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.
Color changing is a graph-theoretic concept which models the spread of a property, represented by the color of a vertex, through a graph over discrete time steps. The concept was formalized in [1] as a deterministic process known as zero forcing. In zero forcing, a blue vertex u of an arbitrary graph G changes the color of an adjacent white vertex w if and only if w is the only white vertex adjacent to u. Determining whether an initial set of blue vertices of G will eventually turn the graph entirely blue is a central question, with extensive work devoted to characterizing such sets of minimum size and their relationship to structural and matrix-theoretic parameters of graphs [22].
The study of zero forcing on graphs was initially motivated by problems in bounding the minimum rank of graphs [1], as well as independently arising in quantum control theory [28,29]. Since its introduction, zero forcing has been extended in many directions by modifying either the forcing rule or the class of allowable forcing vertices. Positive semidefinite zero forcing was introduced to study the minimum positive semidefinite rank problem and differs from standard zero forcing in how forces propagate across components [3]. Skew zero forcing further relaxes the forcing rule by allowing white vertices to force, leading to different extremal behavior and complexity questions [14]. A complementary line of work focuses on propagation time rather than feasibility, including variants that study tradeoffs between the size of an initial forcing set and the time required for complete propagation [10]. Related ideas also appear in power domination, which models monitoring processes in networks and can be viewed as a forcing process with delayed observation [21]. In this setting, electrical networks are monitored using phasor measurement units (PMUs), and the placement of these devices is closely related to the zero forcing process on the underlying network graph. As a result, power domination has been extensively studied across a variety of graph families and graph products [27].
In [24], Kang and Yi modified the zero forcing definition to be probabilistic. In this new rule, called probabilistic zero forcing (PZF), each blue vertex u of an undirected graph G attempts to independently change the color of any incident white vertex w and succeeds with probability, P (u turns w blue) = number of blue vertices which are adjacent to u degree of u . (1.1) Unlike deterministic zero forcing, minimal forcing sets for the probabilistic color-change rule (1.1) are trivial: on any connected graph, starting from any nonempty initial blue set, the process colors the entire graph blue with probability 1 [24]. Consequently, the natural parameters in probabilistic zero forcing (PZF) concern how quickly the all-blue state is reached rather than whether it can be reached. This motivates the study of the expected propagation time ept pzf (G, X), defined as the expected number of update rounds required for PZF to color all of V (G) blue starting from an initial blue set X.
The systematic study of ept pzf was initiated by Geneson and Hogben [18], who established general bounds, exact values on several graph families, and extremal constructions. Algorithmic and computational approaches based on Markov chains were developed in [13]. Subsequent work has sharpened general upper bounds for ept pzf in terms of classical graph parameters (such as order and radius) [25], and has developed asymptotic results for additional families, including random graphs [15] and structured graphs such as grids, regular graphs, and hypercubes [23]. Further refinements and related propagation-time parameters for these families have also been studied (e.g., in tightasymptotic regimes for hypercubes and grids) [4]. Collectively, this body of work emphasizes that while PZF guarantees eventual propagation on connected graphs, the expected time scale captures sensitive structural information about the underlying topology.
In this article, we consider a variation of PZF on directed graphs (edges can be bidirectional) based on a color change rule determined by the in-degree of uncolored vertices. Precisely, suppose w is a white vertex in a directed graph G. Then, at each step of the color change process, the probability that w changes from white to blue is given by, P (w turns blue) = # of incident edges to w with blue source in-degree of w . (1.2)
In the case that w has indegree 0, the probability that w turns blue is 0. The color change rule in equation (1.2) does not imply that eventually every vertex of G will turn blue, even if G as an undirected graph is connected. If it is possible, then, just as in the case of PZF, the time it takes to color all of G is a random variable determined by the initial set of blue vertices. We define the expected propagation time for this new process, which we call randomized zero forcing, to accommodate the chance that the graph is never completely co
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