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Percolation under noise: Detecting explosive percolation using the second-largest component.

Wes Viles1, Cedric E Ginestet2, Ariana Tang1

  • 1Department of Mathematics and Statistics, Boston University, Boston, Massachusetts 02215, USA.

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|June 15, 2016
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Summary
This summary is machine-generated.

Distinguishing percolation rates under noise is challenging. The maximal size of the second-largest component effectively differentiates percolation models, outperforming interquartile range methods, particularly in neuroscience applications.

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Area of Science:

  • Statistical modeling
  • Graph theory
  • Stochastic processes

Background:

  • Percolation models are crucial for understanding network dynamics.
  • Distinguishing between percolation rates under observational noise is a significant challenge.
  • Existing models may not adequately account for dynamic edge changes and observation errors.

Purpose of the Study:

  • To develop and compare statistical criteria for distinguishing between different percolation rates in the presence of noise.
  • To evaluate the effectiveness of two proposed discrimination methods: interquartile range (IQR) of the first component's size and the maximal size of the second-largest component.
  • To explore the application of these criteria in applied neuroscience for detecting clinically relevant percolation.

Main Methods:

  • Construction of a statistical model for percolation incorporating birth/death of edges and observational noise (Type-I and Type-II errors).
  • Development of a hidden Markov graph model representing the latent and observed processes.
  • Comparative analysis of two discrimination criteria using data simulations.

Main Results:

  • Classical (Erdős-Rényi) percolation can become visually indistinguishable from faster percolation under specific noise conditions.
  • The maximal size of the second-largest component demonstrates superior discriminatory power compared to the IQR of the first component's size.
  • The proposed methods are effective under physically motivated conditions for edge dynamics and noise.

Conclusions:

  • The maximal size of the second-largest component is a robust statistic for differentiating percolation rates under noise.
  • This method offers a valuable tool for analyzing complex network dynamics where observations are imperfect.
  • Potential applications exist in neuroscience for identifying critical percolation patterns relevant to clinical conditions.