John P. Boyd

The algebra of group kinship

Group theory and the theory of relations are used to study the kinship of certain kinds of primitive societies. It will be shown that these societies partition their members into classes that are permuted by the relations “class X has fathers (mothers) in class Y” so as to form a regular permutation group. A mathematical characterization of the conditions under which groups become relevant for the study of kinship is given and is related to the theory of structural balance. It is argued that the concept of group extension and its specialization to direct and semidirect products determine the evolutionary sequences and the coding of these kinship systems. These predictions are found to be consistent with certain observed changes, geographical distribution, and habits of usage of the kinship systems under consideration. Generalizations to aspects of behavior outside of kinship are briefly discussed.

Consensus analysis of three‐way social network data

Three‐way social network data occurs when every actor in a social network generates a digraph of the entire network. This paper presents a statistical model based on cultural consensus analysis for aggregating these separate digraphs into a single consensus digraph. In addition, the model allows estimation of separate hit and false alarm rates for each actor that can vary within each actor in different regions of the digraph. Several standard signal detection models are used to interpret the hit and false alarm parameters in terms of knowledge and response bias. A published three‐way data set by Kumbasar, Romney, and Batchelder (American Journal of Sociology, 1994) is analyzed, and the model reveals that both response bias and knowledge decrease with distance from ego.

Relations, residuals, regular interiors, and relative regular equivalence

Given a relation α (a binary sociogram) and an a priori equivalence relation π, both on the same set of individuals, it is interesting to look for the largest equivalence πo that is contained in and is regular with respect to α. The equivalence relation πo is called the regular interior of π with respect to α. The computation of πo involves the left and right residuals, a concept that generalized group inverses to the algebra of relations. A polynomial-time procedure is presented (Theorem 11) and illustrated with examples. In particular, the regular interior gives meet in the lattice of regular equivalences: the regular meet of regular equivalences is the regular interior of their intersection. Finally, the concept of relative regular equivalence is defined and compared with regular equivalence.

Computing continuous core/periphery structures for social relations data with MINRES/SVD

Abstract When diagonal values are missing or excluded, MINRES is a natural continuous model for the core/periphery structure of a symmetric social network matrix. Symmetric models, however, are not so useful when dealing with asymmetric data. Singular value decomposition (SVD) is a natural choice to model asymmetry, but this method also requires the presence of diagonal values. In this paper we offer an alternative, more general, approach to continuous core/periphery structures, the minimum residual singular value decomposition (MINRES/SVD), where each node in the network receives two indices, an “in-coreness” and an “out-coreness.” The algorithm for computing these coreness vectors is a least squares computation similar to, but distinct from the SVD, again because of the missing diagonal values. And in contrast to the standard, symmetric MINRES algorithm, we can more accurately model asymmetric matrices. This allows us to distinguish, for example, countries in the world economy that are more in the exporting core than they are in the importing core. We propose two nested PRE (proportional reduction of error) measures of fit: (1) the PRE from the MINRES vector with respect to the data and (2) the PRE of the product of the two MINRES/SVD vectors. Applying the resulting method to citations between journals and to international trade in clothing, we illustrate insights gained from being able to model asymmetrical flow patterns. Finally, two permutation tests are introduced to test independently for the MINRES and MINRES/SVD results.

Kinship systems and inverse semigroups

Inverse semigroups are introduced as a model for kinship structures. It is shown that semigroups are implied by the traditional anthropological law of uniform descent, and that inverses are suggested by the principle of reciprocity. A separation into a ‘surface’ and a ‘deep’ level of analysis was induced by viewing a kinship system as a partition on an underlying inverse semigroup. Some Omaha‐type kinship systems are decomposed into direct and semidirect products. Additional assumptions are made to deiive a theoretical similarity measure, which is then compared with a judged similarity task on Tenejapan Indians. The results of this preliminary experiment seem to be encouraging.

