On the Shortest Path Problem of Uncertain Random Digraphs

In the field of graph theory, the shortest path problem is one of the most significant problems. However, since varieties of indeterminated factors appear in complex networks, determining of the shortest path from one vertex to another in complex networks may be a lot more complicated than the cases in deterministic networks. To illustrate this problem, the model of uncertain random digraph will be proposed via chance theory, in which some arcs exist with degrees in probability measure and others exist with degrees in uncertain measure. The main focus of this paper is to investigate the main properties of the shortest path in uncetain random digraph. Methods and algorithms are designed to calculate the distribution of shortest path more efficiently. Besides, some numerical examples are presented to show the efficiency of these methods and algorithms.

The critical window in random digraphs

We consider the component structure of the random digraph D(n,p) inside the critical window $p = n^{-1} + \lambda n^{-4/3}$ . We show that the largest component $\mathcal{C}_1$ has size of order $n^{1/3}$ in this range. In particular we give explicit bounds on the tail probabilities of $|\mathcal{C}_1|n^{-1/3}$ .