Action convergence of general hypergraphs and tensors

TL;DR

This paper extends the concept of action convergence from linear operators to multi-linear operators and tensors, providing a unified framework for hypergraph and simplicial complex limits, including sparse and inhomogeneous cases.

Contribution

It introduces a generalized action convergence theory for multi-linear operators and tensors, linking hypergraph limits with quasirandomness and extending to non-uniform hypergraphs and complexes.

Findings

Established existence of convergent subsequences for bounded multi-linear operators.

Linked hypergraph convergence notions with quasirandomness hierarchy.

Connected tensor convergence with hypergraph and simplicial complex limits.

Abstract

Action convergence provides a limit theory for linear bounded operators $A_n:L^{\infty}(\Omega_n)\longrightarrow L^1(\Omega_n)$ where $\Omega_n$ 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 $A_n:L^{\infty}(\Omega_n)\times \ldots \times L^{\infty}(\Omega_n)\longrightarrow L^1(\Omega_n)$. 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.