Tight bounds for expected propagation time of probabilistic zero forcing

TL;DR

This work analyzes the probabilistic zero forcing process on connected graphs, obtaining tight bounds for both propagation time and throttling. It proves that a single starting vertex yields an expected propagation time of n/2 + O(1), confirming a conjecture, and demonstrates th_pzf(G) = O(√n) using an initial set of size O(√n) to achieve O(√n) propagation time, improving previous results. The authors introduce a subgraph-decomposition framework with v-good pairs and an optimized starting-vertex strategy, employing advanced probabilistic tools (Chernoff bounds, Azuma, Doob) to derive tail bounds and expectations. Collectively, the results resolve key conjectures by Narayanan and Sun and advance understanding of how initial resources influence diffusion speed in probabilistic zero forcing.

Abstract

We study the probabilistic zero forcing process, a probabilistic variant of the classical zero forcing process. We show that for every connected graph $G$ on $n$ vertices, there exists an initial set consisting of a single vertex such that the expected propagation time is $n/2 + O(1)$. This result is tight and confirms a conjecture posed by Narayanan and Sun. Additionally, we show tight bounds on the probabilistic throttling number, which captures the trade-off between the size of the initial set and the speed of propagation. Namely, we show that for every connected graph $G$ on $n$ vertices, there exists an initial set consisting of $O(\sqrt{n})$ vertices such that the expected propagation time is $O(\sqrt{n})$. This improves upon previous results by Geneson and Hogben, and confirms another conjecture by Narayanan and Sun.

Theorems & Definitions (38)

• Theorem 1.1: Hogben, Theorem 6.5
• Theorem 1.2
• Theorem 1.3: Sun, Theorem 4.11
• Theorem 1.4
• Theorem 3.1
• Lemma 3.2
• proof
• Definition 3.3: Submartingale
• Theorem 3.4
• Definition 3.5: Stopping Time