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arXiv · 2512.01429

Tight bounds for expected propagation time of probabilistic zero forcing

Mehdi Jelassi·Julien Portier·Rik Sarkar
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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.

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Mehdi Jelassi, Julien Portier, Rik Sarkar. 2025-12-01. Tight bounds for expected propagation time of probabilistic zero forcing. https://arxiv.org/abs/2512.01429

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