The Small Scale - Nodes and Centrality

Network science can be used to understand systems not just in terms of which parts are connected, but also in terms of how their neighbors and their neighbor's neighbors. In other words, network science can be used to understand the structure of a networked system are connected. The following three chapters describe how network science is used to understand network structure at various scales. This chapter focuses on the small-scale, or micro-scale, structure. The micro-scale structure describes the positions and roles played by specific nodes within the greater network. Such analysis can be used to identify influential individuals, bottlenecks in flows, and convenient locations for assembling information or resources. These properties (and others) can be characterized using centrality measures, which quantify various structural properties...

Centrality – finding key nodes

My interest in network science was, in part, inspired by my experience with hackerspaces and makerspaces. These spaces are member-run organizations that provide space and tools for do-it-yourself/technology/art projects. In 2009, I worked with several others in the Detroit area to start a space called i3 Detroit. During my time at i3 Detroit, I was struck by the creativity and innovation happening in i3 Detroit and other spaces, not just in the projects that were being created, but also in the unique types of volunteer-run, non-hierarchical organizations that were being developed. One of the keys to this innovation was the way ideas were spread within and between spaces. Ideas were spread from space to space by participants in national events and by members who went on hackerspace tours. Ideas were spread within spaces by the regulars who...

Bridges, brokers, and bottlenecks – betweenness centrality

In the popular children's game Telephone, one player starts by whispering a message to another, who then whispers that message to another, and so on. At the end, the last player recites the message out loud. Typically, the final message bears no resemblance to the original. So, a message that began as "follow the funky flow" might end up as "if every pork chop were perfect, we wouldn't have hotdogs". Every time a message passes from person to person, it has the possibility of changing, possibly because it was misheard, or possibly because it was deliberately changed. In more complex social networks, such as organizations and social movements, the individuals who connect different parts of the network have the greatest ability to filter, amplify, and alter information. Such individuals...

Hubs – eigenvector centrality

Imagine having an important message that needs to reach an entire group (for example, your employer or school), but only being able to give that message to one person. Who would you tell? You'd want to find someone well-connected to the entire network. You might try the person with the highest degree centrality (the most friends). The downside to that approach is that their friends might not be well connected to the rest of the network. In a hypothetical company, for example, the Director of East Coast Sales might know the most people, but might not know how to reach anyone in other departments or regions. Instead, it would be better to find someone who is highly connected to other highly-connected people, such as the CEO (or, more likely, their assistant). Such individuals are sometimes called hubs, because, like the center of a spoked...

Closeness centrality

The measure known as closeness centrality is one of the oldest centrality measures used in network science, proposed by the sociologist, Alex Bavelas, in 1950. Closeness is defined as the reciprocal of farness. What is farness? It's the reciprocal of closeness, of course! More helpfully, the farness of a node is the sum of distances between that node and all other nodes. So, a node with high closeness centrality is literally close to other nodes. Nodes with high closeness have, on average, short paths to many other nodes, which can be helpful for disseminating resources quickly.

The following example uses the NetworkX closeness_centrality() function to calculate the closeness centrality values for the suffragette network and display the top 10:

closeness = nx.closeness_centrality(G)
sorted(closeness.items(), key=lambda x:x[1], reverse=True)[0:10]

[(&apos...

Local clustering

The last structural measure presented in this chapter is a little different from the ones seen so far. Betweenness, eigenvector, and closeness centrality all characterize a node by its relation to other nodes in the network. The measure presented in this section concerns the relationships between a node's neighbors, rather than those of the node itself. It is often useful to consider whether a node's neighbors tend to be connected to each other. In a social network, this question translates to asking whether the friend of a friend is also your friend, a property known as transitivity [to mathematicians who enjoy polysyllabic words]. The result of such relationships are triangles: three nodes, all mutually connected. The tendency for such triangles to arise is called clustering. When strong clustering is present, it often suggests robustness, and redundancy...

Summary

This chapter has shown how to analyze the microscale structure of networks by calculating centrality measures and other node-based measures of network structure. Betweenness centrality identifies bridges and brokers: edges and nodes that connect otherwise poorly connected parts of a network. Eigenvector centrality identifies nodes that are connected to other well-connected nodes. Closeness centrality identifies nodes that are, on average, closest to other nodes. Finally, the triangle count and local clustering coefficient quantify how well-connected a node's friends are. By examining a historical social network of suffragette activists, we saw that ranking highly on one centrality value doesn't necessarily mean a node ranks highly on others. While sometimes correlated, different centrality values measure different things, so meaningful results require choosing...

References

The following is a list of resources that you can consider to get further knowledge:

• Edwards, G., & Crossley, N. (2009). Measures and meanings: Exploring the ego-net of Helen Kirkpatrick Watts, militant suffragette. Methodological Innovations Online, 4(1).
• Edwards, G. (2014). Infectious innovations? The diffusion of tactical innovation in social movement networks, the case of suffragette militancy. Social Movement Studies, 13(1).