随机时序图中时序模体的尖锐阈值与加倍时间
Sharp Thresholds for Temporal Motifs and Doubling Time in Random Temporal Graphs
AI总结：
研究两种随机时序图模型中δ-时序模体的存在性，证明了所有δ-时序模体的尖锐阈值，并刻画了最大δ-时序团的增长以及加倍时间的上下界。
摘要：
本文研究了两种随机时序图的自然模型。在第一种连续模型中，每条边$e$被赋予$l_e$个标签，每个标签均匀随机地从$(0,1]$中抽取，其中$l_e$是服从相同离散概率分布的独立随机变量。在第二种离散模型中，每条边$e$的$l_e$个标签均匀随机地从集合$\{1,2,\ldots,T\}$中选取。在两种模型中，我们研究$\delta$-时序模体的存在性。这里一个$\delta$-时序模体由一对$(H,P)$组成，其中$H$是一个固定的静态图，$P$是其边上的一个偏序。如果时序图$\mathcal{G}=(G,\lambda)$在$H$的边上有一个简单时序子图，其时间标签按$P$排序，且生命期不超过$\delta$，则称$\mathcal{G}$包含$(H,P)$作为$\delta$-时序模体。我们证明了所有$\delta$-时序模体的尖锐存在阈值，并发现其与Erdos-Renyi随机图中类似静态阈值有本质不同的行为。应用相同技术，我们随后刻画了随机时序图连续变体中最大$\delta$-时序团的增长。最后，我们考虑以随机时序图中以小顶点集为中心的可达球加倍时间作为时序扩张的自然代理，并证明了连续模型中最大加倍时间的尖锐上下界。
英文摘要：
In this paper we study two natural models of random temporal graphs. In the first, the 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 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 $δ$-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 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 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 sharp upper and lower bounds for the maximum doubling time in the continuous model.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
