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Let $$D$$ be a simple digraph (directed graph) with vertex set $V(D)$ and arc set $A(D)$ where $n=|V(D)|$, and each arc is an ordered pair of distinct vertices. If $$(v,u) \in A(D)$$, then $$u$$ is considered an \emph{out-neighbor} of $$v$$ in $$D$$. Initially, we designate each vertex to be either filled or empty. Then, the following color change rule (CCR) is applied: if a filled vertex $$v$$ has exactly one empty out-neighbor $$u$$, then $$u$$ will be filled. If all vertices in $V(D)$$ are eventually filled under repeated applications of the CCR, then the initial set is called a \emph{zero forcing set} (ZFS); if not, it is a \emph{failed zero forcing set} (FZFS). We introduce the \emph{failed zero forcing number} $$\F(D)$$ on a digraph, which is the maximum cardinality of any FZFS. The \emph{zero forcing number}, $$\Z(D)$, is the minimum cardinality of any ZFS. We characterize digraphs that have $$\F(D)<\Z(D)$$ and determine $$\F(D)$$ for several classes of digraphs including de Bruijn and Kautz digraphs. We also characterize digraphs with $$\F(D)=n-1$$, $$\F(D)=n-2$$, and $$\F(D)=0$$, which leads to a characterization of digraphs in which any vertex is a ZFS. Finally, we show that for any integer $$n \geq 3$$ and any non-negative integer $$k$$ with $k 

Abstract In globally distributed deep-sea hydrothermal vent plumes, microbiomes are shaped by the redox energy landscapes created by reduced hydrothermal vent fluids mixing with oxidized seawater. Plumes can disperse over thousands of kilometers and their characteristics are determined by geochemical sources from vents, e.g., hydrothermal inputs, nutrients, and trace metals. However, the impacts of plume biogeochemistry on the oceans are poorly constrained due to a lack of integrated understanding of microbiomes, population genetics, and geochemistry. Here, we use microbial genomes to understand links between biogeography, evolution, and metabolic connectivity, and elucidate their impacts on biogeochemical cycling in the deep sea. Using data from 36 diverse plume samples from seven ocean basins, we show that sulfur metabolism defines the core microbiome of plumes and drives metabolic connectivity in the microbial community. Sulfur-dominated geochemistry influences energy landscapes and promotes microbial growth, while other energy sources influence local energy landscapes. We further demonstrated the consistency of links among geochemistry, function, and taxonomy. Amongst all microbial metabolisms, sulfur transformations had the highest MW-score, a measure of metabolic connectivity in microbial communities. Additionally, plume microbial populations have low diversity, short migration history, and gene-specific sweep patterns after migrating from background seawater. Selected functions include nutrient uptake, aerobic oxidation, sulfur oxidation for higher energy yields, and stress responses for adaptation. Our findings provide the ecological and evolutionary bases of change in sulfur-driven microbial communities and their population genetics in adaptation to changing geochemical gradients in the oceans.