The first article in this series discussed the interconnectedness of people in social networks. The second discussed modeling techniques to visualize these networks. This post goes beyond visualization, presenting various analyses in understanding the spread of a virus within social networks. Through these analyses, we can get a clearer understanding of the characteristics of those who are most likely to spread the virus as well as the groups through which Covid moves the quickest.
The diagram below is taken from the post See the Spread, Stop the Spreadby Valdis Krebs, Chief Scientist at Orgnet, LLC. The analysis is based on a peer-reviewed article he wrote with the Center for Disease Control, Transmission Network Analysis to Complement Routine Tuberculosis Contact Investigations. I would strongly recommend reading both of these for further study on this subject. Although the diagram presents a tuberculosis outbreak ten years ago in the American Southwest, it demonstrates the use of a network diagram when examining the spread of contagion as we discussed in the previous post.
The first thing we notice about Krebs’s diagram is the color-coding, something discussed in my previous post. We can see which of the nodes in the network represent individuals who were Infectious, Tested & Infected, Tested & Not Infected, and Not Yet Tested. We should also recall that in the previous post when studying the spread of a virus through a community we are not necessarily interested in a social network per se, meaning social relationships between people. Rather we are focused on contacts between people. A quick review of this image gives us a clear picture of the impact one infectious individual had on verified contacts.
Beyond visualizations, network analysis provides us with a means of analyzing the behavior and relationships of the nodes within the graph. For example, we can calculate a nodes degree centrality which tells us how well connected a node is to the other nodes in the graph. To calculate the degree of centrality, we simply divide the number of nodes to which the node is directly connected by the number of all possible nodes to which the node can be connected.
Note that the degree of centrality does more than simply look at the number of connections a node has but looks at those connections as a percentage of the size of the overall network. For example, we may have two nodes each in a different network. Node A may have 5 network connections in a network of 15 nodes while Node 1 may have 15 connections in a completely different network of 100 nodes. Although Node 1 has more direct connections, its degree centrality is .15 which is less than Node A whose degree centrality is .33. If this were a social network, we could conclude that Node A is more popular within its network than Node 1.
We can apply degree centrality to the networks formed by contact tracing in several ways. We might assume that the higher the degree centrality of any node increases the probability of getting the virus. This makes sense, the greater the number of connections the greater the probability that an individual will contract the virus. But did we need network analysis to tell us that? After all, who doesn’t know that when there is a contagious virus going around that the number of people with whom you come in contact increases the probability you will catch it. There are more interesting questions that can be asked as a result of this one simple metric.
Is there a point at which we can say that a person definitely will become infected, where their degree centrality predicts a 100% chance they will get sick? What about the people who have a high degree of centrality but have not become sick? Let’s say, for the sake of discussion, that nodes with a degree centrality of .65 have an 80% chance of contracting the virus. What about the other 20%, the people who did not get sick? What do they have in common? Is that commonality simply correlation without causation? The majority of the people in that group happen to have blue eyes. Or do they have something in common that provides some type of immunity? At one point some speculated that having Type O Blood reduced your risk of contracting Covid. If there is something that this group has in common that makes them immune is it something we can replicate? As of this writing, I do not believe there is a way to change one’s blood type, but perhaps there is something else such a group may have in common. Perhaps they may have similar diets or patterns of social interaction.
While network analysis may not be able to answer some of these questions, its power is in its ability to provide those combating the disease enough information to ask. Of course, degree centrality is just one of many types of network analyses that can be performed. It demonstrates how data analytics can be used to better understand how a contagion moves through a community. In my next post, I will discuss cliques.