Difficulties of Prediction

"Chi entra papa in conclave ne esce cardinale." — This Italian proverb ("He who enters the conclave as pope exits as cardinal") highlights the historical unpredictability of papal elections.

For example, bookmakers did not list Jorge Mario Bergoglio (Pope Francis) among the top six candidates before his 2013 election (O'Connell, 2019). However, social network analysis reveals that hidden forms of influence, such as episcopal consecration networks, may offer deeper insight into real power structures.

Historical betting data shows that in the ten conclaves we have betting data for, the betting favorite won only three times. On as many occasions, cardinals with minimal or zero betting odds became pope (Williams & Paton, 2015). Despite the growth of prediction markets, conclave outcomes remain extremely uncertain.

Construction of Network Indicators

Conclave 2025 Network Indicators

• Number of ties — Number of direct consecration links to or from a cardinal.
• Betweenness — Frequency of appearing on shortest paths connecting other cardinals.
• PageRank — Influence score reflecting the connectedness to other influential figures.

All Bishops Network Indicators

• Number of ties — Total consecration links globally among bishops.
• Betweenness — Strategic bridging position within the global Catholic network.
• PageRank — Hierarchical influence across the worldwide episcopacy.

Other Indicators

• Betting probability — Cardinal's chance of election based on aggregated betting markets. Source: OddsChecker (April 29, 2025).
• Reputation — Number of awards received, according to Wikidata profiles.
• Expert opinion — Number of times cited as papabile across major media sources: Detroit Catholic, The Independent, ABC News, Wired Italy, Il Giornale, BBC News, CBS News, Domradio.

Correlation Matrix

Key Findings

• Network centrality measures like ties, betweenness, and PageRank are highly interrelated. This reflects a tightly connected conclave network.
• Betting probability and reputation moderately correlate with centrality, suggesting that more central figures enjoy greater visibility and expectations.
• Pagerank among all bishops and number of ties among all bishops are nearly interchangeable in capturing global influence.

Interpretation

Cardinals with more connections occupy more central network positions, which leads to higher reputation and perceived electability.

While most network measures overlap strongly, Pagerank captures slightly distinct aspects of influence compared to simple tie counts or betweenness.

Betting Odds Aggregation Methodology

Betting odds are sourced from Polymarket, Smarkets, and Kalshi. Every day, the previous day's odds (captured at 11:59 PM) are recorded. To produce a weighted average:

• Each market is weighted according to the logarithm of its market capitalization.
• Missing odds are treated as zero if a cardinal is not listed on a platform.
• The resulting averages are rescaled to sum exactly to 100%.

This method aims to reflect both breadth (all available candidates) and market strength (size of participation).

Literature

• Borgatti, S.P. & Everett, M.G. (2006). "A Graph-Theoretic Perspective on Centrality." Social Networks.
• O'Connell, G. (2019). The Election of Pope Francis. Orbis Books.
• Williams, L.V. & Paton, D. (2015). "Forecasting the outcome of closed-door decisions: Evidence from 500 years of betting on papal conclaves." University of Nottingham Repository.
• Ivereigh, A. (2014). The Great Reformer: Francis and the Making of a Radical Pope. New York: Henry Holt and Company.