🧬 biology

Quo nomine vis vocari? A random-copying model explains the temporal sequence of papal names

This paper demonstrates that the 2,000-year history of papal name selection, despite being driven by individual deliberation, can be accurately modeled as a stochastic process where new names are chosen by randomly copying existing ones in proportion to their frequency, with occasional innovations.

Original authors: Egor Lappo, Noah A. Rosenberg

Published 2026-05-11

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CC BY 4.0

Original authors: Egor Lappo, Noah A. Rosenberg

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine a giant, 2,000-year-old game of "Telephone," but instead of whispering a message, the players are passing down names.

This paper investigates a fascinating question: When a new Pope is elected and chooses a name (like "Leo," "Francis," or "John Paul"), is that choice a deeply calculated, unique decision, or does it follow a simple, almost random pattern?

The authors, Egor Lappo and Noah Rosenberg, suggest that despite the heavy symbolism and careful deliberation involved in picking a papal name, the overall pattern of names chosen over two millennia looks exactly like a game of random copying.

Here is how the paper breaks it down, using some simple analogies:

1. The "Chinese Restaurant" of Popes

To understand the math behind the names, the authors use a famous concept from probability theory called the Chinese Restaurant Process.

Imagine a restaurant with infinite tables.

The Rule: When a new customer (a new Pope) walks in, they have two choices:
1. Sit at a new table: This represents choosing a brand new name that has never been used before (a "mutation").
2. Sit at an existing table: This represents choosing a name that has been used before.
The Twist: The probability of sitting at an existing table isn't equal for everyone. If a table already has 10 people sitting there (meaning the name "Leo" has been used 10 times), that table is 10 times more likely to be chosen than a table with only 1 person.

The Finding: The authors found that the history of Papal names fits this "Chinese Restaurant" model almost perfectly. Even though individual Cardinals are making serious, thoughtful decisions, the aggregate result is that a new Pope is statistically likely to pick a name in proportion to how often that name has been used in the past.

2. The "Popularity Contest" vs. The "Innovation"

The paper looks at 265 Popes and 81 distinct names.

The Popular Names: Just like in a restaurant where the biggest tables get the most new customers, popular names like "Gregory," "Benedict," and "Leo" get reused frequently.
The New Names: Occasionally, someone walks in and sits at a brand new table. This happens when a Pope chooses a name no one has ever used before, like "Francis" (chosen in 2013).
The Trend: The study shows that the "mutation rate" (the chance of picking a totally new name) has gone down over time. In the early days, new names were common. In the last 500 years, it's become very rare to invent a new name; the "restaurant" is mostly full of people sitting at the same old, popular tables.

3. The "Glitch" in the System (The 11th–13th Centuries)

If the system is so random, were there times it broke? Yes.

The authors found a specific period (roughly the 11th to 13th centuries) where the pattern didn't fit the "random copying" rule.

What happened? During this time, Popes seemed to be actively avoiding the most popular names. Instead of sitting at the crowded tables, they were seeking out the "lonely" tables—names that had been used only once or twice centuries ago.
The Analogy: Imagine a restaurant where, for a few years, everyone suddenly decided to sit at the empty tables in the back corner, ignoring the full tables in the front.
Why? The paper suggests this was a form of anti-conformity. Popes wanted to distance themselves from the recent past (which was dominated by names like "John" and "Benedict") and reach back to older, rarer names to signal a fresh start.

4. The Big Conclusion

The most surprising part of the paper is the conclusion about human behavior.

Usually, we think of the Papacy as a place of intense, deliberate strategy. Every name is chosen to honor a saint, a predecessor, or a specific political message. However, the authors argue that when you zoom out and look at the entire 2,000-year timeline, all those individual, careful decisions average out to look like simple, random chance.

In short: Even though every single Pope is making a unique, thoughtful choice, the collection of all those choices follows a simple mathematical law of "copying what's popular, with a small chance of trying something new." It's a reminder that complex human history can sometimes be described by very simple rules.

Technical Summary: Quo nomine vis vocari? A random-copying model explains the temporal sequence of papal names

Problem Statement
The study investigates the long-term cultural evolution of papal name selection within the Catholic Church, a process spanning approximately 2,000 years and involving 265 distinct pontiffs. While individual choices of papal names are deliberate, symbolic, and often reference specific predecessors (e.g., Pope Leo XIV referencing Leo XIII), the aggregate behavior of these choices over two millennia remains uncharacterized by simple mathematical laws. The central question is whether the sequence and frequency distribution of these names follow systematic rules or if they can be described by a stochastic process where variants are selected randomly based on their prior frequency, allowing for the occasional introduction of novel variants ("mutations").

Methodology
The authors assembled a dataset of 265 popes, recording their papal names and the start dates of their terms. The analysis applies frameworks from population genetics and stochastic processes, specifically Ewens sampling theory and the Chinese restaurant process (CRP).

