We introduce randomized zero forcing (RZF), a stochastic color-change process on directed graphs in which a white vertex turns blue with probability equal to the fraction of its incoming neighbors that are blue. Unlike probabilistic zero forcing, RZF is governed by in-neighborhood structure and can fail to propagate globally due to directionality. The model extends naturally to weighted directed graphs by replacing neighbor counts with incoming weight proportions. We study the expected propagation time of RZF, establishing monotonicity properties with respect to enlarging the initial blue set and increasing weights on edges out of initially blue vertices, as well as invariances that relate weighted and unweighted dynamics. Exact values and sharp asymptotics are obtained for several families of directed graphs, including arborescences, stars, paths, cycles, and spiders, and we derive tight extremal bounds for unweighted directed graphs in terms of basic parameters such as order, degree, and radius. We conclude with an application to an empirical input-output network, illustrating how expected propagation time under RZF yields a dynamic, process-based notion of centrality in directed weighted systems.

翻译：本文提出随机化零强迫（RZF），这是一种在有向图上的随机颜色传播过程：白色顶点以与其入邻居中蓝色顶点比例相等的概率转变为蓝色。与概率零强迫不同，RZF受入邻域结构支配，且可能因有向性而无法全局传播。该模型通过将邻居计数替换为入边权重比例，自然地推广到加权有向图。我们研究了RZF的期望传播时间，建立了关于初始蓝色顶点集扩大以及从初始蓝色顶点出发的边权重增加时的单调性性质，以及关联加权与未加权动态的不变性。针对若干有向图族（包括树状图、星形图、路径图、环形图和蛛网图）获得了精确值和尖锐的渐近结果，并基于阶数、度数和半径等基本参数，推导出未加权有向图的紧致极值界。最后，我们将其应用于一个经验性投入产出网络，说明RZF下的期望传播时间如何为有向加权系统提供一种基于过程的动态中心性度量。