Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The contact process on a bipartite spatial network

John Fernley, Christian Hirsch, Daniel Valesin

Abstract

We study the contact process on a random bipartite connection hypergraph generated from two Poisson point processes, with mark-dependent connection thresholds. For asymmetric infection rates and asymmetric power law tail decays of the two degree distributions, we determine the dominant survival strategies in all parameter regimes and provide asymptotics for the epidemic probability up to logarithmic factors.

The contact process on a bipartite spatial network

Abstract

We study the contact process on a random bipartite connection hypergraph generated from two Poisson point processes, with mark-dependent connection thresholds. For asymmetric infection rates and asymmetric power law tail decays of the two degree distributions, we determine the dominant survival strategies in all parameter regimes and provide asymptotics for the epidemic probability up to logarithmic factors.

Paper Structure

This paper contains 19 sections, 35 theorems, 197 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background, model definition and main results
  3. Classical contact process
  4. Random connection hypergraph
  5. Contact process on $\mathcal{G}$ with type-dependent infection rates
  6. Main result
  7. Graphical construction of the contact process
  8. Lower bound
  9. Extinction time on stars
  10. Lower bounds on survival probability for four strategies
  11. Upper bound
  12. Ordered traces and martingale bound
  13. Infection paths classified by length and cardinality
  14. Combinatorial paths and discovery trees
  15. Bound on tree weight function
  16. ...and 4 more sections

Key Result

Proposition 2

If $\gamma_1 + \gamma_2 < 1$, then $\mathcal{G}$ has no infinite connected component almost surely. $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: Random connection hypergraph with two types of vertices (red and blue). Each vertex has a mark (represented by the size of the circle) that determines its connection reach and so its expected degree.
  • Figure 2: Strategies that give the exponent in the survival probability for different values of $(\gamma_1,\gamma_2,a)$. We include both the case where the root is of type 1 (which is given in the statement of Theorem \ref{['thm_main']}) and of type 2. For a root of type 1, "root is $\mathsf S$" means that the exponent is $\mu_{\mathsf S}$; "one step to $\mathsf S$" means that it is $1+\bar{\gamma}_2 \nu_{\mathsf S}$; "one step to $\mathsf B$" means that is it $1+\bar{\gamma}_2 \nu_{\mathsf B}$; "one step to $\mathsf D$" means that is it $1+\bar{\gamma}_2 \nu_{\mathsf D}$.
  • Figure 3: Setting $a=1$, Theorem \ref{['thm_main']} gives a result for the survival probability of the classical contact process on our bipartite spatial model (still of course neglecting polylogarithmic factors). Each of the four depicted regions corresponds to a dominating strategy, and in the above order these are: "one step to $\mathsf S$", "one step to $\mathsf B$"; "one step to $\mathsf D$"; "root is $\mathsf S$".
  • Figure 4: Illustration of the combinatorial path $\pi$ associated to a graph path $\gamma$, and the discovery tree of $\pi$
  • Figure 5: Example of application of three operations to reduce a tree to a line segment of length two.
  • ...and 1 more figures

Theorems & Definitions (77)

  • Remark 1: Metastable density
  • Proposition 2: Sub-critical regime
  • proof
  • Theorem 3
  • Remark 4: Implied additional results
  • Remark 5: Polylogarithmic corrections
  • Proposition 6: Exponent optimisation
  • Proposition 7: Lower bound
  • Proposition 8: Upper bound
  • Lemma 9
  • ...and 67 more