[Paper Review] On the limit of large girth graph sequences

Gábor Elek|ArXiv.org|Nov 7, 2008

Graph theory and applications2 references4 citations

TL;DR

This paper proves that any involution-invariant probability measure on rooted trees with maximum degree $ d $ arises as the local weak limit of a sequence of finite graphs with maximum degree $ d $. The proof constructs such graphs via weighted labeled trees and edge-matching constructions, establishing that large girth graph sequences can realize any such measure as their local limit, answering a question by Bollobás and Riordan.

ABSTRACT

We prove that any involution-invariant probability measure on the space of trees with maximum degrees at most d arises as the local limit of a convergent large girth graph sequence. This answers a question of Bollobas and Riordan.

Motivation & Objective

• To resolve Question 6.8 of Bollobás and Riordan concerning the existence of graph sequences whose local limits realize any involution-invariant measure on trees.
• To establish that every involution-invariant measure on rooted trees with maximum degree $ d $ is the local weak limit of a sequence of finite graphs with maximum degree $ d $.
• To show that such graph sequences must be asymptotically treelike, hence corresponding to large girth sequences.
• To construct finite graphs whose local neighborhood distributions approximate a given involution-invariant measure on trees to arbitrary precision.

Proposed method

• Define the space $ extrm{Gr}_d $ of rooted, countable, connected graphs with maximum degree $ d $, equipped with a metric based on isomorphism of $ r $-balls around the root.
• Introduce the notion of involution-invariance for measures on $ extrm{Gr}_d $, requiring invariance under the involution that reverses the direction of the root edge.
• Construct a finite labeled graph $ H $ encoding the structure of the measure $ u_n $ on $ r+1 $-balls, using rational approximations $ w_ ho $ of the measure on edge-balls.
• Build an edge-less graph $ Q $ with vertex sets $ Q(A) $ of size $ N w_ ho(A) $, partitioned according to edge weights $ w_ ho(A, L_A, B) $.
• Add edges between $ Q(A, L_A, B) $ and $ Q(B, L_B, A) $ via bijections $ Z_{A,B} $, forming a graph $ R $ with controlled local structure.
• Construct the final graph $ G $ by grouping vertices of $ R $ into super-vertices corresponding to labeled rooted trees $ M $, ensuring $ r+1 $-balls around super-vertices match $ M $ if $ M $ is label-separated.

Experimental results

Research questions

• RQ1 Can every involution-invariant probability measure on rooted trees with maximum degree $ d $ arise as the local weak limit of a sequence of finite graphs with maximum degree $ d $?
• RQ2 Is every such measure realizable as the local limit of a large girth graph sequence?
• RQ3 What structural conditions on a measure ensure it can be approximated by finite graphs in the local weak topology?
• RQ4 Can a measure on trees be realized as a local limit if it satisfies involution-invariance, and how can such graphs be explicitly constructed?

Key findings

• Any involution-invariant probability measure $ u $ on $ extrm{Gr}_d $ supported on trees arises as the local weak limit of a sequence of finite graphs with maximum degree $ d $.
• For any $ r o ho $ and $ ho > 0 $, there exists a finite graph $ G $ such that $ |p_G(eta) - u(T( extrm{Gr}_d, eta))| < ho $ for all $ eta o U^{r,d} $, proving approximation in the local weak topology.
• The construction ensures that the proportion of vertices in $ G $ with $ r+1 $-ball isomorphic to a non-label-separated tree is bounded by $ rac{ ho}{10} + ho d |U^{r,d}_n| $, which can be made arbitrarily small.
• For label-separated trees $ M $, the local neighborhood frequency $ p_G(M) $ approximates $ u(T( extrm{Gr}_d, M)) $ within $ d ho $, and this error can be made arbitrarily small by choosing $ ho $ small.
• The method constructs graphs via rational-weighted edge-matching on labeled balls, ensuring that the resulting graph’s local structure matches the target measure to arbitrary precision.
• The key insight is that involution-invariance allows the construction of symmetric edge-weighted graphs $ H $, which can be realized as finite graphs via integer scaling and bijection-based edge matching.