The asymptotic spectrum distance, graph limits, and the Shannon capacity

TL;DR

This paper introduces the asymptotic spectrum distance, a new graph limit theory, to analyze Shannon capacity, providing novel bounds and insights into graph convergence and infinite graph limits.

Contribution

It develops a new graph distance theory based on asymptotic spectrum duality and applies it to Shannon capacity, including constructing converging graph sequences and new capacity bounds.

Findings

Constructed non-trivial converging graph sequences

Established connections between finite graph convergence and infinite graph properties

Derived a new Shannon capacity lower bound for the fifteen-cycle

Abstract

Determining the Shannon capacity of graphs is a long-standing open problem in information theory, graph theory and combinatorial optimization. Over decades, a wide range of upper and lower bound methods have been developed to analyze this problem. However, despite tremendous effort, even small instances of the problem have remained open. In recent years, a new dual characterization of the Shannon capacity of graphs, asymptotic spectrum duality, has unified and extended known upper bound methods and structural theorems. In this paper, building on asymptotic spectrum duality, we develop a new theory of graph distance, that we call asymptotic spectrum distance, and corresponding limits (reminiscent of, but different from, the celebrated theory of cut-norm, graphons and flag algebras). We propose a graph limit approach to the Shannon capacity problem: to determine the Shannon capacity of a graph, construct a sequence of easier to analyse graphs converging to it. (1) We give a very general construction of non-trivial converging sequences of graphs (in a family of circulant graphs). (2) We construct Cauchy sequences of finite graphs that do not converge to any finite graph, but do converge to an infinite graph. We establish strong connections between convergence questions of finite graphs and the asymptotic properties of Borsuk-like infinite graphs on the circle. (3) We observe that all best-known lower bound constructions for Shannon capacity of small odd cycles can be obtained from a "finite" version of the graph limit approach. We develop computational and theoretical aspects of this approach and use these to obtain a new Shannon capacity lower bound for the fifteen-cycle. The theory of asymptotic spectrum distance applies not only to Shannon capacity of graphs; indeed, we will develop it for a general class of mathematical objects and their asymptotic properties.