/W7108641209Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.5281/zenodo.17816726Digital Object Identifier
- Title
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A Discrete Phase–Metric Coupling Model on Graphs: Small–Coupling Regime and a Fully Worked Two–Node PrototypeWork title
- Type
-
articleOpenAlex work type
- Language
-
encPrimary language
- Publication year
-
2025Year of publication
- Publication date
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2025-12-04Full publication date if available
- Authors
-
de Veigy, LucList of authors in order
- Landing page
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https://doi.org/10.5281/zenodo.17816726Publisher landing page
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://doi.org/10.5281/zenodo.17816726Direct OA link when available
- Concepts
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Bounded function, Balanced flow, Convexity, Mathematics, Uniqueness, Hessian matrix, Coupling (piping), Mathematical analysis, Regular polygon, Scalar field, Flow (mathematics), Convergence (economics), Graph, Exponential function, Energy (signal processing), Scalar (mathematics), Domain (mathematical analysis), Term (time), Boundary (topology), Boundary value problem, Convex function, Trajectory, Pure mathematics, Energy functional, Vector field, Energy flow, Field (mathematics), Discrete time and continuous time, Relaxation (psychology)Top concepts (fields/topics) attached by OpenAlex
- Cited by
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0Total citation count in OpenAlex
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| abstract_inverted_index.system | 5 |
| abstract_inverted_index.within | 185 |
| abstract_inverted_index.yields | 86 |
| abstract_inverted_index.Hessian | 112 |
| abstract_inverted_index.analyse | 125 |
| abstract_inverted_index.because | 110 |
| abstract_inverted_index.bounded | 57, 80, 118 |
| abstract_inverted_index.broader | 187 |
| abstract_inverted_index.coupled | 33 |
| abstract_inverted_index.energy. | 34 |
| abstract_inverted_index.towards | 170 |
| abstract_inverted_index.worked, | 181 |
| abstract_inverted_index.Finally, | 123 |
| abstract_inverted_index.Hessian. | 62 |
| abstract_inverted_index.argument | 104 |
| abstract_inverted_index.coupling | 53, 73 |
| abstract_inverted_index.discrete | 12, 174 |
| abstract_inverted_index.emergent | 195 |
| abstract_inverted_index.explicit | 146 |
| abstract_inverted_index.gradient | 4, 29, 91, 168 |
| abstract_inverted_index.identify | 144 |
| abstract_inverted_index.intended | 177 |
| abstract_inverted_index.research | 188 |
| abstract_inverted_index.simplest | 130 |
| abstract_inverted_index.strictly | 49 |
| abstract_inverted_index.together | 26 |
| abstract_inverted_index.vertices | 24 |
| abstract_inverted_index.coercive. | 84 |
| abstract_inverted_index.convexity | 108 |
| abstract_inverted_index.establish | 159 |
| abstract_inverted_index.existence | 150 |
| abstract_inverted_index.geometry. | 196 |
| abstract_inverted_index.programme | 189 |
| abstract_inverted_index.reference | 183 |
| abstract_inverted_index.unbounded | 61 |
| abstract_inverted_index.uniformly | 117 |
| abstract_inverted_index.calculable | 182 |
| abstract_inverted_index.minimiser, | 157 |
| abstract_inverted_index.minimiser. | 172 |
| abstract_inverted_index.nontrivial | 131 |
| abstract_inverted_index.threshold, | 148 |
| abstract_inverted_index.trajectory | 165 |
| abstract_inverted_index.uniqueness | 152 |
| abstract_inverted_index.\varepsilon | 44 |
| abstract_inverted_index.convergence | 162 |
| abstract_inverted_index.exponential | 161 |
| abstract_inverted_index.\varepsilon, | 74 |
| abstract_inverted_index.dissipation. | 96 |
| abstract_inverted_index.perturbative | 103 |
| abstract_inverted_index.prototype” | 142 |
| abstract_inverted_index.sufficiently | 71 |
| abstract_inverted_index.“metric” | 13 |
| abstract_inverted_index.configuration | 121 |
| abstract_inverted_index.nondegenerate | 156 |
| abstract_inverted_index.“two–node | 141 |
| abstract_inverted_index.phase–metric | 191 |
| abstract_inverted_index.small–coupling | 147 |
| abstract_inverted_index.well–posedness | 88 |
| abstract_inverted_index.finite–dimensional | 3 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 1 |
| citation_normalized_percentile.value | 0.64426811 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | False |