In a temporal network with discrete time-labels on its edges, entities and information can only ``flow'' along sequences of edges whose time-labels are non-decreasing (resp. increasing), i.e. along temporal (resp. strict temporal) paths. Nevertheless, in the model for temporal networks of [Kempe, Kleinberg, Kumar, JCSS, 2002], the individual time-labeled edges remain undirected: an edge $e=\{u,v\}$ with time-label $t$ specifies that ``$u$ communicates with $v$ at time $t$''. In this paper we make a first attempt to understand how the direction of information flow on one edge can impact the direction of information flow on other edges. More specifically, naturally extending the classical notion of a transitive orientation in static graphs, we introduce the fundamental notion of a temporal transitive orientation and we systematically investigate its algorithmic behavior. An orientation of a temporal graph is called temporally transitive if, whenever $u$ has a directed edge towards $v$ with time-label $t_1$ and $v$ has a directed edge towards $w$ with time-label $t_2\geq t_1$, then $u$ also has a directed edge towards $w$ with some time-label $t_3\geq t_2$. If we just demand that this implication holds whenever $t_2 > t_1$, we call the orientation strictly temporally transitive, as it is based on the strict directed temporal path from $u$ to $w$. Our main result is a conceptually simple, yet technically quite involved, polynomial-time algorithm for recognizing whether a given temporal graph $\mathcal{G}$ is transitively orientable. In wide contrast we prove that, surprisingly, it is NP-hard to recognize whether $\mathcal{G}$ is strictly transitively orientable. Additionally we introduce and investigate further related problems to temporal transitivity, notably among them the temporal transitive completion problem, for which we prove both algorithmic and hardness results.

翻译：在边带有离散时间标签的时序网络中，实体与信息只能沿着时间标签非递减（或递增）的边序列“流动”，即沿时序（或严格时序）路径传播。然而，在[Kempe, Kleinberg, Kumar, JCSS, 2002]提出的时序网络模型中，单个带时间标签的边仍保持无向性：一条时间标签为$t$的边$e=\{u,v\}$仅表示“$u$在时刻$t$与$v$通信”。本文首次尝试探究一条边上信息流动的方向如何影响其他边上信息流动的方向。具体而言，我们通过自然扩展静态图中传递定向的经典概念，引入时序传递定向这一基本概念，并系统研究其算法性质。当时序图的定向满足以下条件时称为时序传递的：若$u$存在一条指向$v$且时间标签为$t_1$的有向边，且$v$存在一条指向$w$且时间标签为$t_2\geq t_1$的有向边，则$u$也必须存在一条指向$w$且时间标签为$t_3\geq t_2$的有向边。若仅要求当$t_2 > t_1$时该蕴含关系成立，则称之为严格时序传递定向，因其基于从$u$到$w$的严格有向时序路径。我们的主要成果是一个概念简洁但技术层面相当复杂的多项式时间算法，用于判定给定时序图$\mathcal{G}$是否可传递定向。与之形成鲜明对比的是，我们证明了判定$\mathcal{G}$是否可严格传递定向是NP困难的结论。此外，我们引入并研究了与时序传递性相关的其他问题，其中值得注意的是时序传递补全问题，对此我们同时给出了算法结果与复杂性证明。