边具有非常稀疏随机可用性的时间图中的移动
Moving in temporal graphs with very sparse random availability of edges
摘要：
在这项工作中，我们考虑时间图，即图形，其中的每个边缘被分配一组离散的时间标签从一组整数。边缘的标签指示边缘可用的离散时刻。我们还考虑时间图中的时间路径，即其边缘被分配严格递增的标签序列的路径。此外，我们假设统一的情况下（UNI-CASE），其中图的每一个边缘被分配恰好一个时间标签从一组整数和分配给图的边缘的时间标签是随机和独立地选择，与选择以下的均匀分布。我们称一致随机时态图为满足唯一情形的图。我们首先推导出均匀随机时间集团中给定长度的时间路径的预期数量。我们定义了两个顶点的时间距离，它是连接这些顶点的时间路径的到达时间，即最后一条边的时间标签，它在连接这些顶点的所有时间路径中具有最小的到达时间。然后，我们提出并研究了两个统计性质的时间图。一个是最大预期时间距离，如该术语所示，它是图中所有预期时间距离的最大值。另一个是时间直径，不严格地说，它是图中最大时间距离的期望。我们推导出均匀随机时间星图的最大期望时间距离以及均匀随机时间团的归一化版本的最大期望时间距离和时间直径的上界，其中最大的时间标签等于顶点的数量。最后，我们提供了一个算法，解决了一个特定类型的两个顶点的时间（多）图的优化问题。
英文摘要：
In this work we consider temporal graphs, i.e. graphs, each edge of which is assigned a set of discrete time-labels drawn from a set of integers. The labels of an edge indicate the discrete moments in time at which the edge is available. We also consider temporal paths in a temporal graph, i.e. paths whose edges are assigned a strictly increasing sequence of labels. Furthermore, we assume the uniform case (UNI-CASE), in which every edge of a graph is assigned exactly one time label from a set of integers and the time labels assigned to the edges of the graph are chosen randomly and independently, with the selection following the uniform distribution. We call uniform random temporal graphs the graphs that satisfy the UNI-CASE. We begin by deriving the expected number of temporal paths of a given length in the uniform random temporal clique. We define the term temporal distance of two vertices, which is the arrival time, i.e. the time-label of the last edge, of the temporal path that connects those vertices, which has the smallest arrival time amongst all temporal paths that connect those vertices. We then propose and study two statistical properties of temporal graphs. One is the maximum expected temporal distance which is, as the term indicates, the maximum of all expected temporal distances in the graph. The other one is the temporal diameter which, loosely speaking, is the expectation of the maximum temporal distance in the graph. We derive the maximum expected temporal distance of a uniform random temporal star graph as well as an upper bound on both the maximum expected temporal distance and the temporal diameter of the normalized version of the uniform random temporal clique, in which the largest time-label available equals the number of vertices. Finally, we provide an algorithm that solves an optimization problem on a specific type of temporal (multi)graphs of two vertices.
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