Graph Theoretic Measures of Power

Power is a topic of long-standing interest in sociology, and its nature, causes, and consequences are approached through a variety of theoretical perspectives. A particularly fruitful definition, attributable to Max Weber, treats power as the ability of individuals or collectivities to exercise their will over another, even against the will of the other.

Graph theoretic approaches to explaining and measuring power have attracted significant interest due to their conceptual precision and predictive accuracy in experimental tests. Graph theory is a branch of mathematics that focuses on properties of graphs, which it defines in terms of sets of points and lines, or vertices and edges, respectively. This makes it very useful for the description and analysis of social networks that typically represent social actors (individual or collective) as points or nodes, and their relationships (directed or nondirected) as lines or ties.

In general, graph theoretic measures of power determine the relative levels of power for each position in a network. They do so by taking into account certain attributes of the network structure and aspects of a position's location in that structure. This is in contrast to more individualistic approaches to power that focus on attributes of actors such as personal charisma or negotiation skills.

Graph theory provides methods for describing and deriving a wide array of network structural properties. Some of these properties are quite simple and intuitive. For example, the geodesic between two points is simply the minimum number of ties that separates them; the indegree of a network node is the number of its ties from other nodes; and the density of a network is the ratio of the number of existing ties to the number of possible ties. At the other extreme, some graph properties are much more complicated and esoteric. However, theories of power in social networks usually employ relatively simple graph theoretic measures.

There are two important precursors to graph theoretic approaches to power: measures of prestige and measures of centrality. Prestige measures usually entail directed ties, for example A → B. Prestige may be assumed to accrue to one who is chosen by many others, such as to a person who is a key source of information. More sophisticated measures account for an actor's prestige in terms of the level of prestige of the actors who select him or her.

In contrast to prestige measures, centrality measures are designed to capture the degree to which a given actor is well connected to the rest of the network. These measures typically employ nondirected or mutual ties. Similar to the case for prestige, more sophisticated measures may be designed to take into account the centrality of not only the focal actor, but also the centrality of actors with whom the focal actor has ties.

Because power is not identical to centrality or prestige, it stands to reason that it is not measured the same way. Even if we suspect that centrality is essential for power, it would not make sense always to equate them. For example, while being tied to others with high centrality may raise one's own centrality, being tied to others with high power may diminish one's own power.

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