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Structural convergence and algebraic roots

David Hartman, Tomáš Hons, Jaroslav Nešetřil

TL;DR

This work resolves a refined version of the rooting problem in $FO$-convergence: if a sequence of graphs $(G_n)$ converges to a modeling limit $L$ and $r$ is an algebraic vertex of $L$ (i.e., lies in a finite definable set), then there exist $r_n in V(G_n)$ such that $(G_n, r_n)$ FO-converges to $(L, r)$. The authors achieve this through a three-tier strategy: (i) establish rooting for a single FO formula by analyzing the pushforward measure on the finite definable set $\xi(L)$ and ordering atoms by their induced probabilities; (ii) extend to any finite collection of formulas via a carefully constructed combined formula whose parameters separate the contributions; (iii) extend to all FO formulas using compactness and a König’s lemma argument to produce an infinite rooting path witnessing convergence for all formulas. A finite boolean lattice lemma and Newton-type identities are used to guarantee continuity of the relevant probability profiles, enabling a robust passage from finite to infinitary considerations. The paper also provides explicit examples showing the optimality of the finite definable-set condition and discusses gadget constructions that preserve FO-convergence while controlling rooting. Overall, the results advance the understanding of how algebraic definability of roots governs the feasibility of rooted FO-limits in structural convergence, with implications for gadget-based graph constructions and modeling limits.

Abstract

Structural convergence is a framework for convergence of graphs by Nešetřil and Ossona de Mendez that unifies the dense (left) graph convergence and Benjamini-Schramm convergence. They posed a problem asking whether for a given sequence of graphs $(G_n)$ converging to a limit $L$ and a vertex $r$ of $L$ it is possible to find a sequence of vertices $(r_n)$ such that $L$ rooted at $r$ is the limit of the graphs $G_n$ rooted at $r_n$. A counterexample was found by Christofides and Král', but they showed that the statement holds for almost all vertices $r$ of $L$. We offer another perspective to the original problem by considering the size of definable sets to which the root $r$ belongs. We prove that if $r$ is an algebraic vertex (i.e. belongs to a finite definable set), the sequence of roots $(r_n)$ always exists.

Structural convergence and algebraic roots

TL;DR

This work resolves a refined version of the rooting problem in -convergence: if a sequence of graphs converges to a modeling limit and is an algebraic vertex of (i.e., lies in a finite definable set), then there exist such that FO-converges to . The authors achieve this through a three-tier strategy: (i) establish rooting for a single FO formula by analyzing the pushforward measure on the finite definable set and ordering atoms by their induced probabilities; (ii) extend to any finite collection of formulas via a carefully constructed combined formula whose parameters separate the contributions; (iii) extend to all FO formulas using compactness and a König’s lemma argument to produce an infinite rooting path witnessing convergence for all formulas. A finite boolean lattice lemma and Newton-type identities are used to guarantee continuity of the relevant probability profiles, enabling a robust passage from finite to infinitary considerations. The paper also provides explicit examples showing the optimality of the finite definable-set condition and discusses gadget constructions that preserve FO-convergence while controlling rooting. Overall, the results advance the understanding of how algebraic definability of roots governs the feasibility of rooted FO-limits in structural convergence, with implications for gadget-based graph constructions and modeling limits.

Abstract

Structural convergence is a framework for convergence of graphs by Nešetřil and Ossona de Mendez that unifies the dense (left) graph convergence and Benjamini-Schramm convergence. They posed a problem asking whether for a given sequence of graphs converging to a limit and a vertex of it is possible to find a sequence of vertices such that rooted at is the limit of the graphs rooted at . A counterexample was found by Christofides and Král', but they showed that the statement holds for almost all vertices of . We offer another perspective to the original problem by considering the size of definable sets to which the root belongs. We prove that if is an algebraic vertex (i.e. belongs to a finite definable set), the sequence of roots always exists.
Paper Structure (10 sections, 7 theorems, 10 equations)

This paper contains 10 sections, 7 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Notation and tools
  3. Rooting in algebraic sets
  4. Single formula
  5. Finite collection of formulas
  6. All formulas
  7. A lemma about finite boolean lattices
  8. Examples
  9. Proof of Proposition \ref{['prop:example']}
  10. Concluding remarks

Key Result

Theorem 1

Let $(G_n)$ be an $\mathop{\mathrm{FO}}\nolimits$-convergent sequence of graphs with a modeling limit $L$ and $r$ be an algebraic vertex of $L$. Then there is a sequence $(r_n)$, $r_n \in V(G_n)$, such that $(G_n, r_n)$$\mathop{\mathrm{FO}}\nolimits$-converges to $(L,r)$.

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • proof : Proof of Theorem \ref{['thm:rooting_solution_of_algebraic_formulas']}
  • Lemma 6
  • ...and 2 more