Water transport on infinite graphs

TL;DR

This paper investigates water transport on infinite graphs modeled as interconnected barrels, analyzing how water levels can be maximized without pumps, revealing a unique behavior on certain graph structures.

Contribution

It characterizes the distribution of maximum achievable water levels on infinite graphs, identifying the two-sided infinite path as the unique exception.

Findings

Supremum of water levels concentrates on a single value for most infinite graphs.

The two-sided infinite path exhibits a different, non-degenerate distribution.

The model relates to opinion formation processes like the Deffuant model.

Abstract

If the nodes of a graph are considered to be identical barrels - featuring different water levels - and the edges to be (locked) water-filled pipes in between the barrels, consider the optimization problem of how much the water level in a fixed barrel can be raised with no pumps available, i.e. by opening and closing the locks in an elaborate succession. This model is related to an opinion formation process, the so-called Deffuant model. We consider i.i.d. random initial water levels and ask whether the supremum of achievable levels at a given node has a degenerate distribution, i.e. concentrates on a single value. This turns out to be the case for all infinite connected quasi-transitive graphs with exactly one exception: the two-sided infinite path.