We consider random temporal graphs, a version of the classical Erd\H{o}s--R\'enyi random graph G(n,p) where additionally, each edge has a distinct random time stamp, and connectivity is constrained to sequences of edges with increasing time stamps. We study the asymptotics for the distances in such graphs, mostly in the regime of interest where np is of order log n. We establish the first order asymptotics for the lengths of increasing paths: the lengths of the shortest and longest paths between typical vertices, the maxima of these lengths from a given vertex, as well as the maxima between any two vertices; this covers the (temporal) diameter.

翻译：我们考虑随机时序图，即经典厄尔多斯-雷尼随机图 G(n,p) 的一种变体，其中每条边额外带有独立的随机时间戳，且连通性受限于时间戳递增的边序列。我们主要研究此类图中距离的渐近性，重点关注的参数区域为 np 与 log n 同阶。我们建立了递增路径长度的一阶渐近性：典型顶点间最短与最长路径的长度、从给定顶点出发的这些路径长度的最大值，以及任意两顶点间路径长度的最大值——这涵盖了（时序）直径。