Scott A. Boorman

Group selection on the boundary of a stable population

A model of group selection is constructed for the case of differential extinction acting on small boundary populations of a large, fixed population. Consideration is restricted to extinction operators acting at or near to carrying capacity. Under the assumption that the extinction rate is large relative to individual genetic parameters affecting gene frequencies in boundary populations, we discuss the conditions under which differential extinction is most likely to produce a significant effect. In particular, a condition for bimodality in the distribution of gene frequencies in boundary populations (population polymorphism) is that there be some critical allele frequency at which the extinction rate jumps from high to low. An extinction operator linear in allele frequency produces no qualitative effect of this kind. In consequence, we are able to make precise rather limited circumstances under which group selection is likely to have a significant effect.

Constructing blockmodels: How and why

Abstract Blockmodel approaches to network analysis as developed by Harrison White are shown to fall in a broader class of established data analysis methods based on matrix permutations (e.g., clique detection, seriation, permutation algorithms for sparse matrices). Blockmodels are seen as an important generalization of these earlier methods since they permit the data to characterize their own structure, instead of seeking to manifest some preconceived structure which is imposed by the investigator (e.g., cliques, hierarchies, or structural balance). General algorithms for the inductive construction of blockmodels thus occupy a central position in the development of the area. We discuss theoretical and practical aspects of the blockmodel search procedure which has been most widely used (CONCOR algorithm). It is proposed that the distinctive and advantageous feature of CONCOR is that it solves what is initially presented as a combinatorial problem (permutations of matrices to reveal zeroblocks) by representing the problem as a continuous one (analysis of correlation matrices). When this representation strategy receives further development, it is predicted that the fairly crude empirical approach of CONCOR will be supplanted by more powerful procedures within this same class.

Multidimensional scaling of measures of distance between partitions

Abstract The techniques of multidimensional scaling were used to study the numerical behavior of twelve measures of distance between partitions, as applied to partition lattices of four different sizes. The results offer additional support for a system of classifying partition metrics, as proposed by Boorman (1970) , and Boorman and Arabie (1972) . While the scaling solutions illuminated differences between the measures, at the same time the particular data with which the measures were concerned offered a basis both for counterexamples to some common assumptions about multidimensional scaling and for some conjectures as to the nature of scaling solutions. The implications of the latter findings for selected examples from the literature are considered. In addition, the methods of partition data analysis discussed here are applied to an example using sociobiological data. Finally, an argument is made against undue emphasis upon interpreting dimensions in nonmetric scaling solutions.

Metrics on spaces of finite trees

Abstract With the increasing popularity of hierarchical clustering methods in behavioral science, there is a need for ways of quantitatively comparing different tree structures on the same set of items. We employ lattice-theoretic methods to construct a variety of metrics on spaces of trees and to analyze their properties. Certain of these metrics are applied to data from Fillenbaum and Rapoport (1971) on the semantic structure of common English kin terms. This application shows that tree metrics can be used to select a componential analysis which is maximally consistent with an empirically derived set of trees.

Blockmodeling complex statutes : Mapping techniques based on combinatorial optimization for analyzing economic legislation and its stress point over time

Blockmodeling, a combinatorial technique for relational data analysis, is applied to studying texts of complex economic legislation. By making this area a subject for mathematical modeling, using methods related to combinatories, logic, and discrete optimization, we describe a new type of frontier between law and economics.

The network matching principle: A model of efficient resource allocation by informal social networks in non-profit and other non-market social structures☆

Abstract A new mathematical model related to the theory of order statistics is introduced to show how informal social networks promote efficient (second-best) resource use in non-market settings. Such networks may play extremely important — if frequently little recognized — roles in ‘matching’ resource packages to their best users in complex social structures subject to many legal and administrative constraints.