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 et al., 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 $t$". This is a symmetric relation between $u$ and $v$, and it can be interpreted that the information can flow in either direction. 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, we introduce the notion of a temporal transitive orientation and we systematically investigate its algorithmic behavior in various situations. 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 $t_2 > t_1$, the orientation is called strictly temporally transitive. 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：在具有离散时间标签的时序网络中，实体与信息仅能沿时间标签非递减（或严格递增）的边序列流动，即沿时序（或严格时序）路径传播。然而，在[Kempe等人，JCSS，2002]提出的时序网络模型中，每条带有时间标签的边仍保持无向性：边$e=\{u,v\}$及其时间标签$t$表示"$u$在时刻$t$与$v$通信"。这是$u$与$v$之间的对称关系，可理解为信息能沿任一方向流动。本文首次尝试探究单条边上的信息流向如何影响其他边上的信息流向。具体而言，我们提出时序传递定向的概念，并系统研究其在多种情境下的算法性质。时序图的定向称为时序传递的，若任意时刻当$u$在时间$t_1$存在指向$v$的有向边，且$v$在时间$t_2\geq t_1$存在指向$w$的有向边时，则$u$也在某一时间$t_3\geq t_2$存在指向$w$的有向边。若仅要求该蕴含关系在$t_2 > t_1$时成立，则称该定向为严格时序传递的。我们的主要成果是一个概念简洁但技术复杂多项式时间算法，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，我们证明令人惊讶的是，判定$\mathcal{G}$是否可严格传递定向是NP难的。此外，我们引入并研究了时序传递性的若干相关问题，其中特别包括时序传递补全问题，并对此问题给出了算法与难解性结果。