In this situation a plot of N cl,eff versus clustering level tends to exhibit a broad, nearly flat region whose ordinate is roughly indicative of the number of large clusters present. Thus, one can represent a given clustering as a set of dividers or partitions placed in the generically ordered list, separating the clusters. At the first clustering level, where all items are in separate clusters, there is a divider after each item on the list. At each clustering level, a single divider is removed. But between these bounding levels some reorderings break up clusters, and are forbidden.
S re goes through a minimum as clustering level or threshold increases.
It can be shown that there is a tendency for S re to be minimal when one has a large number of small clusters, and in fact we have observed this in chemical examples. In any event, N cl,eff is guaranteed to decrease monotonically as cluster level increases. Inc. All rights reserved.