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Action convergence of general hypergraphs and tensors

Giulio Zucal

Abstract

Action convergence provides a limit theory for linear bounded operators $A_n:L^{\infty}(Ω_n)\longrightarrow L^1(Ω_n)$ where $Ω_n$ are potentially different probability spaces. This notion of convergence emerged in graph limits theory as it unifies and generalizes many notions of graph limits. We generalize the theory of action convergence to sequences of multi-linear bounded operators $A_n:L^{\infty}(Ω_n)\times \ldots \times L^{\infty}(Ω_n)\longrightarrow L^1(Ω_n)$. Similarly to the linear case, we obtain that for a uniformly bounded (under an appropriate norm) sequence of multi-linear operators, there exists an action convergent subsequence. Additionally, we explain how to associate different types of multi-linear operators to a tensor and we study the different notions of convergence that we obtain for tensors and in particular for adjacency tensors of hypergraphs. We obtain several hypergraphs convergence notions and we link these with the hierarchy of notions of quasirandomness for hypergraph sequences. This convergence also covers sparse and inhomogeneous hypergraph sequences and it preserves many properties of adjacency tensors of hypergraphs. Moreover, we explain how to obtain a meaningful convergence for sequences of non-uniform hypergraphs and, therefore, also for simplicial complexes. Additionally, we highlight many connections with the theory of dense uniform hypergraph limits (hypergraphons) and we conjecture the equivalence of this theory with a modification of multi-linear action convergence.

Action convergence of general hypergraphs and tensors

Abstract

Action convergence provides a limit theory for linear bounded operators where are potentially different probability spaces. This notion of convergence emerged in graph limits theory as it unifies and generalizes many notions of graph limits. We generalize the theory of action convergence to sequences of multi-linear bounded operators . Similarly to the linear case, we obtain that for a uniformly bounded (under an appropriate norm) sequence of multi-linear operators, there exists an action convergent subsequence. Additionally, we explain how to associate different types of multi-linear operators to a tensor and we study the different notions of convergence that we obtain for tensors and in particular for adjacency tensors of hypergraphs. We obtain several hypergraphs convergence notions and we link these with the hierarchy of notions of quasirandomness for hypergraph sequences. This convergence also covers sparse and inhomogeneous hypergraph sequences and it preserves many properties of adjacency tensors of hypergraphs. Moreover, we explain how to obtain a meaningful convergence for sequences of non-uniform hypergraphs and, therefore, also for simplicial complexes. Additionally, we highlight many connections with the theory of dense uniform hypergraph limits (hypergraphons) and we conjecture the equivalence of this theory with a modification of multi-linear action convergence.
Paper Structure (16 sections, 28 theorems, 214 equations)

This paper contains 16 sections, 28 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Action convergence
  4. Tensors and hypergraphs
  5. Multi-action convergence for multi-linear operators
  6. Construction of the limit object
  7. Properties of limit objects
  8. Norms and metrics comparison
  9. Multi-action convergence of hypergraphs and tensors
  10. Uniform bounds on sequences of $s-$actions
  11. Properties of $s-$actions as $P-$operators
  12. Generalization of $s-$actions
  13. Examples of hypergraph sequences and action convergence
  14. Finite hypergraphs and action convergence
  15. Sparse and non-uniform hypergraphs and different tensors
  16. ...and 1 more sections

Key Result

Theorem 3.6

Let $p\in [1,\infty)$ and $q\in [1,\infty]$. Let $\{A_i\}_{i=1}^\infty$ be a Cauchy sequence of $P$-operators with uniformly bounded $\|\cdot\|_{p\to q}$ norms. Then there is a $P$-operator $A$ such that $\lim_{i\to\infty} d_M(A_i,A)=0$, and $\|A\|_{p\to q}\leq\sup_{i\in\mathbb{N}}\|A_i\|_{p\to q}$.

Theorems & Definitions (90)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 2.1
  • Definition 3.1
  • Example 3.2
  • Definition 3.3: Lévy-Prokhorov metric
  • Definition 3.4: Hausdorff metric
  • Definition 3.5: Metrization of action convergence
  • Theorem 3.6: Theorem 2.14 in backhausz2018action
  • ...and 80 more