In this paper we study two natural models of \textit{random temporal} graphs. In the first, the \textit{continuous} model, each edge $e$ is assigned $l_e$ labels, each drawn uniformly at random from $(0,1]$, where the numbers $l_e$ are independent random variables following the same discrete probability distribution. In the second, the \textit{discrete} model, the $l_e$ labels of each edge $e$ are chosen uniformly at random from a set $\{1,2,\ldots,T\}$. In both models we study the existence of \textit{$δ$-temporal motifs}. Here a $δ$-temporal motif consists of a pair $(H,P)$, where $H$ is a fixed static graph and $P$ is a partial order over its edges. A temporal graph $\mathcal{G}=(G,λ)$ contains $(H,P)$ as a $δ$-temporal motif if $\mathcal{G}$ has a simple temporal subgraph on the edges of $H$ whose time labels are ordered according to $P$, and whose life duration is at most $δ$. We prove \textit{sharp existence thresholds} for all $δ$-temporal motifs, and we identify a qualitatively different behavior from the analogous static thresholds in Erdos-Renyi random graphs. Applying the same techniques, we then characterize the growth of the largest $δ$-temporal clique in the continuous variant of our random temporal graphs model. Finally, we consider the \textit{doubling time} of the reachability ball centered on a small set of vertices of the random temporal graph as a natural proxy for temporal expansion. We prove \textit{sharp upper and lower bounds} for the maximum doubling time in the continuous model.

翻译：本文研究两种自然的\textit{随机时序图}模型。在第一种\textit{连续}模型中，每条边$e$被分配$l_e$个时间标签，每个标签均从$(0,1]$区间均匀随机抽取，其中数量$l_e$为服从相同离散概率分布的独立随机变量。在第二种\textit{离散}模型中，每条边$e$的$l_e$个标签从集合$\{1,2,\ldots,T\}$中均匀随机选取。在这两种模型中，我们研究\textit{$δ$-时序模式}的存在性问题。此处$δ$-时序模式由二元组$(H,P)$构成，其中$H$为固定静态图，$P$为其边集上的偏序关系。当时序图$\mathcal{G}=(G,λ)$存在以$H$的边构成的简单时序子图，且其时间标签按$P$排序、生存时长不超过$δ$时，称$\mathcal{G}$包含$(H,P)$作为$δ$-时序模式。我们证明了所有$δ$-时序模式的\textit{尖锐存在性阈值}，并发现其与Erdos-Renyi随机图中类似静态阈值存在本质不同的行为特征。应用相同技术，我们进一步刻画了随机时序图模型连续变体中最大$δ$-时序团的增长规律。最后，我们将随机时序图中以小规模顶点集为中心的到达球的\textit{倍增时间}视为时序扩张的自然代理指标，证明了连续模型中最大倍增时间的\textit{尖锐上下界}。