digraph methods and the Perron-Frobenius theory for stochastic matrices, where each $n$-step entry is a Markov chain. We first present two linear models: (1) a network consisting of four sites, each one connected to all the others by two-way arteries. The total number of circulating vehicles is assumed to be constant in time. (2) Thereafter, we extend that network introducing an input of new vehicles and an output of old ones; see the digraph in Fig. \[fig03\]. To each digraph one associates a square matrix $\mathbb{A}$ whose entries contain the information obtained empirically, namely, the number of vehicles established by counts. To forecast the distribution of the vehicles at future moments in the network we normalize the rows of $\mathbb{A}$ to 1, thus getting a matrix $\mathbb{M}$. Its entries are the fractions of vehicles in sites and roads. In this way, the predictions will be probabilistic and we assume that the daily change of the distribution obeys an evolution law based on an $n$-step process – $n$ standing for the discretized time. This approach constitutes an adaptation of a model proposed in Ref. [@harary1966] for human migrations. We present the theory, work out illustrative numerical examples and analyze the results. 