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Large Deviations for Subcritical Bootstrap Percolation on the Erdős-Rényi Graph

Omer Angel1, Brett Kolesnik2

  • 1Department of Mathematics, University of British Columbia, Vancouver, BC Canada.

Journal of Statistical Physics
|November 1, 2021
PubMed

Insights

This study analyzes bootstrap percolation on random graphs, revealing the probability of widespread infection from small initial sets. We identify the most efficient infection pathways for unexpected large-scale spread.

Area of Science:

  • Probability theory
  • Statistical physics
  • Network science

Background:

  • Bootstrap percolation models infection spread on graphs.
  • Erdős-Rényi random graphs are a fundamental model for network structure.
  • Previous work established critical conditions for widespread infection.

Purpose of the Study:

  • Investigate atypical, large-deviation events in bootstrap percolation.
  • Quantify the probability of unexpected infection spread from small initial sets.
  • Identify the most efficient pathways for such large deviations.

Main Methods:

  • Analysis of bootstrap percolation dynamics on Erdős-Rényi random graphs.
  • Large deviation theory to calculate rate functions.
  • Identification of minimum-cost trajectories for rare events.

Main Results:

  • Calculated the rate function for small initial sets causing unexpected large infections.
  • Identified the least-cost trajectory for realizing these large deviations.
  • Characterized the probability of rare, extensive infection spread.

Conclusions:

  • Provides a deeper understanding of extreme events in random network percolation.
  • Offers insights into the dynamics of rare but significant spread phenomena.
  • Contributes to the statistical analysis of complex systems.
Keywords:
Bootstrap percolationDiscrete calculus of variationsLarge deviationsPhase transitionRandom graphs

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