Water transport on infinite graphs

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.