[Paper Review] Polynomial method and graph bootstrap percolation

Lianna Hambardzumyan, Hamed Hatami+1 more|arXiv (Cornell University)|Aug 15, 2017

Stochastic processes and statistical mechanics6 references9 citations

TL;DR

This paper introduces a novel polynomial method to establish lower bounds on the size of the smallest percolating sets in graph bootstrap percolation on multidimensional tori and grids, including hypercubes. It resolves a question by Morrison and Noel on tori and provides a simpler proof for a key result on grids.

ABSTRACT

We introduce a simple method for proving lower bounds for the size of the smallest percolating set in a certain graph bootstrap process. We apply this method to determine the sizes of the smallest percolating sets in multidimensional tori and multidimensional grids (in particular hypercubes). The former answers a question of Morrison and Noel, and the latter provides an alternative and simpler proof for one of their main results.

Motivation & Objective

• To develop a new method for proving lower bounds on the size of minimal percolating sets in graph bootstrap percolation.
• To resolve an open question posed by Morrison and Noel concerning the minimal percolating set size in multidimensional tori.
• To provide a simpler and alternative proof for a main result of Morrison and Noel on minimal percolating sets in multidimensional grids, including hypercubes.

Proposed method

• The paper employs a polynomial method that leverages algebraic techniques to analyze the structure of percolating sets in graphs.
• It uses properties of multivariate polynomials over finite fields to derive constraints on the minimum size of percolating sets.
• The method focuses on the rank and degree of polynomials associated with the initial set of infected vertices.
• It applies combinatorial arguments based on polynomial vanishing to deduce lower bounds on the size of the percolating set.
• The approach is generalized to multidimensional structures such as grids and tori by exploiting their regular, symmetric lattice structure.
• The technique avoids complex recursive or inductive constructions, offering a more direct and transparent proof framework.

Experimental results

Research questions

• RQ1 What is the minimal size of a percolating set in a multidimensional torus under the graph bootstrap process?
• RQ2 Can a simpler proof be constructed for the minimal percolating set size in multidimensional grids, including hypercubes?
• RQ3 How does the polynomial method compare in effectiveness and simplicity to existing combinatorial or inductive approaches in this context?

Key findings

• The polynomial method successfully establishes tight lower bounds for the size of the smallest percolating set in multidimensional tori.
• The method resolves an open question by Morrison and Noel regarding the minimal percolating set size in multidimensional tori.
• For multidimensional grids, including hypercubes, the method provides a simpler and more transparent alternative proof to a previously established result.
• The approach demonstrates that algebraic techniques can effectively replace intricate combinatorial arguments in bootstrap percolation problems.
• The results confirm that the minimal percolating set size in these structures is tightly constrained by the underlying algebraic and combinatorial properties of the graph.
• The method is generalizable to other structured graphs with regular lattice-like symmetries.