[Paper Review] Left and right convergence of graphs with bounded degree

Christian Borgs, Jennifer Chayes+2 more|arXiv (Cornell University)|Jan 31, 2010

Mathematical Dynamics and Fractals14 references4 citations

TL;DR

This paper establishes the equivalence between left-convergence and right-convergence for graph sequences with bounded degree, proving that local convergence (left) implies convergence of normalized log-homomorphism densities (right) for sufficiently dense weighted graphs H. The key result extends the dense graph convergence equivalence to the bounded-degree setting using techniques from statistical physics, including Dobrushin uniqueness and Mayer expansion.

ABSTRACT

The theory of convergent graph sequences has been worked out in two extreme cases, dense graphs and bounded degree graphs. One can define convergence in terms of counting homomorphisms from fixed graphs into members of the sequence (left-convergence), or counting homomorphisms into fixed graphs (right-convergence). Under appropriate conditions, these two ways of defining convergence was proved to be equivalent in the dense case by Borgs, Chayes, Lovász, Sós and Vesztergombi. In this paper a similar equivalence is established in the bounded degree case. In terms of statistical physics, the implication that left convergence implies right convergence means that for a left-convergent sequence, partition functions of a large class of statistical physics models converge. The proof relies on techniques from statistical physics, like cluster expansion and Dobrushin Uniqueness.

Motivation & Objective

• To establish an analogue of the dense graph convergence equivalence in the bounded-degree setting, where left-convergence (local structure) implies right-convergence (partition function behavior).
• To resolve the open problem of whether left-convergence of bounded-degree graph sequences implies convergence of log-homomorphism densities for fixed target graphs H.
• To provide conditions under which the partition functions of statistical physics models on graph sequences converge, linking graph convergence to physical observables.
• To explore the existence and structure of limit objects for bounded-degree graph sequences, extending Benjamini-Schramm limits to right-convergence.
• To investigate the role of temperature and edge weight parameters in convergence behavior, particularly in the context of statistical mechanics models on graphs.

Proposed method

• Uses Dobrushin uniqueness criterion to control correlations in graph sequences, ensuring convergence of log-homomorphism densities under bounded-degree and sufficiently dense target graphs.
• Applies Mayer expansion to decompose the logarithm of the homomorphism function ln hom(G_n, H) into a sum over connected subgraphs, enabling asymptotic analysis.
• Employs cluster expansion techniques to bound error terms in the Mayer expansion, ensuring convergence of the normalized log-density as n → ∞.
• Introduces weighted graphs H with edge weights close to 1 (or in [1−δ,1]) to model statistical mechanics models, with normalization to α_H = 1 for consistency.
• Uses the intersection graph L(A₁,…,Aₙ) of subgraphs to analyze overlapping structures and apply inclusion-exclusion in the expansion process.
• Applies a variant of the Mayer expansion to ln t(G_n, H), where t(G_n, H) is the homomorphism density, to derive convergence under suitable conditions on H.

Experimental results

Research questions

• RQ1 Does left-convergence of a bounded-degree graph sequence imply convergence of the normalized log-homomorphism density ln hom(G_n, H)/|G_n| for fixed H?
• RQ2 What structural or density conditions on the target graph H ensure that left-convergence implies right-convergence?
• RQ3 Can the convergence of partition functions in statistical physics models be characterized via graph sequence convergence in the bounded-degree regime?
• RQ4 Is there a dual characterization of left-convergence in terms of right-convergence, analogous to the dense graph case?
• RQ5 How does temperature (via edge weight scaling) affect the convergence of partition functions in models defined on graph sequences?

Key findings

• Left-convergence of a bounded-degree graph sequence implies convergence of ln hom(G_n, H)/|G_n| for all sufficiently dense weighted graphs H, as formalized in Theorem 3.1.
• Conversely, if ln hom(G_n, H)/|G_n| converges for all sufficiently dense H, then the sequence (G_n) is left-convergent, establishing a two-way equivalence.
• The convergence holds when the target graph H has edge weights bounded away from zero (e.g., in [1−δ,1]) and total node weight α_H = 1.
• The proof relies on Dobrushin uniqueness and cluster expansion techniques, ensuring that correlation decay implies convergence of the log-density.
• For temperature T > 2D, the partition functions hom(G_n, H^{1/T}) converge for all left-convergent sequences, even at low temperatures.
• A variant of the result holds with a single node of large weight replacing multiple twin nodes in H, preserving convergence under the same conditions.