In this continuing series of posts taken from the notes for my Intro to Political Analysis class, we’ll look now at networks and what they tell us about power.

Networks are composed of nodes and ties. Nodes are like points. Each represents an actor. An actor can be any individual, institution, nation-state, or social group with a distinct personality. It can even be a chemical.

Ties are like lines connecting nodes. They can represent any relationship—economic, romantic, religious, chemical. These ties have characteristics, like strength, direction, and elements. Strength can be dichotomous (binary) or cardinal (being represented by a number on a scale.) A tie can be undirected (two-way) or directed (one-way, and assymetric.) The elements of a tie can be univalent (just one relationship), or multivalent (with multiple strands of relationships.)

This is what a total (left) and partial (right) dominance hierarchy look like drawn as networks. The arrows point to the dominators (so A has the most dominance in both networks.)

The ties in these networks are directed, but for simplification we will focus on dichotomous, undirected, univalent ties.

So here are some of those types of networks:

So what do networks tell us about power?

The more central actors are more powerful, which makes sense. If someone is at the center of a network, they aren’t very far removed from any part of it. They also probably have a lot of connections, and messages are more likely to run through more central nodes. One way to quantify this it to look at the number of nodes the actor is tied to. This measure of centrality is called degree centrality, and it is calculated by finding the number of ties over the number of every other node—or

DC_a = Ties_a/(nodes_T-1)

Where “a” is the actor and “T” signifies the total number of nodes.

Centrality can be measured in other ways though, like closeness. An actor with one tie may be more powerful than an actor with two, if that one tie is to a very central node. So,

CC_a = (n-1)/(sum of distances from “a” to other nodes)

In the above scenarios, it would be calculated as (5-1)/(ab+ac+ad+ae).

You can also think of it as 1/(average distance from “a” to another node).

The third way of looking at the centrality of a node is betweenness centrality. No, really. That’s what it’s called.

Here’s how you find it:

For a given node (we’ll say “a”) look at the other pairs of nodes, and find the shortest paths between all of them—and there can be more than one shortest path, so find all of them. Then determine how many of those paths have to go through “a.” For each pairing, find the average of shortest paths crossing the node. Then add up all those averages, and divide them by the total number of pairings (take the average of the average.) And that’s your betweenness centrality.

Who’s confused? Everybody? Let’s break this down then.

Let’s say we have a four kids in a classroom whose parents have named all of them letters of the alphabet. Naturally these kids are not well-adjusted, and they like to pass notes in class—but they’re all in a single row of desks, and they can only pass notes to the student directly in front of or behind them. We could model this network like this:

A—B—C—D

Now let’s say we want to find the betweenness centrality of “B.” This one’s pretty simple—we only have three possible pairings (since we’re not looking at paths that go to B), A to C, A to D, and C to D. We could include the reciprocals of those, but it wouldn’t change the betweenness centrality, so let’s not.

There’s only one shortest path between A and C, and it runs through B—so that’s a 1/1.

The same is true for A to D, so that’s another 1/1.

However, if C wants to pass a note to D, the shortest path is to just give it directly to D, and never go through B. So that’s a 0/1, or just a 0.

Now we add all of these up (1+1+0=2) and divide that by the total number of pairings (3) and we get 2/3, or 0.67 as the betweenness centrality. By comparison, because A and D are on the outside of this network, no messages will ever pass through them, and they have a degree centrality of 0. These are small numbers, but bear in mind that the highest betweenness or closeness or degree centrality possible is 1.

And the only type of network where a node can have a value of 1 for betweenness centrality, closeness centrality, and degree centrality is a star network. The most powerful an actor can be is if they are the hub of that network, and by looking at the above graph it’s pretty easy to see why. You can also see that satellites (the nodes sticking off of the hub of the network) will be weaker the more of them there are.

This is why totalitarian governments attempt to break all ties between members of the populace (for example, by prohibiting large gatherings of people), except the spokes connecting to the central hub of the government. This makes the citizens weak, and the government powerful.

So that’s networks. There’s an assumption here that we haven’t talked about—that is, that networks and hierarchies are just a given. But that’s not the case. Networks are created by choice—and choice will be the topic of the next post.