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Constrained Multi-Relational Hyper-Graphons with Maximum Entropy

Juan Alvarado, Jan Ramon, Yuyi Wang

TL;DR

The most typical constrained multi-relational hyper-graphons (the most typical possible worlds) are computable by proving the solutions of the maximum entropy subjected by quantum sub-hypergraph densities in the space of multi-relational hyper-graphons are step functions.

Abstract

This work has two contributions. The first one is extending the Large Deviation Principle for uniform hyper-graphons from Lubetzky and Zhao \cite{lubetzky2015replica} to the multi-relational setting where each hyper-graphon can have different arities. This extension enables the formulation of the most typical possible world in Relational Probabilistic Logic with symmetric relational symbols in terms of entropy maximization subjected to constraints of quantum sub-hypergraph densities. The second contribution is to prove the most typical constrained multi-relational hyper-graphons (the most typical possible worlds) are computable by proving the solutions of the maximum entropy subjected by quantum sub-hypergraph densities in the space of multi-relational hyper-graphons are step functions except for in a zero measure set of combinations of quantum hyper-graphs densities with multiple relations. This result proves in a very general context the conjecture formulated by Radin et al.\ \cite{radin2014asymptotics} that states the constrained graphons with maximum entropy are step functions.

Constrained Multi-Relational Hyper-Graphons with Maximum Entropy

TL;DR

The most typical constrained multi-relational hyper-graphons (the most typical possible worlds) are computable by proving the solutions of the maximum entropy subjected by quantum sub-hypergraph densities in the space of multi-relational hyper-graphons are step functions.

Abstract

This work has two contributions. The first one is extending the Large Deviation Principle for uniform hyper-graphons from Lubetzky and Zhao \cite{lubetzky2015replica} to the multi-relational setting where each hyper-graphon can have different arities. This extension enables the formulation of the most typical possible world in Relational Probabilistic Logic with symmetric relational symbols in terms of entropy maximization subjected to constraints of quantum sub-hypergraph densities. The second contribution is to prove the most typical constrained multi-relational hyper-graphons (the most typical possible worlds) are computable by proving the solutions of the maximum entropy subjected by quantum sub-hypergraph densities in the space of multi-relational hyper-graphons are step functions except for in a zero measure set of combinations of quantum hyper-graphs densities with multiple relations. This result proves in a very general context the conjecture formulated by Radin et al.\ \cite{radin2014asymptotics} that states the constrained graphons with maximum entropy are step functions.
Paper Structure (39 sections, 44 theorems, 132 equations, 4 figures, 1 table)

This paper contains 39 sections, 44 theorems, 132 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Outline
  3. Notations
  4. Mathematical Background
  5. Riemannian Geometry
  6. Exponential Map on Riemannian Manifolds.
  7. Theory of Constrained Optimization
  8. Preservation of critical Points by Diffeomorphisms
  9. Hypergraphons
  10. Subgraph density
  11. Quantum graphs
  12. Step functions in $\mathcal{W}_{\mathbb{R}}^{(r,d)}$
  13. Partial Derivatives of $t(F, (A,\pi))$
  14. Labeled Graphs and Partial Subgraph Densities
  15. Partial Derivatives of $t(F,)$ in $\mathcal{W}^{({m,r,d})}_{\mathbb{R}}$
  16. ...and 24 more sections

Key Result

Theorem 2.3

If $x^* \in M_h$ is a local minimum of $\min_{x \in M_h} f(x)$ then there is a Lagrange multiplier vector $\beta^* \in \mathbb{R}^k$ such that thus Hence $x^* \in M_h$ must be a critical point of $f:M_h \to \mathbb{R}$.

Figures (4)

  • Figure 1: $F^{\bullet(ab)}$ and $F^{\bullet(ba)}$ are not equivalent.
  • Figure 2: A split map transforms the representation of a $3$ step function into a $4$ step function
  • Figure 3: Theorem dependencies to prove the computability of $W^{*(r,d)}(\mathcal{F},u)$
  • Figure 4: Since $(A,\pi) \in \partial {\mathcal{W}}^{({m,r,d})} \cap {{S}^{(m,r,d)}(\mathcal{F},u)}$, $t(\mathcal{F},(A,\pi)) \in \partial T^{(m)}(\mathcal{F})$

Theorems & Definitions (92)

  • Definition 2.1: Definition $1.4.2$ in jost2008riemannian
  • Definition 2.2: Lagrangian of a function
  • Theorem 2.3: First-Order Necessary Conditions
  • Definition 2.4: The set of linearized feasible directions
  • Theorem 2.5: Second-order necessary conditions
  • Theorem 2.6: Second-order sufficient conditions
  • Lemma 2.1
  • Definition 3.1: Quantum graphs
  • Definition 4.1: Labeled Graphs
  • Definition 4.2: Linear combination of partial subgraph densities.
  • ...and 82 more