Limits of action convergent graph sequences with unbounded $(p,q)$-norms

TL;DR

This paper explores the limits of graph sequences under action convergence when only the $( ext{infinity,}1)$-norm is bounded, revealing non-uniqueness and non-self-adjointness of limit operators, thus showing the space of graphops is not compact.

Contribution

It constructs graph sequences with only bounded $( ext{infinity,}1)$-norms that still converge, demonstrating the non-uniqueness and non-self-adjointness of their limits.

Findings

Limit operators are not unique in this setting.

Limit operators may not be self-adjoint or positivity-preserving.

The space of graphops is not compact under action convergence.

Abstract

The recently developed notion of action convergence by Backhausz and Szegedy unifies and generalises the dense (graphon) and local-global (graphing) convergences of graph sequences. This is done through viewing graphs as operators and examining their dynamical properties. Suppose $(A_n)_n^\infty$ is a sequence of operators representing graphs, Cauchy with respect to the action metric. If $(A_n)_n^\infty$ has uniformly bounded $(p,q)$-norms where $(p,q)$ is any pair in $[1,\infty)\times(1,\infty)$, then Backhausz and Szegedy prove that $(A_n)_n^\infty$ has a limit operator which, moreover, must be self-adjoint and positivity-preserving. In the present work, we construct a large class of graph sequences whose only uniformly bounded $(p,q)$-norm is the $(\infty,1)$-norm, but which converge nonetheless. We show that the limit operators in this case are not unique, not self-adjoint, and need not be positivity-preserving. In particular, in the action convergence language, this means that the space of graphops is not compact. By identifying these multiple limits, we also demonstrate that $c$-regularity is not invariant under weak equivalence, where $c$ is the eigenvalue of the identity function, when the identity function is an eigenfunction.