Social networks are the roadmaps in which infectious diseases travel. In these networks, people or hosts, more generally, are represented by nodes. And potential disease transmission events are represented by edges between the nodes. If we know the structure of a network, we're much better able to understand how the disease will spread on that network. And how it can best be controlled. Large networks are very complex structures. But we can reduce their complexity by looking at the general structural properties of the network. Among the most important is a property called the degree distribution. The degree of the given node is the number of edges that this node has. In a social network then, the degree corresponds to the number of contacts that a person has. More contacts translates into more disease transmission, which is why the degree is an important measure on its own. Now, we can count the number of contacts for each node, and then we can take a look at the distribution of these degrees over the entire network. Let's take a look at a concrete example. This is a simple, rather small network. It has 500 nodes and 2500 edges. But it helps to illustrate the point. First, we would count the number of edges that are attached to each node, that's the degree of a node. Then we would want to look at the degree distribution over the entire network. For this particular network, the degree distribution looks like this. As you can see, the average degree is 10. The lowest degree is 8, and the highest degree is 13. In order to understand why the degree distribution is important, we need to understand how it affects the basic productive number R nought. Imagine a node in our network that is infectious. Let's say this is the first infected node in the network. First, we need to know the probability of disease transmission per contact. That's straightforward. It is simply the total time that the node is infectious, L times the probability of disease transmission per unit time beta. So the product of these two numbers is the per contact transmission probability. The more contacts you have therefore, the more individuals you would infect. And this brings us right back to the degree distribution. The average degree, which is the average number of contacts, is important because a high average degree means that there are many potential disease transmission possibilities. Which will aid the spread of the disease. If everyone had the exact same number of contacts, we could write R nought equals beta times L times K, where K is the average degree. Again, beta times L is the probability of disease transmission per contact. And we multiply this by the number of contacts K, which gives us the number of infections caused by an infectious individual. In reality, though, not everybody has the exact same number of contacts. Some people have more contacts than average, others have fewer contacts. In a distribution, the variance is a measure of how far a distribution is spread out. It turns out that when all else is equal, an increase in the variance leads to an increase of R nought. Mathematically, we can write the following. Here, sigma squared is the variance of the degree distribution. What this means is that the degree distribution can actually have a very large effect on R nought. Let me show you an example. This network here is almost identical to the one I showed before. It has the same number of nodes, the same number of edges, and the same average degree. However, we can see that the degree distribution looks rather different. This network's degree distribution clearly has a higher variance than the other network. It turns out that if we calculate the R naught for both of these networks, the R naught of this high variance network is about four times higher than the R naught of this low variance network. That's a substantial change in R naught, and as you know, a higher R-naught means more transmission. More people getting sick, and control is going to be more difficult. This is one of many reasons why understanding the social network on which diseases spread is so important. I'd like to end by pointing out that what we just learned, is important for control too. Controlling the spread of disease means to reduce its R naught. We can now understand that reducing the number of contacts or reducing the variance in the degree distribution are both ways of reducing our node. An efficient strategy is to target nodes at very high degrees, and to somehow protect them from getting and spreading the disease. This strategy is efficient because it reduces both the average degree, and the variance in a network. And by doing so it reduces R naught.