Covid-19, Network Analysis, and Network Models

In part 1 of this post, I discussed the interconnectivity of social networks highlighting the closeness we all share. I also noted the impact this has on the spread of a virus such as Covid-19. In this post, I explore how to visualize networks to better understand its transmission, enabling us to combat its spread.

Network analysis is a set of methodologies used to represent relationships between objects. Through visualization, we can better understand the dynamics of these networks.  A network can be anything from the neurons in a brain to computers across to the internet. In this post, however, we are concerned with social networks, networks formed by the relationships between people, and how to analyze these networks to better understand the spread of disease.

A social network starts simply enough, we have two people who have some association. We could represent these networks in several ways. Perhaps we create a huge table with a row for each individual with a unique ID for that row. The columns of the row would have the id of the other people in the table with whom they have a relationship. We might even try some sort of matrix with people in both the columns and rows, marking the intersection to indicate a relationship between the two. While these methods of storing the data may be helpful, they don’t facilitate visualization.

Network diagrams cannot only capture the relationship between individuals in a network, they also aid in visualizing the nature of a network. The individuals in the relationship are shown as nodes in the network diagram. The relationship between them is shown as a line which is referred to as an edge. These relationships can be either directed, such as Person A reports to Person B in an organizational structure, or undirected, Person C lives on the same street as Person D. The edge of a directed relationship is represented with an arrow. The figure below presents both a directed and undirected network diagram.  While the networks in the figure below are very basic, simply think of an organization chart at any company to see how quickly they can become very complex.

Figure 1

In applying this to the issue at hand, the spread of a disease through a community, we are interested in not a person’s relationships, but the people with whom they have come in contact. Think of your typical pre-Covid day. Think of all the people with whom you typically came in contact. When you get up in the morning you go to the gym and then stop for a coffee on the way to the office. You might attend a morning status meeting. For lunch, you grab a burger. Our contact diagram captures all of the things in a typical day where we encounter all sorts of people in a variety of places. A network diagram of your typical day could easily look like the one presented below.

Figure 2 – Daily contact diagram

The figure above shows the result of contact tracing for an individual for a single day. We make some interesting modifications to this network. We may color code the nodes to indicate if someone has tested positive for the virus or displays symptoms. We might change the boldness of the edges to indicate the frequency of contact between the nodes. All of these things will better enable us to understand the dynamics of the network.

Up to this point in this post, I have presented the most common method for network diagraming. However, several others can be quite useful. For example, the figure below presents a chord diagram that is popular in the area of genomic research. In the figure below, the various network entities are represented by the nodes along the circumference of the circle with a line to show the relationship between the nodes. The boldness of the lines gives us an indication of the number of relationships for that node. We should note that there are a variety of circos plots, from chord diagrams to circular phylogenic trees. As fascinating as an exploration of circos plots may be, they are beyond the scope of this post, but I would suggest further research on this subject.

Figure 3

A third alternative is the arc graph. Basically, we transform the circle in the chord diagram shown above into a straight line. The figure below provides an example of an arc graph.

Figure 4

Another way to add information to these diagrams is to vary the size of the nodes representing the entities in the network. The greater the number of connections the larger the node representing that entity. A second variation is to label the nodes in the diagram. In the figure below I have provided an example of each. More details on these two diagrams can be found at Data-to-vis.com along with some other interesting visualizations.

Figure 5

Each of the visualizations presented in this post is relatively simple. We can easily understand what they are attempting to communicate. What happens, though, when we get into something more complex. The figure below is a network diagram of the Six Degrees of Kevin Bacon we discussed in the previous article. This image was created by Seok-Hee Hong at the University of Sydney. While the visualization is interesting, we may not be able to gather a great number of insights from this image. When we begin to address networks as complex as the spread of Covid-19, we are going to need tools that go past mere visualization, which is the subject of the third post in this series.

Figure 6

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