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Transitive closure in a polluted environment

Janko Gravner, Brett Kolesnik

TL;DR

This work analyzes a transitive-closure process on a fixed vertex set with an initial edge set $E_0$ and a polluted-open edge set $E_{ ext{open}}$, where an open edge $i\to j$ is added if $i\to k\to j$ holds for some $k$. It identifies three regimes for the unoriented linear initial graph, depending on left/right open-edge probabilities, and proves a subcritical-to-saturated transition in the bounded-degree setting at $p_{ ext{open}}\asymp (\log n)^{-1/2+o(1)}$, with Catalan-percolation phenomena arising in the oriented subcase. The intermediate regime is handled via the tilde process to control cross-direction interactions, while nucleation drives the supercritical regime, yielding saturation when $p_{ ext{open}}\ge C\log\log n/\sqrt{\log n}$ for bounded-degree initial graphs, and extending to $R$-unoriented graphs. The paper connects transitive-closure growth under pollution to bootstrap and jigsaw percolation frameworks, establishing precise thresholds and outlining rich open problems for broader graph classes and higher-dimensional settings.

Abstract

We introduce and study a new percolation model, inspired by recent works on jigsaw percolation, graph bootstrap percolation, and percolation in polluted environments. Start with an oriented graph $G_0$ of initially occupied edges on $n$ vertices, and iteratively occupy additional (oriented) edges by transitivity, with the constraint that only open edges in a certain random set can ever be occupied. All other edges are closed, creating a set of obstacles for the spread of occupied edges. When $G_0$ is an unoriented linear graph, and leftward and rightward edges are open independently with possibly different probabilities, we identify three regimes in which the set of eventually occupied edges is either all open edges, the majority of open edges in one direction, or only a very small proportion of all open edges. In the more general setting where $G_0$ is a connected unoriented graph of bounded degree, we show that the transition between sparse and full occupation of open edges occurs when the probability of open edges is $(\log n)^{-1/2+o(1)}$. We conclude with several conjectures and open problems.

Transitive closure in a polluted environment

TL;DR

This work analyzes a transitive-closure process on a fixed vertex set with an initial edge set and a polluted-open edge set , where an open edge is added if holds for some . It identifies three regimes for the unoriented linear initial graph, depending on left/right open-edge probabilities, and proves a subcritical-to-saturated transition in the bounded-degree setting at , with Catalan-percolation phenomena arising in the oriented subcase. The intermediate regime is handled via the tilde process to control cross-direction interactions, while nucleation drives the supercritical regime, yielding saturation when for bounded-degree initial graphs, and extending to -unoriented graphs. The paper connects transitive-closure growth under pollution to bootstrap and jigsaw percolation frameworks, establishing precise thresholds and outlining rich open problems for broader graph classes and higher-dimensional settings.

Abstract

We introduce and study a new percolation model, inspired by recent works on jigsaw percolation, graph bootstrap percolation, and percolation in polluted environments. Start with an oriented graph of initially occupied edges on vertices, and iteratively occupy additional (oriented) edges by transitivity, with the constraint that only open edges in a certain random set can ever be occupied. All other edges are closed, creating a set of obstacles for the spread of occupied edges. When is an unoriented linear graph, and leftward and rightward edges are open independently with possibly different probabilities, we identify three regimes in which the set of eventually occupied edges is either all open edges, the majority of open edges in one direction, or only a very small proportion of all open edges. In the more general setting where is a connected unoriented graph of bounded degree, we show that the transition between sparse and full occupation of open edges occurs when the probability of open edges is . We conclude with several conjectures and open problems.

Paper Structure

This paper contains 11 sections, 20 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Acknowledgments
  5. Subcritical regime for bounded-degree initial graphs
  6. Catalan percolation
  7. Intermediate regime for linear initial graphs
  8. The tilde process
  9. Supercritical regime for bounded-degree initial graphs
  10. $R$-unoriented initial graphs
  11. Open problems

Key Result

Theorem 1.1

Assume that $G_0=(V,E_0)$ is a connected unoriented graph on $V=[n]$ with vertex degrees bounded by a constant $D$, and that open (oriented) edges are chosen independently (from amongst those not in $E_0$) with probability $p_{\text{\rm open}}$. Fix a constant $\alpha>0$. Then there exist constants

Figures (5)

  • Figure 1.1: Illustration of the three regimes in Theorem \ref{['tc2']} when $n=300$: subcritical (left, with $p_{\text{\rm left}}=0.24$, $p_{\text{\rm right}}=0.36$), intermediate (middle, with $p_{\text{\rm left}}=0.2$, $p_{\text{\rm right}}=0.4$; note the non-monotone fashion in which edges are occupied), and supercritical (right, with $p_{\text{\rm left}}=p_{\text{\rm right}}=0.35$; note the nucleation). The dynamics are represented as the evolution of the adjacency matrix, with edges exhibited as sites in the square. Initially occupied sites next to the diagonal are black, closed sites are grey and open sites are white. After the transitive closure process is complete, the initially white sites that become occupied are colored according to the time of occupation, from blue (the earliest) to yellow (the latest).
  • Figure 2.1: The four ways $v$ can be the base and $y$ the tip of a horn. Initially occupied edges in $E_0$ are represented by solid arrows, open edges in $E_{\text{\rm open}}$ by dotted arrows, and edges in $E_0\cup E_{\text{\rm open}}$ by dashed arrows.
  • Figure 3.1: At left: The open edge $(2,11)$ becomes occupied due to a pair of oriented paths of open sites, from $(3,4)$ to $(2,5)$ and from $(8,9)$ to $(5,11)$. At right: The same occupation process whereby edges are represented as usual. Note that, until the very last step, each transitive step involves at least one initially occupied edge.
  • Figure 4.1: A leftward open edge $i\leftarrow j$ becomes occupied due to occupied edges $i\leftarrow k\leftarrow j$ of opposite orientations, for some $k\notin[i,j]$.
  • Figure 6.1: Left: nucleation in $K_4$-percolation with $p_{\text{\rm open}}=0.39$. Middle and right: illustration of Open Problem \ref{['con-oriented']}, with $r=50$, and respective probabilities $p_{\text{\rm open}}=0.3$, $p_{\text{\rm open}}=0.37$. In all figures, $n=400$ and the coloring scheme is similar to the one in Fig. \ref{['fig-tra']}.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 28 more