This paper is concerned with a discrete Markov process related to an underlying weighted graph. With this graph we associate an urn containing p types of balls which correspond to the p vertices of the graph, and such that the number of balls of each type is proportional to the weight of the related vertex. A drawing scheme from the urn is defined which leads to Markovian non-homogeneous transition probabilities. Many kinds of weighted graphs are found to have a strong convergence property; this is that the number of balls drawn from the urn at the end of any single draw converges in probability to the total number of balls in the urn, as this tends to infinity.