An explicit definition of a (k, r)-cluster is proposed. Each (k, r)-cluster has the property that each of its elements is within a distance r of at least k other elements of the same cluster and the entire set can be linked by a chain of links each less than or equal to r. Some exact distributional results are derived under a nonmetric hypothesis for the case k = 1. An example is given to illustrate the use of probability theory in identifying significant clustering structures in terms of compactness and isolation.