Understanding Eigenvector Centrality and Its Uses

Eigenvector centrality is a concept used in many industries to analyze connectivity patterns. Professionals can use this type of algorithm to predict social patterns, rank web pages, and even analyze brain patterns in imaging scans. Eigenvector centrality is also a key algorithm that can analyze neural network structures in developing deep learning algorithms. Centrality measures help develop artificial intelligence (AI) models by optimizing and studying how machine learning systems acquire information.

Explore eigenvector centrality, common applications across fields, and how you can deepen your understanding of this concept. 

What is eigenvector centrality?

Eigenvector centrality is a measure used to determine the influence of a node (an individual element) within a network. The value of this eigenvector is weighted not only on the number of connections a node has but also on the quality or importance of these connections. In other words, a node is more influential if it connects to other influential nodes.

For a simpler way to think about the concept, imagine you are at a party full of people. If you connect with many people at the party, you might initially appear influential. But what if the people you’re connected to hardly know anyone else in the room? Being connected to fewer people but having influential connections (such as being connected to the party's hosts) would likely make you more influential than having many less-weighted connections. Eigenvector centrality is not just about the quantity of connections but the quality, too.

Mathematical explanation

In mathematical terms, the eigenvector centrality formula calculates the centrality of a node by taking the sum of the centrality scores of its neighbors (the nodes it connects to), considering the strength of these connections. These centrality scores form an eigenvector—a non-zero vector that only changes by a scalar factor when you apply a linear transformation to it—of the network’s adjacency matrix.

However, most network analysis software will compute eigenvector centrality independently, so it is not generally necessary to understand how to compute eigenvectors yourself. By understanding the general principles of the process behind the creation of node weights and centrality scores, you can get a sense of the applications of this concept across industries.

What makes a node important?

When weighing nodes and determining eigenvector centrality, the algorithm will look at several metrics. A node’s centrality is based on how many connections it has (out-going links) and whether its out-going links have a high volume of out-going links themselves. The algorithm also looks for incoming links from nodes with high degrees of incoming links. In essence, the algorithm wants to know how many links a node has and whether these links are influential. 

Eigenvector centrality examples

Eigenvector centrality finds use across several industries but is popularly referenced for its uses in the Google PageRank system and in social networking algorithms. Some popular examples where eigenvalue centrality plays a role include: 

Online search page ranking

The basic principle of the PageRank algorithm is to determine which websites are of the highest quality and, consequently, the most useful to browsers. If a web page links to several high-quality websites, it is thought to be likely that this page is also high-quality. 

In a model format, websites on the internet can be thought of as nodes. These nodes link through arrows, representing the direction of web traffic flow. For example, if your website had a link to an external page, an arrow would point from your page to the external page. The relative importance of the websites connected to your site, as well as the volume of web pages, will influence how the PageRank algorithm classifies your page. PageRank will also calculate the probability that someone browsing the internet will end up on your site by clicking links or randomly “jumping” between sites. 

In essence, this algorithm looks at:

• How many sites link to your website

• The quality of the sites that link to your website

• The number of sites these linked sites link to

Social networking

Each person (“actor”) in a social media network links to one or more others, and the pattern of these connections can tell you a lot about the network's structure and dynamics. Eigenvector centrality helps us understand each person's importance by counting the number of connections they have and examining the quality of these connections. 

This type of social network analysis can find who is influential within certain social circles. You can use this information to drive marketing strategies, product development, trend analysis, and many other functions. 

Sociology

In sociology, measuring relative social power can help sociologists understand the relationship dynamics and intricacies of social networks. Sociologists use eigenvector centrality to determine whether people have favorable positions, opportunities, or constraints and how this relates to their relative influence. 

For example, someone with highly influential connections is likely to have access to additional resources through resource sharing, while someone without connections to a certain resource might experience limitations. Measures of closeness and betweenness can also show how structural advantages come into play in social dynamics. 

Neural architecture

Eigenvector centrality has also been applied in neuroscience to better understand the brain's architecture. This approach, known as eigenvector centrality mapping (ECM), creates detailed maps of the brain that can then identify differences between various states of brain activity.

Researchers used this mapping technique in a study to design an algorithm that could analyze brain scans to identify networks of hubs, which are key areas of the brain with high connectivity with other areas [1]. The results showed a network of hubs. By applying statistical tests to the ECM data, researchers could identify significant differences in the centrality scores between the two scans. During the first scan, eigenvector centrality scores were significantly higher in the left and right thalamus and in the cerebellum. These results demonstrate the potential of ECM to capture the brain's architecture at a detailed level.

Limitations of eigenvector centrality

You should be aware of a few limitations of eigenvector centrality. For one, the PageRank algorithm weights all nodes the same when determining whether someone would land on a webpage through following links or “jumping” between pages. However, humans may be more likely to select one page than another in practice, which the algorithm doesn’t account for.

Eigenvector centrality also does not account for localized influence. For instance, a node might significantly influence a small network cluster, even though it may not be very well connected to the overall network. In such cases, eigenvector centrality might underestimate the particular node's importance.

Learn eigenvector centrality

To learn more about the math behind eigenvector centrality, you must first become comfortable with linear algebra, which is the mathematical basis. 

1. Get comfortable with algebra and calculus: Before you start with linear algebra, make sure you have a good understanding of basic mathematics, especially algebra, statistics, and calculus.

2. Find a textbook or online course: You can learn independently using one of the many good textbooks and online courses available. A classic textbook to consider is Gilbert Strang’s Linear Algebra and Its Applications. 

3. Start with the basics: Start by understanding the fundamental concepts such as vectors, matrices, and systems of linear equations. Practice problems that involve these concepts.

4. Learn about vector spaces: Once you understand the basics, move on to more complex topics like vector spaces, eigenvalues, and eigenvectors.

5. Practice: Like any other area of mathematics, the key to mastering linear algebra is through practice. Solve lots of problems and examples to reinforce your understanding.

Next steps

Eigenvector centrality scores the quantity and quality of a node's connection within a network, allowing for analysis of influence in social media, ranking of pages on the internet, and the effectiveness of a machine learning neural network. You can learn more about network analysis and eigenvector centrality through courses on Coursera. 

Consider taking the beginner course Social Network Analysis offered by UC Davis, or gain a more comprehensive understanding by completing the entire Computational Social Science Specialization. To learn more about the mathematics behind eigenvectors and linear algebra, try the Linear Algebra from Elementary to Advanced Specialization from Johns Hopkins University, also on Coursera.