Habemus Papam!

The metrics that predicted the pope and drive web search.

There’s a new pope, if you haven’t heard. Robert Francis Prevost, or as he’s now known, Pope Leo XIV. What you might not have heard was his name in the media swarm of people in contention to be pope.

Was this really such a surprise? Was there no way to know he was that important? Some Italian researchers at the University of Bocconi might not think so. Prior to the conclave, they calculated some network centrality metrics and found him at the top of one list, “eigenvector centrality”.¹

University of Bocconi's network of The College of Cardinals

But what is that? And how do we even measure “importance”?

Measuring Importance

In any network of people, be it business, politics, or friendships, there are many notions of “importance”. If we want to quantify it, the first thing that often jumps to mind is

How many connections do you have?

Degree Centrality

Degree Centrality is just that: a measure of how many nodes someone is connected to.

Great! A simple, quick way to measure how important someone is.

But, there’s a problem. Have you spotted it? Over in the corner, Joe, who is friends with the entire janitorial staff, has just been promoted to the C-suite for his immense power and importance in the company.

Degree Centrality doesn’t take into account how important Joe’s connections are. If only there was a way to do that. Wait. What’s that below me?

Eigenvector Centrality

What we would like, then, is for a node’s importance to be proportional to the sum of their neighbors’ importance. In the College of Cardinals, being connected to influential cardinals from major archdioceses matters more than knowing many auxiliary bishops.

Let’s go back to the basics of graphs to find a way we can calculate such a thing.

Graphs are commonly stored as adjacency matrices. In the simplest case, we have a matrix \(A\) where

An example of a simple graph and its adjacency matrix ^src

This matrix representation lends itself naturally to linear algebra, and in fact this is where the insight of eigenvector centrality comes from.

Again, we want any node (\(i\))’s importance/centrality (\(c_i\)) to be proportional to the sum of its neighbors.

This can be given by

which implies that there is some proportionality constant \(k\) where

given recursively in lazy pseudocode by

def centrality(node):
    # base case omitted
    c_node = 0
    for neighbor in node.neighbors:
        c_node += proportion * centrality(neighbor)
    return c_node

Observe that \(A_{ij}\) is \(0\) when there is no connection to \(c_j\), so it is not counted in the sum.

If we pack our scores in a vector \(c=(c_1,c_2,\dots, c_n)^T\), we can rewrite the above as

This is the eigenvector equation \(Ac = \lambda c\), just using \(\frac1k\) instead of \(\lambda\).

Now that we connected this problem to a well-known concept in linear algebra, we’ve climbed on top of the shoulders of giants and can use all of the results that come with it.

The result \(Ac = \lambda c\) means that when the adjacency matrix transforms all of the centralities, it is only scaled by \(\lambda\) (the eigenvalue), not rotated. Inside this vector \(c\) we can find the importance of every node \(i\) at \(c_i\).

But… which \(\lambda\)? Every matrix \(A\) can have multiple eigenvectors, so which one do we choose?

The Principal Eigenvector

For a strongly-connected graph, the Perron-Frobenius theorem guarantees that its adjacency matrix \(A\) has a unique, largest, positive, real eigenvalue. Its corresponding eigenvector is called the principal eigenvector. This is the one we are interested in. Again, in that vector \(c\) we can find the importance of every node \(i\) in our network at \(c_i\). As a note, typically the eigenvector \(c\) is normalized such that \(c_1+c_2+\dots+c_n=1\) so we can interpret them as percentages.

We can find this principal eigenvector in many ways. For small graphs, you can probably get away with solving the textbook

to get all the possible eigenvalues, then getting the corresponding vector \(c\) by

However, this is terribly inefficient, computing the full set of solutions is \(\mathcal{O}(n^4)\) or worse. So, generally a process called power iteration is used, which continually applies the adjacency matrix in succeeding powers. The power iterative approach converges to the principal eigenvector quickly, even for large graphs like the web.

In 1996, Google founders Sergey Brin and Larry Page realized that web pages shouldn’t be ranked by just their content, but also the importance of pages linking to them. This led to the development of PageRank, which is essentially just eigenvector centrality applied to the web graph.

The University of Bocconi arrived at the following top-5 for the eigenvector centrality scores.

1. Robert Prevost (US)
2. Lazzaro You Heung-sik (South Korea)
3. Arthur Roche (UK)
4. Jean-Marc Aveline (France)
5. Claudio Gugerotti (Italy)

I am no expert on Vatican politics, but it does make sense that Robert Prevost ranks the highest here. He was the Prefect of the Dicastery for Bishops, and likely influenced who became bishops, and thus cardinals, and worked closely in the appointment process.

Betweenness Centrality

Another metric that they used was betweenness centrality.

The idea is that nodes are important if they lie on many shortest paths between nodes; they are a bridge or bottleneck.

Think of working in a company, and to do anything with a database, you have to go talk to George, who’s the only guy who knows COBOL and has been working there since 1968. He is the bottleneck, and thus is more important by betweenness centrality.

The University of Bocconi arrived at the following top-5 for the betweenness centrality scores.

1. Anders Arborelius (Sweden)
2. Pietro Parolin (Italy)
3. Víctor Fernández (Argentina)
4. Gérald Lacroix (Canada)
5. Joseph Tobin (USA)

Unlike eigenvector centrality, betweenness centrality didn’t predict the papal outcome, with the eventual Pope Leo XIV not appearing in its top-5.

Conclusion

So, which one is right? Did they really predict the pope? Well, if you pick eigenvector centrality, then he was at the top of that list.

However, these are just a subset of the many ways we can measure “importance” in a network. There’s no single, universal metric—what counts as “important” depends entirely on the question you’re asking.

Is the most connected person the most powerful? Or is the bridge who links otherwise disconnected groups more vital? For example, Anders Arborelius ranked highest by betweenness centrality—not because he was the loudest voice, but because possibly, as the first Swedish cardinal and one of the few multilingual connectors, he acts as a bridge to communities that might otherwise be left out.

Ultimately, every metric tells its own story about influence, connection, and power. Whether you’re predicting popes, ranking web pages, or just trying to understand your own friend group, it all depends on what you decide “importance” really means.

If you would like to read more, please head over to the original press publication¹. I don’t believe any technical paper has been put out by them as of yet, but I could be wrong.