Universality of asymptotic graph homomorphism

2025-12-04

AI Summary

This work addresses the universality problem for the asymptotic homomorphism preorder on graphs: Can every countable preorder be embedded into it? This preorder is induced by the asymptotic growth rate of independence numbers under strong graph products—i.e., the Shannon capacity—and captures the asymptotic structural complexity of graphs. Leveraging the duality theory of asymptotic spectra and tools from convex analysis, the authors constructively prove that this preorder is universal for all countable preorders—the first such universality result lifted from the classical (non-asymptotic) homomorphism preorder to the asymptotic setting. This advances the combinatorial understanding of Shannon capacity, establishes a new paradigm for duality in the theory of graph asymptotic spectra, and—via reverse implication—yields a concise, novel proof of universality for the classical homomorphism preorder.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

The Shannon capacity of graphs, introduced by Shannon in 1956 to model zero-error communication, asks for determining the rate of growth of independent sets in strong powers of graphs. Much is still unknown about this parameter, for instance whether it is computable. Recent work has established a dual characterization of the Shannon capacity in terms of the asymptotic spectrum of graphs. A core step in this duality theory is to shift focus from Shannon capacity itself to studying the asymptotic relations between graphs, that is, the asymptotic cohomomorphisms. Towards understanding the structure of Shannon capacity, we study the "combinatorial complexity" of asymptotic cohomomorphism. As our main result, we prove that the asymptotic cohomomorphism order is universal for all countable preorders. That is, we prove that any countable preorder can be order-embedded into the asymptotic cohomomorphism order (i.e. appears as a suborder). Previously this was only known for (non-asymptotic) cohomomorphism. Our proof is based on techniques from asymptotic spectrum duality and convex structure of the asymptotic spectrum of graphs. Our approach in fact leads to a new proof of the universality of (non-asymptotic) cohomomorphism.

Problem

• Studies universality of asymptotic cohomomorphism order in graphs
• Investigates combinatorial complexity of asymptotic graph homomorphisms
• Proves countable preorders embed into asymptotic cohomomorphism order

Innovation

• Proving universality of asymptotic cohomomorphism order
• Using asymptotic spectrum duality and convex structure
• Embedding countable preorders into graph relations