Ego‐centered and local roles: A graph theoretic approach

Structural equivalence (Lorrain and White, 1971) and automorphic equivalence (Everett, 1985) are generalized to define neighborhood‐ and ego‐centered equivalences. It is shown that local versions of these equivalences can then be formulated quite naturally. In addition to these natural localizations, a generalized procedure capable of localizing any model of role equivalence is presented. From a theoretical point of view, local roles are recommended by the notion that network influences on ego diminish with distance. From a practical point of view, local roles help find structure in graphs where global equivalences find no two actors equivalent.

Measuring Centrality and Power Recursively in the World City Network: A Reply to Neal

In a recent article, Zachary Neal (2011) distinguishes between centrality and power in world city networks and proposes two measures of recursive power and centrality. His effort to clarify oversimplistic interpretations of relational measures of power and position in world city networks is appreciated. However, Neal’s effort to innovate methodologically is based on theoretical reasoning that is dubious when applied to world city networks. And his attempt to develop new measures is flawed since he conflates ‘eigenvector centrality’ with ‘beta centrality’ and then argues that ‘eigenvector-based approaches’ to recursive power and centrality are ill-suited to world city networks. The main problem is that his measures of ‘recursive’ centrality and power are not recursive at all and thus are of very limited utility. It is concluded that established eigenvector centrality measures used in past research (which Neal critiques) provide more useful gauges of power and centrality in world city networks than his new indexes.

The universal semigroup of relations

Abstract Semigroup theory is shown to be a very flexible and appropriate language for the study of social relations. Some of the difficulties with the application of the Lorrain and White (1971) approach to social relations are discussed. It is pointed out that the ‘universal’ semigroup of relations is often more suitable for studying relations than the more commonly used ‘semigroup of relations’. Several standard examples and techniques are presented for the study of semigroups. The concept of structural equivalence is generalized to structural similarity modulo a congruence relation. A Galois connection, between the set of relations on the social relations themselves and the lattice of congruences on the universal semigroup, is applied to the evolution of kinship systems.

Are social equivalences ever regular? Permutation and exact tests

Abstract A regular equivalence on a relation induces matrix blocks that are either 0-blocks or regular-blocks, where a regular-block contains at least one positive entry in each row and column. The authors devise both a permutation test and an exact statistical test that separates these two aspects of regular equivalence, 0-blocks and regular-blocks. To test for the regular-block property, the natural test statistic is the number of rows and columns within each purported regular-block that fail to meet the criteria of having at least one positive entry. This statistic is computed for permutations that fix each regular-block as a whole (alternatively, within each sub-row), except for diagonal blocks, for which the diagonal entries are individually fixed. The exact test is derived by assuming that the number of zeros in each block is fixed and that each permutation of zeros is uniformly distributed. This implies that the probability of finding, say, k zeros in a given set of rows and columns follows the hypergeometric distribution, known in physics as the Fermi–Dirac statistics. These results from the separate blocks are combined by convolution to give the distribution of k zero vectors in the matrix as a whole. These tests were applied to data sets from Sampson’s Monastery, Wasserman and Faust’s Countries Trade Networks, Krackhardt’s High-Tech Managers, and B.J. Cole’s Dominance Hierarchies in Leptothorax ants. In all four cases, the 0-blocks were very significant, having only a tiny fraction of permutations with fewer errors than was found in the data. With the regular-blocks, however, there was no significant relation in the Countries data and a significant overall tendency in the other three data sets toward having more departures from regular 1-blocks in the data than in the permuted matrices.

ITERATED ROLES: MATHEMATICS AND APPLICATION

Abstract Recent work by Borgatti and Everett (1989) has shown that the collection of regular equivalences described by White and Reitz (1983) forms a lattice. In this paper, we present a procedure called iterated roles for tracing systematic paths through the lattice. At the heart of iterated roles is the proof that the regular equivalence of a regular equivalence is itself regular. The procedure enables us to find several otherwise unknown regular equivalences, including an extension of automorphic equivalence (Everett 1985) that is not sensitive to degree. A key benefit of iterated roles is the generation of sequences of hierarchically nested equivalences. This capability suggests an approach to role structure analysis in which one examines not just one blocking of actors but a series of increasingly broad simplifications of the data. Consequently, we are able to (a) choose the level of simplification that proves most illuminating, and (b) view both to broad structural outlines of the data and the finer details simultaneously.