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Optimal Unimodular Matching

Nathanaël Enriquez, Mike Liu, Laurent Ménard, Vianney Perchet

TL;DR

This work analyzes optimal weight matchings on sparse random graphs that converge locally to unimodular i.i.d. weighted trees. It develops a belief-propagation–style framework on Unimodular Bienaymé-Galton-Watson (UBGW) trees, introducing a recursive variable Z that satisfies $Z \overset{law}{=} \max\left(0,\max_{1\le i\le N}(W_i-Z_i)\right)$ with $N\sim \hat{\pi}$ and $W_i\sim \omega$, and proves the existence and uniqueness (in law) of an optimal unimodular matching on the limit tree. The main result shows that, under mild moment and atomlessness assumptions, the maximal weight matchings of finite graphs converge in the local topology to this limiting unimodular optimal matching, with explicit asymptotics for the average weight and density given by $\mathbb E[W\mathbf 1_{Z+Z'<W}]$ and $\mathbb P(Z+Z'<W)$, respectively. The paper also provides a detailed analysis of the associated fixed-point equation, introduces self-loops to handle partial matchings, proves the uniqueness of the limiting message-passing distribution, and discusses extensions to multi-type trees, vertex weights, and capacity-constrained subgraphs, highlighting both the method's reach and its limitations.

Abstract

We consider sequences of finite weighted random graphs that converge locally to unimodular i.i.d. weighted random trees. When the weights are atomless, we prove that the matchings of maximal weight converge locally to a matching on the limiting tree. For this purpose, we introduce and study unimodular matchings on weighted unimodular random trees as well as a notion of optimality for these objects. In this context, we prove that, in law, there is a unique optimal unimodular matching for a given unimodular tree. We then prove that this law is the local limit of the sequence of matchings of maximal weight. Along the way, we also show that this law is characterised by an equation derived from a message passing algorithm.

Optimal Unimodular Matching

TL;DR

This work analyzes optimal weight matchings on sparse random graphs that converge locally to unimodular i.i.d. weighted trees. It develops a belief-propagation–style framework on Unimodular Bienaymé-Galton-Watson (UBGW) trees, introducing a recursive variable Z that satisfies with and , and proves the existence and uniqueness (in law) of an optimal unimodular matching on the limit tree. The main result shows that, under mild moment and atomlessness assumptions, the maximal weight matchings of finite graphs converge in the local topology to this limiting unimodular optimal matching, with explicit asymptotics for the average weight and density given by and , respectively. The paper also provides a detailed analysis of the associated fixed-point equation, introduces self-loops to handle partial matchings, proves the uniqueness of the limiting message-passing distribution, and discusses extensions to multi-type trees, vertex weights, and capacity-constrained subgraphs, highlighting both the method's reach and its limitations.

Abstract

We consider sequences of finite weighted random graphs that converge locally to unimodular i.i.d. weighted random trees. When the weights are atomless, we prove that the matchings of maximal weight converge locally to a matching on the limiting tree. For this purpose, we introduce and study unimodular matchings on weighted unimodular random trees as well as a notion of optimality for these objects. In this context, we prove that, in law, there is a unique optimal unimodular matching for a given unimodular tree. We then prove that this law is the local limit of the sequence of matchings of maximal weight. Along the way, we also show that this law is characterised by an equation derived from a message passing algorithm.
Paper Structure (30 sections, 29 theorems, 233 equations, 19 figures)

This paper contains 30 sections, 29 theorems, 233 equations, 19 figures.

Table of Contents

  1. Keywords:
  2. Mathematics Subjects Classification:
  3. Introduction
  4. Organisation of the paper
  5. Acknowledgements.
  6. Unimodular matched random graphs
  7. Rooted graphs and local convergence
  8. Vertex-rooted and edge-rooted unimodularity
  9. Unimodular Bienaymé-Galton-Watson trees
  10. Matchings, optimality and unimodularity
  11. Optimal unimodular matchings on UBGW trees : Theorem \ref{['unicityUBGW']}
  12. A message passing algorithm
  13. Adding self-loops
  14. Optimality
  15. Uniqueness
  16. ...and 15 more sections

Key Result

Theorem 1

Let $(\mathbb{T},\overset{\rightarrow}{o},w)$ be an edge-rooted Unimodular Bienaymé-Galton-Watson (UBGW) tree with reproduction law $\pi$ and i.i.d edge-weights with law $\omega$ such that: There exists a unique (in law) optimal matching $\mathbb{M}_{\mathrm{opt}}(\mathbb{T})$ on $(\mathbb{T},\overset{\rightarrow}{o},w)$. Furthermore, the distribution of $(\mathbb{T},\overset{\rightarrow}{o},w,\m

Figures (19)

  • Figure 1: A 2-Neighbourhood of an edge-rooted UBGW tree with the law of the number of children drawn on every vertex.
  • Figure 2: A $3-$Neighbourhood of a vertex-rooted UBGW tree with the law of the number of children drawn on every vertex.
  • Figure 3: Definitions of $Z$.
  • Figure 4: Illustration of the deduction of the recursive equation assuming $v_i$ is matched to $v$.
  • Figure 5: Construction of the law of $Z_h$ on a $1-$neighbourhood.
  • ...and 14 more figures

Theorems & Definitions (72)

  • Theorem 1: Existence and uniqueness of the optimal matching for iid weighted UBGW
  • Theorem 2: Local convergence of maximal matchings
  • Corollary 1
  • Corollary 2
  • Theorem 3: Uniqueness of the belief propagation stationary measure
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: Vertex-rooted unimodularity
  • Definition 2.5: Edge-rooted unimodularity
  • ...and 62 more