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Percolation on dense random graphs with given degrees

Lyuben Lichev, Dieter Mitsche, Guillem Perarnau

Abstract

In this paper, we study the order of the largest connected component of a random graph having two sources of randomness: first, the graph is chosen randomly from all graphs with a given degree sequence, and then bond percolation is applied. Far from being able to classify all such degree sequences, we exhibit several new threshold phenomena for the order of the largest component in terms of both sources of randomness. We also provide an example of a degree sequence for which the order of the largest component undergoes an unbounded number of jumps in terms of the percolation parameter, giving rise to a behavior that cannot be observed without percolation.

Percolation on dense random graphs with given degrees

Abstract

In this paper, we study the order of the largest connected component of a random graph having two sources of randomness: first, the graph is chosen randomly from all graphs with a given degree sequence, and then bond percolation is applied. Far from being able to classify all such degree sequences, we exhibit several new threshold phenomena for the order of the largest component in terms of both sources of randomness. We also provide an example of a degree sequence for which the order of the largest component undergoes an unbounded number of jumps in terms of the percolation parameter, giving rise to a behavior that cannot be observed without percolation.
Paper Structure (20 sections, 27 theorems, 77 equations, 2 figures)

This paper contains 20 sections, 27 theorems, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and terminology.
  3. Our results.
  4. Main ideas and plan of the paper.
  5. Preliminaries and tools
  6. The switching method
  7. Connection between average and minimum degree in graphs
  8. Stochastic comparison of the components of vertices with different degrees
  9. Further probabilistic ingredients
  10. Proofs of the main results: upper bounds
  11. Proof of Part (a) in Theorems \ref{['thm 1']} and \ref{['thm 6']}
  12. Proof of Theorem \ref{['thm 7']}
  13. Proof of the main results: lower bounds
  14. Proof of Part (b) in Theorems \ref{['thm 1']} and \ref{['thm 4']}
  15. Proof of Part (b) in Theorem \ref{['thm 6']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\mathcal{D}_n$ be a degree sequence such that there exist $\delta\ge 0$ and $d = d(n)$ satisfying $|S_n(d)|= \delta n +o(n)$. We have

Figures (2)

  • Figure 1: The switching of $vz$ and $xy$ with $vx$ and $zy$. First, $v$ and $z$ may move from $S_{iso}^2\cup S_{iso}^3$ to $S_{iso}^1$ (note that $z$ may be outside $S_{iso}$ in general while $v\in S_{iso}$ by definition), and therefore for every vertex $w\in S_{iso}$, the edges $wv$ and $wz$ may contribute to $N_2(w)$ instead of $N_3(w)$ after the switching. Apart from that, $z$ loses one edge towards $S_{iso}$ and $v$ loses at most one edge towards $S_{iso}$. Hence, $X_3$ changes by at most $e(v, S_{iso}\setminus v) + e(z, S_{iso}\setminus z)\le 2|S_{iso}|$ but $X_2+X_3$ changes by 1 or 2.
  • Figure 2: The function $f(\alpha)$ for $\alpha\in(0,1]$.

Theorems & Definitions (57)

  • Theorem 1.1: Percolation threshold for $\delta$-large
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: Sublinear size: no jump
  • Remark 1.6
  • Theorem 1.7: Linear size in wide interval: coarse threshold with linear jumps
  • Corollary 1.8: Linear size in narrow interval: sharp threshold with linear jumps
  • Theorem 1.9: Sublinear size thresholds
  • Theorem 1.10
  • ...and 47 more