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Moving in temporal graphs with very sparse random availability of edges

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.