Model Framework: The study posits a "random-copying" model where a new pope selects a name with probability proportional to the frequency of that name's previous usage. A "mutation" event corresponds to the selection of a previously unused name. The probability of a novel name is governed by a parameter $\theta$ (mutation rate), while the probability of copying an existing name $i$ (which has appeared $i$ times previously) is proportional to $i$ .
Statistical Testing (Slatkin Exact Test): To assess the fit of the observed name frequency distribution to the Ewens distribution, the authors employed Slatkin's exact test. This test evaluates the tail probability ( $P_E$ $P_{E}$ ) of the observed frequency vector $c_0$ $c_{0}$ conditional on the number of distinct names ( $k$ $k$ ) and the sample size ( $n$ $n$ ). The null hypothesis is that the observed frequencies represent a realization of the infinitely-many-alleles model (random copying).
- For large $n$ and $k$ , where exhaustive enumeration of frequency vectors is impractical, the authors utilized the Feller coupling algorithm to simulate frequency vectors from the Ewens distribution.
- The test was performed cumulatively for each pope from the 44th (Sixtus III) to the 265th, truncating the dataset at each step to observe the fit over time.
Conditional Analysis: To detect temporal deviations, a conditional Slatkin exact test was developed. This method tests sequences of name choices within 200-year moving windows, conditioning on all prior names being fixed. This allows for the assessment of whether specific historical periods deviated from the global random-copying trend.
Parameter Estimation: Maximum Likelihood Estimation (MLE) was used to estimate the scaled mutation rate $\hat{\theta}$ for the entire dataset and for specific temporal windows.
Diversity Metrics: Shannon entropy was calculated for 200-year windows to measure the "evenness" of the name distribution, distinguishing between conformist (high-frequency names chosen more often) and anti-conformist (rare names chosen more often) deviations.

Key Results

Global Fit: The Slatkin exact test reveals that throughout the entire history of the papacy (from the 44th pope to the present), the tail probability $P_E$ remains close to 1 and never approaches the significance level of $\alpha = 0.05$ . This indicates that the aggregate sequence of papal names is statistically indistinguishable from a random-copying process.
Mutation Rate ( $\theta$ ): The estimated mutation rate $\hat{\theta}$ is high in the early periods but decreases over time, stabilizing at approximately $\hat{\theta} \approx 40$ for the last 500 years. This implies that prior to the election of Leo XIV (2025), the model predicted a $\approx 13\%$ probability of a new pope choosing a novel name.
Temporal Deviations: While the global model fits well, the conditional analysis identified a specific period of deviation between 1013 and 1261. During this window, the tail probability dropped significantly (rejecting the null hypothesis at $p < 0.05$ $p < 0.05$ ).
- Nature of Deviation: Analysis of Shannon entropy during this period showed a peak, indicating the frequency vector became more "even" than predicted. Rare, previously used names were chosen disproportionately often. This suggests a negative frequency-dependent bias (anti-conformity), where popes actively sought to avoid the most common names (like John and Benedict) in favor of rarer, older names.
- Historical Context: This period aligns with historical observations (e.g., Poole, 1917) of streaks of popes choosing names as the "second," "third," or "fourth" of their respective names, potentially as a strategy to disassociate from a specific era of church leadership.
Recent Trends: The most recent windows show a return to the random-copying model, with the introduction of novel names like "John Paul" (1978) and "Francis" (2013) contributing to a slight increase in $\hat{\theta}$ .

Significance and Claims
The paper claims that despite the high degree of individual deliberation and symbolic intent inherent in selecting a papal name, the aggregate cultural behavior over two millennia conforms to a simple stochastic law: random copying with mutation.

Emergent Simplicity: The study demonstrates that a complex cultural process involving specific, identified individuals and a single institution can be described by simple mathematical models derived from population genetics.
Methodological Contribution: The authors introduce a conditional Slatkin exact test that allows for the application of Ewens sampling theory to explicitly temporal processes. By conditioning on prior sequences, this method can detect transient deviations from random copying that global tests might obscure.
Cultural Evolution: The findings suggest that even in highly structured institutions, cultural variants (names) can evolve via a mechanism where the probability of adoption is proportional to frequency, punctuated by occasional innovations. The temporary deviation in the 11th–13th centuries highlights that while random copying is a robust baseline, specific historical contexts can induce anti-conformist biases that temporarily alter the trajectory of cultural evolution.

The authors conclude that the random-copying model provides a powerful null hypothesis for cultural evolution, against which specific historical anomalies (like the 11th–13th-century anti-conformity) can be rigorously identified and analyzed.

1. The "Chinese Restaurant" of Popes

2. The "Popularity Contest" vs. The "Innovation"

3. The "Glitch" in the System (The 11th–13th Centuries)

4. The Big Conclusion

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