Phillip Bonacich

Power and Centrality: A Family of Measures

Although network centrality is generally assumed to produce power, recent research shows that this is not the case in exchange networks. This paper proposes a generalization of the concept of centrality that accounts for both the usual positive relationship between power and centrality and Cook et al.s recent exceptional results.

Factoring and weighting approaches to status scores and clique identification

(1972). Factoring and weighting approaches to status scores and clique identification. The Journal of Mathematical Sociology: Vol. 2, No. 1, pp. 113-120.

Eigenvector-like measures of centrality for asymmetric relations

Abstract Eigenvectors of adjacency matrices are useful as measures of centrality or of status. However, they are misapplied to asymmetric networks in which some positions are unchosen. For these networks, an alternative measure of centrality is suggested that equals an eigenvector when eigenvectors can be used and provides meaningfully comparable results when they cannot.

Some unique properties of eigenvector centrality

Abstract Eigenvectors, and the related centrality measure Bonacichs c(β), have advantages over graph-theoretic measures like degree, betweenness, and closeness centrality: they can be used in signed and valued graphs and the beta parameter in c(β) permits the calculation of power measures for a wider variety of types of exchange. Degree, betweenness, and closeness centralities are defined only for classically simple graphs—those with strictly binary relations between vertices. Looking only at these classical graphs, where eigenvectors and graph–theoretic measures are competitors, eigenvector centrality is designed to be distinctively different from mere degree centrality when there are some high degree positions connected to many low degree others or some low degree positions are connected to a few high degree others. Therefore, it will not be distinctively different from degree when positions are all equal in degree (regular graphs) or in core-periphery structures in which high degree positions tend to be connected to each other.

TECHNIQUE FOR ANALYZING OVERLAPPING MEMBERSHIPS

Sociologists study the structure or pattern of relationships among individuals and among groups. The sharing of common members is an important relationship, as is the pattern of overlapping members. I will first discuss few instances of sociological concern with the pattern of overlapping memberships in order to clarify the problems that this chapter will solve. The pattern of interlocking directorates among business organizations can give information about the power centralization in a society. Lieberson (1971) suggests that an analysis of the pattern of interlocking directorates among the largest business organizations sheds light on whether the power-elite view or the pluralist view of American society is the more accurate. Lieberson confines himself to a few selected facts (for instance, that the boards of the seven largest New York City banks in 1965 included officials from 51 of the largest 500 industrial companies). Although suggesting the value of a thorough analysis of interlocking directorates, Lieberson does not attempt to provide it.

Cooperation and Group Size in the N-Person Prisoners' Dilemma

This study explores the relationship between group size and member cooperation when individual and group interests conflict. Existing theories incorrectly imply that larger groups should experience more difficulty in providing “public goods” (goods whose benefits are not confined to the purchaser). A theoretically based typology that identifies nine systematic patterns associated with changes in group size and parameters of individual and group payoff structure is presented. The typology describes nine different ways in which group size can affect cooperation in the provision of public goods. In a computer-administered experiment using three of these nine types and group sizes of three, six, and nine, 90 college students were given sequences of opportunities to choose between “cooperation” (which benefited everyone) and “noncooperation” (which benefited the chooser but harmed everyone else). For one of these types, a counter-intuitive but theoretically predicted positive relationship between group size and cooperation was found. Two explanations for a theoretically unexpected negative relation between group size and cooperation in the other conditions are developed, one based on Steiners concept of a conjunctive task, the other based on the decreasing information value of responses in larger groups.

The core as a solution to exclusionary networks

Abstract The core is a game theory solution concept that is applicable to exclusionary (negatively connected) exchange networks. The core makes only the minimal assumptions of individual and coalitional rationality. The core describes who should exchange with whom and what the terms of the exchanges should be. Unlike other algorithms that have been applied to exclusionary exchange networks, the core applies to a wider variety of problems than any single competing procedure previously proposed. It can easily be extended to handle a variety of situations: more than one exchange per round; relationships having unequal values; positions having sources of reward other than exchange; and situations where more than one exchange is required. Moreover, networks in which the core is empty should experience particular difficulty in reaching agreement about exchanges.

Simultaneous group and individual centralities

Abstract A completely symmetric pair of measures of individual and group centrality is described. The centralities of groups are a function of the centralities of their members. The centralities of individuals are a function of the centralities of the groups to which they belong. It is shown that the standard approach to controlling for variations in group size is inadequate. It produces measures of centrality that are roughly proportional to the square roots of group sizes. A better way to control for variations in group size or in the numbers of individual memberships is presented. Comparisons are made with correspondence analysis.

When Rationality Fails Unstable Exchange Networks with Empty Cores

Power differences within networks can arise when some positions have many alternative transaction partners who themselves have few alternatives. Recent social psychological models developed to predict power in these situations have assumed that stable transaction patterns will emerge in these networks. These models depend on the assumption that stable exchange patterns will develop. Using the game theoretic concept of the core, we show that in some networks, stable patterns should not emerge. Who transacts with whom and terms of the transaction should never reach equilibrium. Three types of instability are distinguished in this article: unreliability, insecurity, and variety. These types of instability are affected by the existence and size of the core. Experiments and simulations comparing networks with and without core solutions and with different-sized cores show that the predicted differences exist.

Calculating status with negative relations

Abstract A new measure of status or network centrality that takes into account both positive and negative relationships is suggested. This measure is based on the eigenvector measure of centrality, a standard measure in network research. Its use is illustrated with data from Sampsons well-known study of monks in a monastery.