The isolation principle of clustering: structural characteristics and implementation.

The isolation principle rests on defining internal and external differentiation for each subset of at least two objects. Subsets with larger external than internal differentiation form isolated groups in the sense that they are internally cohesive and externally isolated. Objects that do not belong to any isolated group are termed solitary. The collection of all isolated groups and solitary objects forms a hierarchical (encaptic) structure. This ubiquitous characteristic of biological organization provides the motivation to identify universally applicable practical methods for the detection of such structure, to distinguish primary types of structure, to quantify their distinctiveness, and to simplify interpretation of structural aspects. A method implementing the isolation principle (by generating all isolated groups and solitary objects) is proven to be specified by single-linkage clustering. Basically, the absence of structure can be stated if no isolated groups exist, the condition for which is provided. Structures that allow for classifications in the sense of complete partitioning into disjoint isolated groups are characterized, and measures of distinctiveness of classification are developed. Among other primary types of structure, chaining (complete nesting) and ties (isolated groups without internal structure) are considered in more detail. Some biological examples for the interpretation of structure resulting from application of the isolation principle are outlined.