Banach-valued graph limits: Graphon representability and Banach-space structure

2026/07/29 by Motoki Otsuka
Mathematics · #math.CO #math.FA #msc:05C80 #msc:46B20 #msc:46B22 #msc:46G10

paper · pdf

31 pages, 3 figures

arxiv created 2026/07/29 · arxiv updated 2026/07/30

Abstract

We study a graph-limit problem for Banach-decorated graphs. Given a sequence of X-decorated graphs whose homomorphism densities converge against all X^*-decorated test graphs, we ask whether the limiting densities are represented by an X-valued graphon. The results connect this graph-limit problem with Banach-space structure. If X^* is separable, then the graphon representation property for graph sequences uniformly bounded in L^p for every finite p holds if and only if X is reflexive. For Banach lattices, it is equivalent to the Radon--Nikodým property. For dual Banach spaces, it is equivalent to the conjunction of the Radon--Nikodým property and weak sequential completeness. In the bounded setting, the same characterization extends to arbitrary Banach spaces: for every Banach space X, the representation property for uniformly L^∞-bounded graph sequences holds if and only if X has the Radon--Nikodým property and is weakly sequentially complete.

Citations

• Graphons, cut norm and distance, couplings and rearrangements
• The large deviation principle for the Erdős-Rényi random graph
• Testing properties of graphs and functions
• Moments of Two-Variable Functions and the Uniqueness of Graph Limits
• Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing
• Limits of dense graph sequences

Related

• Uniqueness of Banach space valued graphons
• Regularity partitions and the topology of graphons
• Limits of compact decorated graphs
• Weak regularity and finitely forcible graph limits
• Existence of Modeling Limits for Sequences of Sparse Structures