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This study investigates graph community structures using the stochastic block model, focusing on component sizes down to extremely low probabilities. Sophisticated simulations reveal insights into large-deviation principles and differences from Erdős-Rényi models.

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Area of Science:

  • Network Science
  • Statistical Physics
  • Computer Science

Background:

  • The stochastic block model is a key tool for analyzing community structures in networks.
  • Understanding component size distributions is crucial for community detection algorithms.
  • Extreme probability events in network models require advanced analytical and simulation techniques.

Purpose of the Study:

  • To analyze the size distributions of the largest connected and biconnected components in a two-block stochastic block model.
  • To investigate tail probabilities down to 10^{-800} for these component sizes.
  • To compare these distributions with the Erdős-Rényi model and explore large-deviation principles.

Main Methods:

  • Utilizing sophisticated Markov chain Monte Carlo simulations to sample graphs from the stochastic block model ensemble.
  • Employing advanced statistical methods to analyze component size distributions, including extreme tails.
  • Comparing simulation results with theoretical predictions and the Erdős-Rényi random graph model.

Main Results:

  • Detailed characterization of the size distributions for the largest connected and biconnected components in the two-block stochastic block model.
  • Empirical evidence supporting the conjecture that a large-deviation principle holds for these distributions.
  • Identification of subtle but significant differences between the stochastic block model and Erdős-Rényi model distributions, particularly around the percolation threshold.

Conclusions:

  • The study provides deep insights into the behavior of component sizes in stochastic block models, especially in extreme regimes.
  • Findings suggest that the stochastic block model exhibits distinct properties compared to the Erdős-Rényi model, impacting community detection and network analysis.
  • The research validates the use of advanced simulation techniques for exploring rare events in complex network models.