On the motion of substance in a channel and growth of random networks

We discuss the connection between the theory of motion of substance in a channel and the theory of growth of large random networks. The channel contains nodes connected by edges. The substance is located in the nodes of the channel. It can move between the nodes of the channel through the edges which connect the nodes. We study probability distributions connected to distribution of substance for selected cases of stationary motion of substance in a channel. The corresponding model equations contain as specific cases the model equations for the degree distributions of several classes of growing random networks. We consider two scenarios for stationary motion of substance in the channel. Scenario 1 is connected to stationary motion of substance in direction to nodes labeled by larger numbers. Scenario 2 is connected to stationary motion of substance in the opposite direction. Probability distributions are obtained for different cases connected to the two scenarios. The results for the Scenario 1 are connected to the theory of degree distributions of nodes of a random network growing in presence of preferential attachment (as specific case we obtain also the degree distribution for the Barabasi-Albert case). Corresponding results for Scenario 2 can be connected to the theory of growing random networks too. The kind of attachment for these results can be classified as negative preferential attachment or preferential attachment with preference of attachment to nodes of smaller degree. (C) 2021 Elsevier B.V. All rights reserved.