Everyone, now that we've formed the foundation for the basics with network analysis, we will look at analyzing influence with regards to customer satisfaction. Here are some things that we have to think about with regards to analyzing influence in regards to customer satisfaction. We want to identify characteristics about people and network structure that could be related to influence. Now, influence is of course, a concept that is far more involved sociologically. But we are just going to handle it in this lesson from a basic structural standpoint. Let's think about it with regards to network analysis or network graphs. Remember this diagram or this graph from the previous lesson that we were talking about clusters. But I don't know if you noticed that when you saw this graph, there's this interesting phenomenon that's happening in this labeled graph between E and F. That between E and F, if that edge was not there, this graph would be two graphs. There'd be two subgraphs or two separate graphs. This phenomenon is usually called a bridge. A type of link that connects two different groups in a graph. We can suggest that these individuals, E and F, they hold some power in this graph because if it wasn't for their connection, these two subgraphs would be on their own. We can see from a network graph structure, potential influence or potential power positions at play here. Another concept with regards to analyzing influence when it comes to network analysis is what is called centrality. Centrality is the importance of a node to a network graph. Now there's various forms of centrality. We'll look at degree centrality first. Degree centrality is the number of links connected to a node. What do I mean by that? Here are some examples. Let's look at C on the top left-hand corner of this slide. How many links are coming into or edges are coming into C? Two. The degree centrality of C is two. How many edges are coming into E? Four. The degree centrality of E is four. H has five edges coming into it, and so the degree centrality is five and with J degree of centrality is three. This could be a measure of influence with regards to any one of these nodes that maybe we say H, because it's got a high degree centrality, there's some power influence that H has on this overall network as compared to other nodes. Now degree centrality can be broken out into in-degree and out-degree. In-degree is the number of links in a directional graph that are coming into a node and out-degree is a number of links or edges coming out of a node, again in a directed graph. Let's look at B. Node B has two edges coming into it, and so it's in-degree is two. But it has three edges coming out of it. We have B to E, B to C, and then it's hard to see B to D. There's three coming out of B. The total degree of centrality of B is five, but in-degree is two, out-degree is three. Now let's look at node C. You have two edges coming in, so its two in-degree, zero edges coming out so it's zero out-degree. But the total degree is two. Now if you're looking at total degree, B seems to have more influence or power than C. But let's say for whatever reason that you're looking at a phenomenon where in-degree is the most important thing. Then B is actually equal to C because B has two in-degree and C has two in-degree. You see how when you're looking at in-degree versus out-degree with a greater degree centrality, you could look at different ways to slice and dice influence or positions of power within a network. Now, there is another centrality measure called closeness centrality. Now closeness centrality is the average of the shortest path links from one node to all other nodes in the graph. Basically how quickly can one node get to all the other nodes within a graph? Let's look at the closeness centrality for B. What's the shortest path from B to A? You see this table on the right-hand side signifying shortest paths from B to the node on the first column, which B to A is one. There's an edge that goes directly to it. B to C is one, B to D is one, B to E is one. How you calculate closeness centrality is you say 1 plus 1 plus 1 plus 1, which is all the shortest paths from B to the other nodes in the graph and you divide it by the number of nodes minus one because we're not counting B to B. 1 plus 1 plus 1 plus 1 divided by 4 equals 1. The closeness centrality of B equals 1. Now what about the closeness centrality of E? It's slightly higher than the closest reality of B because from E to A is one, E to B is one. But what about E to C? We see in this table that it's actually two. E cannot get to C without going through either D or B. You have 1 plus 1 plus 2 plus 1 divided by 4 equals 1.25. Less is better. B is better in a sense than E in closeness centrality. This all sounds theoretical, but let's say that you want to get a message out to everybody in the graph as quickly as possible. Let's say its emergency message. You ask yourself, who's the best person to place that message with first? I can tell you right now it's probably B over E because E's got to take way too many steps with regards to getting to C than B does. You might place a message with B and say, we need this emergency message to get to everybody. B is closest to everybody compared to E. That's a measure of closeness centrality. Now there's a bunch of other measures of centrality, betweenness, eigenvector, etc. We're not going to cover those, but you can read up on those. If this is interesting knowledge to you, eigenvector is what actually calculates the relationship of pages on the Internet and Google originally used eigenvector centrality to understand how influential pages were to each other and then that changed how Google search ranked various suggestions when you search something. Eigenvector centrality is very much used online. With all of this said, I've talked about some measures, bridges, we've looked at centrality as various ways to use network structure to understand influence. I'd like to close with this example, this very real-world example with regards to how network structure was shown to explain why a certain family, The Medici Family, which was one of the most powerful families in Italy, held so much power within the 14th century. Now, in this network graph, the nodes are families, and the links of the edges in this graph are marriages across families. Apparently at that time, they would strategically marry off their family members to other families to form a trust relationship with those families. What you see here in this network graph is by that lower right-hand quadrant where The Medici Family lives. That there are certain families that cannot get to the rest of the other families in this Italian family network without going through the trust network of The Medici Family. When you look at this graph, many scholars say that the way that they made their marriage alliances, when you look at it from a network structure perspective, they became influential. They became powerful with regards to their relationships with everybody else, all the other telling families. The Medici family actually created the largest and most respected bank in Europe during their prime, and it was called The Medici Bank. We can see that if we use network structure to understand influence, that we could potentially use that understanding of influence to understand how people could affect customer satisfaction towards your company, your product, your services, or your brand.