Martin G. Everett

Models of core/periphery structures

A common but informal notion in social network analysis and other fields is the concept of a core/periphery structure. The intuitive conception entails a dense, cohesive core and a sparse, unconnected periphery. This paper seeks to formalize the intuitive notion of a core/periphery structure and suggests algorithms for detecting this structure, along with statistical tests for testing a priori hypotheses. Different models are presented for different kinds of graphs (directed and undirected, valued and nonvalued). In addition, the close relation of the continuous models developed to certain centrality measures is discussed.

A Graph-theoretic perspective on centrality

The concept of centrality is often invoked in social network analysis, and diverse indices have been proposed to measure it. This paper develops a unified framework for the measurement of centrality. All measures of centrality assess a nodes involvement in the walk structure of a network. Measures vary along four key dimensions: type of nodal involvement assessed, type of walk considered, property of walk assessed, and choice of summary measure. If we cross-classify measures by type of nodal involvement (radial versus medial) and property of walk assessed (volume versus length), we obtain a four-fold polychotomization with one cell empty which mirrors Freemans 1979 categorization. At a more substantive level, measures of centrality summarize a nodes involvement in or contribution to the cohesiveness of the network. Radial measures in particular are reductions of pair-wise proximities/cohesion to attributes of nodes or actors. The usefulness and interpretability of radial measures depend on the fit of the cohesion matrix to the one-dimensional model. In network terms, a network that is fit by a one-dimensional model has a core-periphery structure in which all nodes revolve more or less closely around a single core. This in turn implies that the network does not contain distinct cohesive subgroups. Thus, centrality is shown to be intimately connected with the cohesive subgroup structure of a network.

Network analysis of 2-mode data

Network analysis is distinguished from traditional social science by the dyadic nature of the standard data set. Whereas in traditional social science we study monadic attributes of individuals, in network analysis we study dyadic attributes of pairs of individuals. These dyadic attributes (e.g. social relations) may be represented in matrix form by a square 1-mode matrix. In contrast, the data in traditional social science are represented as 2-mode matrices. However, network analysis is not completely divorced from traditional social science, and often has occasion to collect and analyze 2-mode matrices. Furthermore, some of the methods developed in network analysis have uses in analysing non-network data. This paper presents and discusses ways of applying and interpreting traditional network analytic techniques to 2-mode data, as well as developing new techniques. Three areas are covered in detail: displaying 2-mode data as networks, detecting clusters and measuring centrality.

Ego network betweenness

In this paper, we look at the betweenness centrality of ego in an ego network. We discuss the issue of normalization and develop an efficient and simple algorithm for calculating the betweenness score. We then examine the relationship between the ego betweenness and the betweenness of the actor in the whole network. Whereas, we can show that there is no theoretical link between the two we undertake a simulation study, which indicates that the local ego betweenness is highly correlated with the betweenness of the actor in the complete network.

The centrality of groups and classes

This paper extends the standard network centrality measures of degree, closeness and betweenness to apply to groups and classes as well as individuals. The group centrality measures will enable researchers to answer such questions as ‘how central is the engineering department in the informal influence network of this company?’ or ‘among middle managers in a given organization, which are more central, the men or the women?’ With these measures we can also solve the inverse problem: given the network of ties among organization members, how can we form a team that is maximally central? The measures are illustrated using two classic network data sets. We also formalize a measure of group centrality efficiency, which indicates the extent to which a groups centrality is principally due to a small subset of its members.

Notions of position in social network analysis

The notion of position is fundamental in structural theory. However, at least two profoundly different conceptions of position exist. The two basic types of position have radically different characteristics, making them appropriate for different theoretical applications. We present examples in which scholars have operationalized one type of position but drawn conclusions as if the other type had been used. We compare the two notions of position in terms of their applicability in several research areas, including power in exchange networks, role theory, world-system theory, and social homogeneity.

Parallel Dynamic Graph Partitioning for Adaptive Unstructured Meshes

A parallel method for the dynamic partitioning of unstructured meshes is described. The method introduces a new iterative optimization technique known as relative gain optimization which both balances the workload and attempts to minimize the interprocessor communications overhead. Experiments on a series of adaptively refined meshes indicate that the algorithm provides partitions of an equivalent or higher quality to static partitioners (which do not reuse the existing partition) and much more rapidly. Perhaps more importantly, the algorithm results in only a small fraction of the amount of data migration compared to the static partitioners.

Regular equivalence: General theory

The theory of regular equivalence has advanced over the last 15 years on a number of different fronts. Notation and terminology have developed often making it difficult to obtain a coherent view of the area as a whole. This paper attempts to provide a framework in which to develop and explore the general mathematical theory of regular equivalence and to place a number of the more important results into that framework.

LS sets, lambda sets and other cohesive subsets

Abstract Seidman (1983a) has suggested that the engineering concept of LS sets provides a good formalization of the intuitive network notion of a cohesive subset. Some desirable features that LS sets exhibit are that they are difficult to disconnect by removing edges, they are relatively dense within and isolated without, they have limited diameter, and individual members have more direct links to other members than to non-members. Unfortunately, this plethora of features means that LS sets occur only rarely in real data. It also means that they do not make good independent variables for structural analyses in which greater-than-expected in-group homogeneity is hypothesized with respect to some substantive dependent variable, because it is unclear which aspect of the LS set was responsible for the observed homogeneity. We discuss a variety of generalizations and relations of LS sets based on just a few of the properties possessed by LS sets. Some of these simpler models are drawn from the literature while others are introduced in this paper. One of the generalizations we introduce, called a lambda set, is based on the property that members of the set have greater edge connectivity with other members than with non-members. This property is shared by LS sets. Edge connectivity satisfies the axioms of an ultrametric similarity measure, and so LS sets and lambda sets are shown to correspond to a particular hierarchical clustering of the nodes in a network. Lambda sets are straightforward to compute, and we have made use of this fact to introduce a new algorithm for computing LS sets which runs an order of magnitude faster than the previous alternative.

The class of all regular equivalences: Algebraic structure and computation

In this paper, we explore the structure of the set of all regular equivalences (White and Reitz 1983) proving that it forms a lattice, and suggest a general approach to computing certain elements of the lattice. The resulting algorithm represents a useful complement to the White and Reitz algorithm, which can only find the maximal regular equivalence of a graph. Using this algorithm, it is possible to compute several well-known equivalences, such as structural equivalence (Lorrain and White 1971), automorphic equivalence (Everett and Borgatti 1988) and Winship-Pattison equivalence (Winship and Mandel 1983). In addition, any number of other useful equivalences may be generated, providing suitable mathematical descriptions of them are available.