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

TL;DR

This work develops a graph-limit approach to Shannon capacity by introducing the asymptotic spectrum distance $d(G,H)$ derived from the asymptotic spectrum $\\mathcal{X}$, and showing that convergence in this distance preserves capacity via $\\Theta(G)=\min_{F\in\\mathcal{X}} F(G)$. It constructs nontrivial converging sequences of finite fraction graphs $E_{p/q}$ that realize right-continuity for spectrum functionals and investigates their infinite-limit points as circle graphs on the circle, including a detailed open/closed dichotomy. The paper also presents a unified orbit-based framework for lower bounds on the Shannon capacity of odd cycles, culminating in a new bound for $\Theta(C_{15}) \ge 7.30139$, and studies the structure of independence numbers in products of fraction graphs, including a comprehensive discontinuity analysis for $\alpha_3$. It further demonstrates a qualitative difference between graphs and tensors in the asymptotic spectrum distance, where tensor sequences are discrete, while graph sequences can exhibit genuine convergence and limit points. Overall, the results offer a continuous-analytic toolkit for Shannon-capacity questions, connect finite constructions to infinite models, and raise open problems about the completeness and density of graph-limit spaces.

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.

Figures (5)

• Figure 1: Beginning of non-trivial sequence converging to the seven-cycle (\ref{['th:intro:rational-right-cont']}).
• Figure 2: From left to right, these are the fraction graphs $E_{3/1}$, $E_{5/2}$ and $E_{8/3}$.
• Figure 3: The neighbourhood of a vertex (in red) in the circle graphs $E^\mathrm{o}_3$, $E^\mathrm{c}_3$ and $E^\mathrm{c}_{5/2}$ (from left to right). We take the circles to have unit circumference.
• Figure 4: Hasse diagram of the poset of discontinuity points of \ref{['th:discont']}. Here $u v$ means $u \leq v$, and by definition we have $u \leq v$ if there is a permutation of $u$ that is pointwise at most $v$.
• Figure 5: Graph of the right-continuous step function $\alpha(E_{p/q}^{\boxtimes 3})$ for $p/q \in \mathbb{Q} \cap [2,3)$ with discontinuity points as described in \ref{['th:symm-disc']}.

Theorems & Definitions (124)

• Theorem 1: \ref{['th:rational-right-cont']}
• Theorem 2: \ref{['thm: continuous at irrationals']}
• Theorem 3: \ref{['th:irr-closed-open-equiv']}
• Theorem 4: \ref{['th:op-cl-left']}
• Theorem 5: \ref{['th:open-closed-noneq']}
• Theorem 6: \ref{['th:C15-4']}
• Theorem 7: \ref{['th:discont']}, \ref{['fig:hasse']}
• Theorem 8: Asymptotic spectrum duality
• Remark 9
• Remark 10