How the Math That Powers Google Foresaw the New Pope

The mathematics that predicted the new Pope

TO Jack Murtagh edited by Ginna Bryner

When Pope Francis died on Easter Monday in April, the news sparked not only a flood of mourners, but also a centuries-old tradition shrouded in secrecy: the papal conclave. Two weeks later, 133 cardinal electors closed in the Vatican's Sistine Chapel to choose the next pope. Outside the Vatican, soothsayers of all stripes tried to predict which name would be announced from the basilica's balcony. Among expert experts, crowdsourced prediction markets, bookmakers, platforms similar to fantasy sports And advanced artificial intelligence modelsalmost no one expected Robert Prevost.

Where all known divination methods seemed to fail, a team of researchers from the Bocconi University in Milan found a clue in a mathematical technique that has been around for decades, a cousin of the algorithm that made Google a household name.

Even with polling data, primaries and historical trends, predicting the winners of traditional political elections is difficult. Papal elections, by contrast, are held infrequently and rely on the votes of cardinals sworn to secrecy. To construct their crystal ball in such circumstances, Giuseppe Soda, Alessandro Iorio and Leonardo Rizzo from the Bocconi University School of Management turned to social media. The team combed public records to map a network that documented the personal and professional relationships among the College of Cardinals (senior members of the clergy who are both electors and candidates for the papacy). Think of it as a church LinkedIn. For example, the network included connections between cardinals who worked together in Vatican departments, between those who had ordained or been ordained by others, and between those who were friends. The researchers then applied methods from a branch of mathematics called network science rank cardinals according to three measures of influence within the network.

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Prevost, known to most analysts as an outsider and now known as Pope Leo XIV, ranked first in the first measure of influence, a category called “status.” An important caveat is that he did not make the top five in two other indicators: “brokering power” (how well the cardinal connects disparate parts of the network) and “coalition building” (how effectively the cardinal can form large alliances). Whether this indicator of “status” can shed light on future elections (papal or otherwise) remains to be seen. The study's authors were not attempting to directly predict a new pope, but rather hoped to demonstrate the importance of network approaches in analyzing conclaves and similar processes. Regardless, their success in this case, coupled with the broad applicability of the mathematical basis of their method, makes this model worth understanding.

How do mathematicians make “status” strict? The easiest way to find influential people online is called “degree centrality”—simply count the number of connections for each person. According to this measure, the cardinal who communicates with the greatest number of other cardinals will be called the most influential. While degree centrality is easy to calculate and useful for basic contexts, it does not capture global information about the network. It treats every link the same. In fact, relationships with influential people affect your status more than relationships with uninfluential people. A cardinal with only a few close colleagues can wield enormous influence if those colleagues are powerful Vatican officials. It's the difference between knowing everyone at the local coffee shop and being on a first-name basis with a few senators.

Enter eigenvector centralitya mathematical measure reflecting the recursive nature of influence. Instead of simply counting connections, it assigns each person a score proportional to the sum of their friends' scores in the network. In turn, the results of these friends depend on the results of their friends, which depend on the results of their friends, and so on. Calculating this circular definition requires some mathematical sophistication. To calculate these points, you can assign a value of 1 to each and then proceed in rounds. Each round, everyone updated their scores to the sum of their friends' scores. They then divided their scores by the network's current maximum score. (This step ensures that the scores remain between 0 and 1 while maintaining their relative sizes; if one person's score is twice that of another, this remains true after division.) If you continue to iterate in this manner, the numbers will eventually converge to the desired eigenvector centrality scores. For those who have studied linear algebra, we have just calculated the eigenvector corresponding to the largest eigenvalue adjacency matrix networks.

Google uses a similar measure to rank web pages in search results. When you enter a search query, Google's algorithm assembles a collection of relevant sites and then decides in what order to present them. What makes one website better than another for the end user? At its core, the Internet is a large network of web pages connected by hyperlinks. Google founders Larry Page and Sergey Brin wanted nodes in this network to have a certain “status” to decide how to rank search results. They realized that the link was from a well-connected, influential site such as Scientific American carries more weight than a link from someone's personal blog. They developed the PageRank algorithm, which uses a variation of eigenvector centralization to calculate the importance of web pages based on the importance of the pages that link to them. In addition to providing high-quality search results, this method prevents search engine fraud; artificially boosting your web page by having thousands of pages linking to it won't do much if those pages have low status. PageRank is more complex than eigenvector centrality, in part because links on the Internet are unidirectional, whereas friendships on a social network are bidirectional, and this symmetry simplifies the math.

Eigenvector centrality and its relatives arise wherever researchers need to identify influential nodes in complex networks. For example, epidemiologists use it to find superspreaders in disease networks, and neuroscientists are applying this to brain imaging data identify patterns of neural connections.

The new Pope will likely appreciate Team Bocconi's efforts because he studied mathematics as an undergraduate before donning his vestments. Time will tell whether eigenvector centrality can reliably influence future papal elections. Success this time could have been an accident. But as white smoke poured out of the Sistine Chapel chimney, it became clear that advanced artificial intelligence models and prediction markets had failed. They missed the wisdom of the old mathematical formula: influence comes not only from the people you know, but also from those who They know.

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