Coffman, Courtois, Gilbert and Piret (1991) have introduced a flow process in graphs, where each vertex gets an initial random resource, and where at each time vertices with large resources tend to attract resources from neighbours. The initial resources are assumed to be i.i.d., with a continuous distribution.

We are mainly interested in the following question: does, with probability 1, the resource of each vertex change only finitely many times?

Coffman et al. concentrate mainly on the case where the graph corresponds with the integer points on the line, in which case it is easily seen that the answer to the above question is positive. For higher-dimensional lattices they make general remarks which suggest that the answer to the above question is still positive. However, no proof seems to be known.

We restrict to the case of the square lattice, and, by a percolation approach, we reduce the question above to the question whether a certain quantity, which can be obtained from a finite computation, is sufficiently small. This computation is, however, still too long to be executed in an acceptable time. We therefore apply Monte Carlo simulation for this finite problem, which gives overwhelming evidence that, for the square lattice, the answer to the main question is positive.

The points of a graph G will form clusters as a result of a flow process. Initially, points i of G own resources xi which are i.i.d. random real numbers. Afterwards, resources flow between points, but always from a point to a neighbor that has accumulated a larger total resource. Thus points with small resource tend to lose it and points with large resource tend to gain. Eventually the flow stops with only two kinds of points, nulls with no resource left and absorbers with such large resource that no neighbor can take it. The final resource at an absorber is a sum of certain initial resources xi, and the corresponding points i form one cluster. Analytical results are obtainable when G is the chain of integer points on the line. Probability distributions are derived for the distance between consecutive absorbers and the size of a cluster. Indeed these distributions do not involve the given distribution for the xi. The Laplace transform of the distribution of final resources at absorbers is derived but the distribution itself is obtained by a simulation. For general graphs G only partial results are obtained.