[Paper Review] Weak Convergence of Laws of Finite Graphs

Igor Artemenko|arXiv (Cornell University)|Mar 29, 2011

Topological and Geometric Data Analysis2 references3 citations

TL;DR

This paper introduces a metric ρ on the space of isomorphism classes of locally finite rooted graphs, enabling weak convergence of laws (empirical measures) of finite graphs. It establishes that while rooted graph convergence does not imply weak convergence of laws, certain sequences—like those of finite subtrees of an infinite 3-regular tree—can converge weakly to a unimodular limit measure. The key contribution is constructing a non-degenerate unimodular limit measure as the weak limit of laws of finite graphs.

ABSTRACT

The law of a finite graph is a probability measure induced by the orbits of the graph under its automorphism group. Every law satisfies the intrinsic mass transport principle, which is also known as unimodularity. We discuss the convergence of sequences of laws of finite graphs. Of particular importance is a conjecture proposed by Aldous and Lyons that claims every unimodular measure is a limit of a sequence of laws. Aside from this open problem, other directions of research are also mentioned. We work out in detail a number of results and examples, some of which are new, and others that have been previously stated without proofs. These results include a new characterization of laws of finite connected graphs, a description of the topological space of paths, and a proof that the compact space of weak limits of laws is convex.

Motivation & Objective

• To formalize a metric ρ on the space of isomorphism classes of locally finite rooted graphs to study convergence of graph sequences.
• To investigate the relationship between convergence of rooted graphs and weak convergence of their empirical measures (laws).
• To explore whether unimodular measures can arise as weak limits of laws of finite graphs.
• To construct explicit examples of such limits, particularly in the context of trees and hyperfinite graph families.
• To examine the role of unimodularity and invariance under automorphisms in the limiting behavior of graph sequences.

Proposed method

• Define a metric ρ on bG, the space of isomorphism classes of locally finite connected rooted graphs, based on the largest r such that the r-ball around the root is isomorphic.
• Prove ρ is an ultrametric, ensuring strong topological properties for convergence.
• Define the law Ψ(G) of a finite graph G as the uniform empirical measure over its vertices.
• Use continuous bounded functions f on bG to test weak convergence via ∫f dΨ(Gn) → ∫f dμ.
• Construct a sequence of finite subtrees Tk of the infinite 3-regular tree T∞, rooted at t, and analyze the weak limit of Ψ(Tk).
• Prove that the weak limit μ satisfies μ[S, ui] = 1/2^i for the infinite path S rooted at the i-th vertex along the ray from a leaf to t.

Experimental results

Research questions

• RQ1 Does convergence of rooted graphs imply weak convergence of their laws?
• RQ2 Can unimodular measures arise as weak limits of laws of finite graphs?
• RQ3 What is the structure of the weak limit of laws of finite subtrees of an infinite regular tree?
• RQ4 How do hyperfinite graph families relate to the existence of unimodular limits?
• RQ5 Is the Dirac measure on any vertex-transitive infinite graph unimodular?

Key findings

• The metric ρ is well-defined and forms an ultrametric on the space bG of isomorphism classes of locally finite rooted graphs.
• The sequence of finite subtrees Tk of the infinite 3-regular tree T∞ converges in the ρ-metric to [T∞, t], but the laws Ψ(Tk) do not converge weakly to δ[T∞,t].
• The laws Ψ(Tk) converge weakly to a unimodular probability measure μ on bG3 with μ[S, ui] = 1/2^i for the infinite path S rooted at the i-th vertex.
• The limit measure μ is non-degenerate and supported on a countable set of rooted paths, each with exponentially decaying mass.
• The Dirac measure δ[P∞,·] on the infinite path satisfies the invariance condition for unimodularity, suggesting broader unimodularity in vertex-transitive graphs.
• The collection of finite paths is hyperfinite, and the union of finitely many hyperfinite graph families remains hyperfinite.