$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 10.1: Introduction to Centrality and Power

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## The Several Faces of Power

All sociologists would agree that power is a fundamental property of social structures. There is much less agreement about what power is, and how we can describe and analyze its causes and consequences. In this chapter, we will look at some of the main approaches that social network analysis has developed to study power, and the closely related concept of centrality.

Network training has contributed a number of important insights about social power. Perhaps more importantly, the network approach emphasizes that power is inherently relational. An individual does not have power in the abstract, they have power because they can dominate others - ego's power is alter's dependence. Because power is a consequence of patterns of relations, the amount of power in social structures can vary. If a system is very loosely coupled (low density), not much power can be exerted; in high density systems, there is the potential for greater power. Power is both a systemic (macro) and relational (micro) property. The amount of power in a system and its distribution across actors are related, but are not the same thing. Two systems can have the same amount of power, but it can be equally distributed in one and unequally distributed in another. Power in social networks may be viewed either as a micro property (i.e. it describes relations between actors), or as a macro property (i.e. one that describes the entire population); as with other key sociological concepts, the macro and micro are closely connected in social network thinking.

Network analysts often describe the way that an actor is embedded in a relational network as imposing constraints on the actor, and offering the actor opportunities. Actors that face fewer constraints, and have more opportunities than others are in favorable structural positions. Having a favored position means that an actor may extract better bargains in exchanges, have greater influence, and that the actor will be a focus for deference and attention from those in less favored positions.

But, what do we mean by "having a favored position" and having "more opportunities" and "fewer constraints"? There are no single correct and final answers to these difficult questions. But, network analysis has made important contributions in providing precise definitions and concrete measures of several different approaches to the notion of the power that attaches to positions in structures of social relations.

To understand the approaches that network analysis uses to study power, it is useful to first think about some very simple systems. Consider the three simple graphs of networks in Figures 10.1, 10.2, and 10.3, which are called the "star", "line", and "circle".

Figure 10.1: "Star" network

Figure 10.2: "Line" network

Figure 10.3: "Circle" network

A moment's inspection ought to suggest that actor A has a highly favored structural position in the star network, if the network is describing a relationship such as resource exchange or resource sharing. But, exactly why is it that actor A has a "better" position than all of the others in the star network? What about the position of A in the line network? Is being at the end of the line an advantage or a disadvantage? Are all of the actors in the circle network really in exactly the same structural position?

We need to think about why structural location can be advantageous or disadvantageous to actors. Let's focus our attention on why actor A is so obviously at an advantage in the star network.

Degree: In the star network, actor A has more opportunities and alternatives than other actors. If actor D elects to not provide A with a resource, A has a number of other places to go to get it; however, if D elects to not exchange with A, then D will not be able to exchange at all. The more ties an actor has, the more power they (may) have. In the star network, Actor A has degree six, all other actors have degree one. This logic underlies measures of centrality and power based on actor degree, which we will discuss below. Actors who have more ties have greater opportunities because they have choices. This autonomy makes them less dependent on any specific other actor, and hence more powerful.

Now, consider the circle network in terms of degree. Each actor has exactly the same number of alternative trading partners (or degree), so all positions are equally advantaged or disadvantaged.

In the line network, matters are a bit more complicated. The actors at the end of the line (A and G) are actually at a structural disadvantage, but all others are apparently equal (actually, it's not really quite that simple). Generally, though, actors that are more central to the structure, in the sense of having higher degree or more connections, tend to have favored positions, and hence more power.

Closeness: The second reason why actor A is more powerful than the other actors in the start network is that actor A is closer to more actors than any other actor. Power can be exerted by direct bargaining and exchange. But power also comes from acting as a "reference point" by which other actors judge themselves, and by being a center of attention who's views are heard by larger numbers of actors. Actors who are able to reach other actors at shorter path lengths, or who are more reachable by other actors at shorter path lengths have favored positions. This structural advantage can be translated into power. In the star network, actor A is at a geodesic distance of one from all other actors; each other actor is at a geodesic distance of two from all other actors (but A). This logic of structural advantage underlies approaches that emphasize the distribution of closeness and distance as a source of power.

Now consider the circle network in terms of actor closeness. Each actor lies at different path lengths from the other actors, but all actors have identical distributions of closeness, and again would appear to be equal in terms of their structural positions. In the line network, the middle actor (D) is closer to all other actors than are the set C, E, the set B, F, and the set A, G. Again, the actors at the ends of the line, or at the periphery, are at a disadvantage.

Betweenness: The third reason that actor A is advantaged in the star network is because actor A lies between each other pairs of actors, and no other actors lie between A and other actors. If A wants to contact F, A may simply do so. If F wants to contact B, they must do so by way of A. This gives actor A the capacity to broker contacts among other actors - to extract "service charges" and to isolate actors or prevent contacts. The third aspect of a structurally advantaged position then is in being between other actors.

In the circle network, each actor lies between each other pair of actors. Actually, there are two pathways connecting each pair of actors, and each third actor lies on one, but not on the other of them. Again, all actors are equally advantaged or disadvantaged. In the line network, our end points (A, G) do not lie between any pairs, and have no brokering power. Actors closer to the middle of the chain lie on more pathways among pairs, and are again in an advantaged position.

Each of these three ideas - degree, closeness, and betweenness - has been elaborated on in a number of ways. We will examine three such elaborations briefly here.

Network analysts are more likely to describe their approaches as descriptions of centrality than of power. Each of the three approaches (degree, closeness, betweenness) describe the locations of individuals in terms of how close they are to the "center" of the action in a network - though the definitions of what it means to be at the center differ. It is more correct to describe network approaches this way - measures of centrality - than as measures of power. But, as we have suggested here, there are several reasons why central positions tend to be powerful positions.

This page titled 10.1: Introduction to Centrality and Power is shared under a not declared license and was authored, remixed, and/or curated by Robert Hanneman & Mark Riddle.