In today's SNA Wednesday post, we will be talking about the concept of centrality. There are two ways to analyze networks:

1. Holistically: From every- / no- one's perspective
2. Egotistically: From a particular node's perspective

Centrality measures are egoistic. They aim to give a measure of a node's "place" in the network. These measures might mean:

• Behavioral: The more central a node is might give it more or less impact on the flow (of something) between nodes. Think of any related words: Control, power, influence, facilitation.
• Structural: The more central a node is might make it more or less important to the connectivity between nodes. Think of any related words: fragility, robustness, cohesiveness, similarity.

Generally, we prefer centrality measures that are important on both of these axises. But, of course, context is important.

So, lets begin! We might assume that the more connections a node has, the more central it is. This certainly makes sense along the behavioral axis: We can easily imagine a node that is well connected can influence a large part of the network. And so, this is 1^st centrality measure we will look at:

Degree Centrality

Definition: A node's degree centrality is the number of connections (edges) it has.

Lets imagine a network of 4 nodes. I think 3 networks immediately come to mind:

• A completely unconnected network
• A network with Ego connected to every other node, and no node connected to any other
• A complete network

What is Ego's degree centrality in these 3 cases?

• 0
• 3
• 3

Certainly having a degree centrality of 0 implies no impact with regard to behavior or structure of the network. Given the network has only 4 nodes, a degree centrality of 3 means a connection with each other node. But, between the two networks, the degree of influence by Ego is quite different: In the 2^nd network Ego is the node to which all other nodes link. It holds great power in this network. In the 3^rd, Ego is just 1 among 3 connections for other nodes, certainly not a position of distinction.

Formally, degree centrality is:

C^D_i(g) : d_i(g) / (n - 1)

g: the network we are investigating

i: the node we care about, ie. ego

n: the total number of nodes in g

d: degree of node i in network g

So, it seems degree centrality tells us more when it is low: A lower degree centrality implies less behavioral and structural importance. A higher degree centrality cant tell us as much without knowing about other nodes' degree centrality too.

Well, certainly direct connections are important, but degree centrality leaves other nodes out of the measure. So, how can we bring them in? Well, the concept of "distance" is pretty obvious, and direct connections clearly are a "distance" of 1. Nodes connected to nodes connected to Ego would be at a "distance" of 2 and so on. This is the next centrality measure we will explore.

Closeness Centrality

Definition: A node's closeness centrality is the inverse of the average distance from Ego to every other node.

While we didn't specify any characteristics of the networks we are analyzing, in general we care about connected networks: A connected network is a network where there is a path between any 2 nodes. Basically, you can get to any node from any other.

Lets take the same 4 nodes. The unique set of connected networks are:

Formally, closeness centrality is:

C^C_i(g): (n - 1) / sum(i, j | length(i -> j))

g: the network we are investigating

i: the node we care about, ie. ego

n: the total number of nodes in g

j: another node in g

What is Ego's closeness centrality in these 4 cases?

• 1
  • n = 4; n - 1 = 3
  • length from Ego to A, B & C: 1 in each case; summed is 3
  • 3 / 3 = 1
• 3/4
  • n = 4; n - 1 = 3
  • length from Ego to A & B: 1, C: 2; summed is 4
  • 3 / 4
• 3/5
  • n = 4; n - 1 = 3
  • length from Ego to A: 1, B & C: 2; summed is 5
  • 3 / 5
• 1/2
  • n = 4; n - 1 = 3
  • length from Ego to A: 1, B: 2, C: 3; summed is 6
  • 3 / 6 = 1 / 2

These numbers certainly make some sense (and of course they conform to everyone's mathematical needs: normalized from 0 to 1! ). The more direct connectionsEgo makes causes the sum of the lengths to be smaller, ultimately equally the number of other nodes when Ego is directly connected to each. So, the "closer" a node is to others the higher its closeness centrality. This measure does seem to capture a bit of a node's behavioral importance in a network: Nodes with highercloseness can influence more of the network. But, its not clear that it captures a node's structural importance: Ego's closeness centrality in the 3^rd and 4^thnetworks are different, but if Ego were to be removed from the networks it wouldnt matter to the other nodes' influence. Better than degree centrality but not good enough; we need a measure which takes Ego's network into account like closeness centrality but also takes Ego's structural importance to the network into account.

These are the two simplest network measures of centrality: A measure of a node's power, influence and importance both behaviorally and structurally.

Next SNA Wednesday we will review a classic paper on centrality .. exciting!!