 In recent years, graph theory has become of particular interest to people who study organizations. A sociogram is a graph where the nodes are people and an edge connects people who know each other. For example, you can graph your Facebook relationships. An extremely important question for many, who's the most important person in your network? This information is useful for advertisers who could use it to identify the taste-makers – epidemiologists who could use it to identify potential sources of infection, and law enforcement who could use it to disrupt a criminal network, and many others. So this leads to the concept of a centrality measure. A centrality measure is a way to measure the importance of a node to a network. Now there are as many ways of measuring the importance of a node as there are networks, and so the important question is, what is important? A simple measure of centrality is degree centrality. This is just the number of edges that meet a node. The greater the number of edges, the more connected the node is to the network, and we might view a connected node as someone who knows many people. In fact, we say someone is well connected. So we might have a graph that looks like this, and we can find the node with the greatest degree centrality in the graph. So we measure the degree of every node. So A has two edges coming into it, so A has degree two, B has one, two, three, four edges coming into it, so B has degree four, C has degree five, and we can find the degrees of the remaining nodes, and we see that C has the greatest degree centrality. Degree centrality captures how well connected a node is, but another feature we might want to capture is how far a node is from all other nodes. We might look at it this way. We might expect a node that is far from the others to be less influenced by and less able to influence the rest of the network. So we define the far-ness to be the sum of the distances to all other nodes in the network, and we'll define the closeness centrality to be the reciprocal of this sum. Since we'd expect an important node to be close, we want the node with the least far-ness, and so the greatest closeness. So for example, let's find the closeness centrality of A in the graph, and so we want to figure out how far the rest of the network is from A, and so we can imagine expanding outward from A. This is actually Dijkstra's algorithm. So if I'm at A, there are two nodes, B and C, at distance one. Now if we expand outward from B, we go to D or E, and if we expand outward from C, we go to F or G. So there are four nodes at distance two, and finally this last node H is at distance three. So the far-ness lets two distances of one, four distances of two, one distance of three, for total far-ness of thirteen, and the closeness is the reciprocal one-thirteenth. And we can find the closeness centrality of all of the nodes and find the node that has the greatest closeness centrality. So we already found that node A has closeness centrality one-thirteenth. Let's take a look at B. So if we look at B, we see that we have four nodes, A, C, E, and D, at distance one, and the remaining three nodes, F, G, and H, are at distance two. So we have four at one, three at two, and if we add them all together, we get a total far-ness of ten and a closeness of one-tenth. If we look at node C, there's one, two, three, four, five nodes at distance one, and there's two more nodes at distance two. So our total far-ness, five ones plus two twos, and so our closeness centrality is one-ninth. If we go to node D, we see there are two nodes, B and H, at distance one. We have four nodes at distance two, and one node at distance three. So our total far-ness will be two ones, four twos, and one three, which add two, thirteen, and a closeness of one-thirteenth. For E, we see there's three nodes at distance one, and four nodes at distance two, for total far-ness of, and closeness, for F, we see two nodes at distance one, four nodes at distance two, and one node at distance three. Our total far-ness will be, and our closeness. We'll compute the far-ness of G, and finally H, and we see that of all of these values, C has the greatest closeness centrality. With the smaller the denominator, the larger the fraction.