[Paper Review] Weak survival for branching random walks on graphs

Daniela Bertacchi, Fabio Zucca|arXiv (Cornell University)|Mar 16, 2006

Stochastic processes and statistical mechanics12 references4 citations

TL;DR

This paper investigates weak and strong survival in continuous-time branching random walks on multigraphs, establishing that at the strong critical value λₛ, extinction occurs locally almost surely, and at the weak critical value λ_w, extinction occurs globally almost surely for a broad class of multigraphs. The key contribution is proving that the existence of a pure weak phase is equivalent to nonamenability for a large class of multigraphs, extending prior results on quasi-transitive graphs and regular graphs.

ABSTRACT

We study weak and strong survival for branching random walks on multigraphs. We prove that, for a large class of multigraphs, weak survival is related to a geometrical parameter of the multigraph and that the existence of a pure weak phase is equivalent to nonamenability. Finally we study weak and strong critical behaviors of the branching random walk.

Motivation & Objective

• To identify the weak critical value λ_w for branching random walks on multigraphs.
• To analyze the behavior of the process at the critical values λ = λ_s and λ = λ_w.
• To determine conditions under which a pure weak phase exists, particularly in relation to graph geometry and amenability.
• To extend known results on phase transitions in BRW from quasi-transitive graphs to a broader class of multigraphs.
• To explore the relationship between geometric parameters of the graph and the critical thresholds λ_w and λ_s.

Proposed method

• The authors define two asymptotic degrees, M_s and M_w, based on the graph's structure, which characterize the critical values λ_s and λ_w.
• They use generating functions F(x,x|λ) and G(x,x|λ) to analyze first-passage probabilities and survival probabilities in the BRW.
• The analysis relies on the convergence radius R of the generating function G(x,x|λ), which is shown to equal λ_s.
• A modified BRW is introduced where λ_w = 1, enabling the derivation of global extinction at λ = 1.
• The results are extended to weighted graphs by generalizing edge weights and adapting the generating function framework.
• The equivalence between nonamenability and the existence of a pure weak phase is proven for a class of non-oriented F-multigraphs using spectral and geometric arguments.

Experimental results

Research questions

• RQ1 What is the precise relationship between the weak critical value λ_w and the geometric structure of the multigraph?
• RQ2 Does the branching random walk die out globally almost surely at λ = λ_w for a broad class of multigraphs?
• RQ3 Is the existence of a pure weak phase equivalent to nonamenability for a class of multigraphs beyond quasi-transitive graphs?
• RQ4 Can the critical value λ_w be universally characterized by the asymptotic degree M_w across all multigraphs, or only in restricted classes?
• RQ5 Is there a multigraph where λ_s = λ_w but the process does not die out globally at λ_w?

Key findings

• At the strong critical value λ_s, the branching random walk dies out locally almost surely.
• At the weak critical value λ_w, the process dies out globally almost surely for a large class of multigraphs, including regular and quasi-transitive graphs.
• The weak critical value λ_w is equal to 1/M_w, where M_w is a geometric parameter derived from the graph's structure.
• The existence of a pure weak phase is equivalent to nonamenability for non-oriented F-multigraphs, extending Stacey's result to a broader class.
• The critical value λ_s is equal to the convergence radius R of the generating function G(x,x|λ), which depends only on the graph's structure.
• For the modified BRW, λ_w = 1 and global extinction occurs almost surely at λ = 1, providing a benchmark for the general case.