A Distributed Clustering Process

The points of a graph G will form clusters as a result of a flow process. Initially, points i of G own resources x(i) 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 x(i), 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 x(i). 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.