Stephen B. Seidman

Network structure and minimum degree

Abstract Social network researchers have long sought measures of network cohesion, Density has often been used for this purpose, despite its generally admitted deficiencies. An approach to network cohesion is proposed that is based on minimum degree and which produces a sequence of subgraphs of gradually increasing cohesion. The approach also associates with any network measures of local density which promise to be useful both in characterizing network structures and in comparing networks.

A graph‐theoretic generalization of the clique concept*

For at least twenty‐five years, the concept of the clique has had a prominent place in sociometric and other kinds of sociological research. Recently, with the advent of large, fast computers and with the growth of interest in graph‐theoretic social network studies, research on the definition and investigation of the graph theoretic properties of clique‐like structures has grown. In the present paper, several of these formulations are examined, and their mathematical properties analyzed. A family of new clique‐like structures is proposed which captures an aspect of cliques which is seldom treated in the existing literature. The new structures, when used to complement existing concepts, provide a new means of tapping several important properties of social networks.

Brand Congruence in Interpersonal Relations: A Social Network Analysis

Previous studies dealing with the notion of brand congruence suffer from questionable methods of group determination, suspect demonstrations of brand congruence effects, and inadequate attention paid to types of social relation. To overcome these shortcomings, the present study uses graph-theoretic social network techniques to examine interpersonal relationships and brand choice behavior in natural environments. The brand choices of individuals in a social relationship were compared to those of unrelated individuals across various products, types of social relation, and types of basic sociological structure (dyad, clique, and 2-plex). While significant brand congruence effects were obtained, they were clustered in a few products mediated by types of social relation. Conspicuousness of the product, as traditionally defined, was found to be insufficient to account for these findings.

The hull number of a graph

Abstract A set of points S of a graph is convex if any geodesic joining two points of S lies entirely within S . The convex hull of a set T of points is the smallest convex set that contains T . The hull number ( h ) of a graph is the cardinality of the smallest set of points whose convex hull is the entire graph. Characterisations are given for graphs with particular values of h , and upper and lower bounds for h are derived.

A formal model for module interconnection languages

A model is proposed that formalizes the design of hierarchical module structures. The model is specified by a collection of Z schema type definitions that is invariant across all applications. A particular application is described by specifying the values of generic parameters and adding application-specific declarations and constraints to the schema definitions. As applications, the definitions in the model are used to describe the Conic configuration language and the STILE graphical design and development environment. >

Internal cohesion of ls sets in graphs

Abstract Let G be a finite connected graph. A set of vertices H ⊂ V ( G ) is called a LS set if for every proper subset K ⊂ H , there are more edges linking K to H − K than there are linking K to V ( G )- H . Since “cliques” in social networks have usually been seen informally as sets of individuals more closely tied to each other than to outsiders, LS sets provide a natural realization of the “clique” concept. In this paper, it is shown that LS sets in social networks have cohesive properties that make them even more useful for empirical analyses. In particular, subgraphs induced by LS subsets remain connected even after several edges have been removed. Results bounding the number of edges that can be so removed are used to get an upper bound for the diameter of subgraphs induced by LS subsets.

Structures induced by collections of subsets: a hypergraph approach

Abstract During recent years, much attention has been paid by anthropologists and sociologists to the analysis of social networks. These networks arise from dyadic relationships such as kinship or friendship and they have been studied using techniques derived from graph theory. Although the study of such networks can cast much light on the social structure of a population, many important aspects of this structure cannot be addressed using dyadic relationships alone. For example, group membership gives rise to natural non-dyadic relationships which will be distorted if they are forced into a dyadic mold. The purpose of this paper is to propose an analytical scheme which will permit the study of structure induced by non-dyadic relationships. The concepts used derive from the theory of hypergraphs, and it is shown that these concepts permit a wide variety of structural questions to be posed.

A note on the potential for genuine cross-fertilization between anthropology and mathematics☆

Although there has been much discussion in recent years of the applicability of mathematical methods in the social sciences, little attention has been paid to the interaction of the sociological analysis and the mathematics. In this paper, an example of an extremely fruitful interaction between anthropology and mathematics is described. The interaction has, on the one hand, greatly increased the subtlety and sophistication of the sociological analysis, while on the other hand it has posed interesting mathematical questions.

A formal model for SIMD computation

A formal model for single-instruction multiple-data (SIMD) computation is presented that captures the essential operating features of current SIMD computers, yet allows for extensions and variations of existing architectures. The fundamental components of the model are a host computer, a set of processing elements, a set of control units, a set of input/output controllers, and a set of external devices. Each component sends or receives data or instructions to/from other components are described by six networks, each of which governs the communication between a single pair of components. The networks are represented as functions or collections of functions with formally specified mathematical properties that have natural interpretations in the context of SIMD computation. Using the functional approach, a set of four assumptions for SIMD computers is proposed, and consequences of these assumptions are explored.<<ETX>>

Analysis of competition-based spreading activation in connectionist models

Abstract In this paper we analyse a connectionist model of information processing in which the spread of activity in the network is controlled by the nodes actively competing for available activation. This model meets the needs of various artificial-intelligence tasks and has demonstrated several useful properties, including circumscribed spread of activation, stability of network activation following termination of external influences, and context-sensitive “winner-take-all” phenomena without explicit inhibitory links between nodes representing mutually exclusive concepts. We examine three instances of the competition-based connectionist model. For each instance, we show that the differential equations modelling the changes in the activation level of each node has a solution, and we prove that given any initial activity values of the nodes, certain equilibrium activation levels are reached. In particular, we demonstrate that lateral inhibition, i.e. mutually exclusive activity for nodes in the same layer, is possible without explicitly including links between nodes in the same layer. We believe that our results for these instances of the model give important insights into the behaviour observed in the general model.