Human migration and the motion of substance in a channel of a network

TL;DR

This paper models the movement of a substance through network channels with leakage and attraction variations, analyzing both stationary and non-stationary regimes, and applies the theory to human migration distribution.

Contribution

It introduces a generalized distribution model for substance flow in network channels with leakage and attraction, extending existing distributions like the Waring distribution.

Findings

Distribution follows a generalized Waring distribution in stationary regime.

Non-stationary regime shows exponential growth or decay of substance.

Asymptotic distribution remains stationary despite non-stationary dynamics.

Abstract

We study the motion of a substance in a channel of a network that consists of chain of nodes of a network (the nodes can be considered as boxes) and edges that connect the nodes and form the way for motion of the substance. The nodes of the channel can have different "leakage", i.e., some amount of the substance can leave the channel at a node and the rate of leaving may be different for the different nodes of the channel. In addition the nodes close to the end of the channel for some (construction or other) reason may be more "attractive" for the substance in comparison to the nodes around the entry node of the channel. We discuss channels containing infinite or finite number of nodes and obtain the distribution of the substance along the nodes. Two regimes of functioning of the channels are studied: stationary regime and non-stationary regime. The distribution of the substance along the nodes of the channel for the case of stationary regime is a generalization of the Waring distribution (for channel with infinite number of nodes) or generalization of the truncated Waring distribution (for channel with finite number of nodes). In the non-stationary regime of functioning of the channel one observes an exponential increase or exponential decrease of the amount of substance in the nodes. Despite this the asymptotic distribution of the substance among the nodes of the channel in this regime is stationary. The developed theory is applied for a study of the distrribution of migrants in countries that form migration channels.