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Percolation phase transition in weight-dependent random connection models

Peter Gracar, Lukas Lüchtrath, Peter Mörters

Abstract

We investigate spatial random graphs defined on the points of a Poisson process in $d$-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a nontrivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. We apply this result to characterize robustness of age-based spatial preferential attachment networks.

Percolation phase transition in weight-dependent random connection models

Abstract

We investigate spatial random graphs defined on the points of a Poisson process in -dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a nontrivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. We apply this result to characterize robustness of age-based spatial preferential attachment networks.

Paper Structure

This paper contains 12 sections, 13 theorems, 86 equations, 3 figures.

Table of Contents

  1. Introduction and statement of results
  2. Existence of a subcritical phase
  3. Shortcut-free paths
  4. Skeleton of a path
  5. Graph surgery
  6. Strategy of the proof
  7. Connecting two old vertices
  8. BK-inequality
  9. Proof of the subcritical phase
  10. Absence of a subcritical phase
  11. Proof of Theorem 1.2
  12. Integration results

Key Result

Theorem \oldthetheorem

Suppose $\rho$ satisfies RegularVarying for some $\delta>1$. Then, for the weight-dependent random connection model with preferential attachment kernel, sum kernel or min kernel and parameters $\beta>0$, $0<\gamma<1$, we have that

Figures (3)

  • Figure 1: A path where a vertex's birth time is denoted on the $t$-axis. The vertices of the skeleton are in black. We successively remove all local maxima, starting with the youngest, and replace them by direct edges until the path, only containing the skeleton vertices, is left.
  • Figure 2: On the left the path $P$ where the $t$-axis denotes the vertices' birth times. The vertices $\mathbf{y}_1$ and $\mathbf{y}_0$, which will not appear in the tree, are in grey. We insert the vertex $\mathbf{y}_6$ at the end of the branch that goes left at $\mathbf{y}_2$, right at $\mathbf{y}_3$, and right at $\mathbf{y}_4$.
  • Figure 3: On the left the binary tree $T$. The grey vertices are already explored by depth-first search. The black vertex $\mathbf{v}$ is the vertex currently being explored. The white vertices have not been discovered yet. On the right, the path $P$ corresponding to the already explored tree. The $t$-axis denotes the vertices' birth times. Start and end vertex, $\mathbf{x}$ and $\mathbf{y}$, do not appear in the tree. Since $\mathbf{v}$ is the right child of $\mathbf{w}$, we insert $\mathbf{v}$ as a local maximum between $\mathbf{w}$ and $\mathbf{y}$ in the path $P$.

Theorems & Definitions (25)

  • Theorem \oldthetheorem: Percolation phase transition
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • proof : Proof of Theorem \ref{['ThmPercolationPhase']}(a)
  • Lemma \oldthetheorem
  • ...and 15 more