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Simultaneous Cutoff on the Multitype Configuration Model

John Fernley, Balázs Gerencsér

Abstract

We find Gaussian cutoff profiles for the total variation distance to stationarity of a random walk on a multiplex network: a finite number of directed configuration models sharing a vertex set, each with its own bounded degree distribution and edge probability. Further we consider the minimal total variation distance over this space of possible doubly stochastic edge probabilities at each point in time. Looking at all possible dynamics simultaneously on one realisation of the random graph, we find that this sequence of minimal distances converges in probability to the same cutoff profile as the chain with entropy maximising transition probabilities.

Simultaneous Cutoff on the Multitype Configuration Model

Abstract

We find Gaussian cutoff profiles for the total variation distance to stationarity of a random walk on a multiplex network: a finite number of directed configuration models sharing a vertex set, each with its own bounded degree distribution and edge probability. Further we consider the minimal total variation distance over this space of possible doubly stochastic edge probabilities at each point in time. Looking at all possible dynamics simultaneously on one realisation of the random graph, we find that this sequence of minimal distances converges in probability to the same cutoff profile as the chain with entropy maximising transition probabilities.
Paper Structure (6 sections, 29 theorems, 231 equations, 1 figure)

This paper contains 6 sections, 29 theorems, 231 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Approximation
  4. Cutoff
  5. Optimisation
  6. Appendix

Key Result

Theorem 2.4

Write $w_\bigstar$ and $t_\bigstar$ for the shape parameters of eq_mu_def and eq_sigma_def with probabilities then with Assumptions assumptions

Figures (1)

  • Figure 1: The asymptotic constant multiple of $\log N$ of the cutoff times of chains in Example \ref{['example']}.

Theorems & Definitions (73)

  • Definition 2.1
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 2.6
  • Example 2.7
  • Remark 3.1
  • Definition 3.2: Relaxation Time
  • Corollary 3.3
  • proof
  • ...and 63 more