# Probabilistic Models for Manuscript Transmission

Jacques Istas

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Probabilistic Models for Manuscript
Transmission

Jacques Istas*

December 11, 2025

**Abstract**

Traditional textual criticism often seeks to reconstruct the past: to identify the sources of a manuscript, detect possible contamination, or determine whether an original archetype existed. Such reconstructions generally assume that significant textual changes, whether the disappearance of a variant, the stabilization of a reading, or the extinction of an entire work, result from deliberate human choices, whether by scribes or by authorities directing the copying process. In this article, we reverse the perspective. Instead of moving backward in time, we move forward: starting from an initial “generation zero,” we explore how manuscripts might evolve under the influence of randomness alone. To do so, we first present several key examples drawn from the corpus of biblical manuscripts, illustrating cases of textual divergence, coexistence of variants, extinction of readings, and survival of otherwise marginal works. We then examine classical probabilistic models, Markov chains, Pólya urn processes, branching processes, and Erdős–Rényi random graphs, and evaluate their qualitative ability to account for these examples without invoking intentional human agency. The goal is not to claim that manuscript transmission is governed by chance, but to show that randomness can, in principle, produce phenomena traditionally attributed to deliberate decisions. Recognizing this possibility invites a broader methodological openness: some features of textual history may arise not from purposeful intervention, but from stochastic dynamics inherent in the copying process.

**Keywords:** biblical manuscripts, transmission, Markov chains, branching processes, Pólya urns, Erdős-Rényi random graphs.

<footnote>*Jacques Istas, Laboratoire Jean Kuntzmann, 150 Place du Torrent F-38400 Saint Martin d'Hères jacques.istas@univ-grenoble-alpes.fr</footnote>
# Contents

<table><tr><td>1 Introduction</td><td>3</td></tr><tr><td>2 Some key examples of manuscript transmissions</td><td>4</td></tr><tr><td>3 Textual drift</td><td>6</td></tr><tr><td>4 Pólya Urns and Textual Dynamics</td><td>7</td></tr><tr><td>5 Modeling Manuscript Extinction</td><td>8</td></tr><tr><td>6 The small world of manuscripts</td><td>9</td></tr></table>

# 1 Introduction

The study of manuscript transmission often relies on the assumption that meaningful textual change reflects meaningful human action. When a variant disappears, when two contradictory readings coexist, or when an entire work falls into oblivion, scholars typically infer deliberate behavior—choices made by scribes, theological intentions, or institutional pressures. Much of textual criticism is therefore oriented backward in time: it seeks to reconstruct sources, identify contamination events, and recover, when possible, an original text.

Yet the idea that significant outcomes may arise without deliberate intention is not foreign to the biblical textual traditions themselves. As 1 Kings 22:34–35 recounts: “But a certain man drew his bow at random and struck the king of Israel between the joints of the armor. [...] The king was propped up in his chariot facing the Syrians, and he died at evening.” This striking episode, in which an apparently random gesture produces a decisive historical consequence, offers a vivid metaphor for our inquiry: can chance, too, shape the fate of texts?

In this article, we take the opposite approach to classical textual criticism. Instead of reconstructing the past, we model the future. Beginning with a simple initial configuration, our “generation zero”, we investigate how manuscript traditions might evolve over successive generations if randomness plays a significant role in the copying process. Our central question is straightforward: can certain features of manuscript transmission arise purely by chance? Surprisingly, this question seems to have been little studied, with the notable exception of Guidi et al. [2025], Godreau et al. [2025], who used a birth-death process approach for extinct manuscripts; we will return to later.

If a variant disappears from the record, must this reflect intentional suppression, or could ordinary stochastic fluctuations suffice? If two divergent readings persist for centuries, does this necessarily imply sustained ideological conflict, or can probabilistic dynamics alone explain their coexistence?

To examine such questions, we first present a set of key examples, chosen for homogeneity from the corpus of Biblical manuscripts. These examples illustrate a range of phenomena: persistent contradictory readings, long-term coexistence
of rival variants, disappearance of minority forms, and the extinction or survival of entire works. We then revisit several classical probabilistic models, Markov chains, Pólya urns, branching processes, and random graphs, and test their ability to reproduce the qualitative features observed in these examples.

Our aim is not to argue that manuscript transmission is random. Rather, it is to demonstrate that randomness offers a plausible and often surprisingly powerful explanatory framework. By constructing the simplest possible stochastic models capable of reproducing key qualitative patterns, we show that some aspects of textual history need not be attributed solely to intentional human agency. Chance, too, may have left its imprint on the manuscripts that have reached us.

## 2 Some key examples of manuscript transmissions

### 2.1 David's destroyed chariots

Let us start with an example related to one word/number. This example is taken from the Hebrew Bible: numbers and letters use the same writing system. Two books (2 Samuel and 1 Chronicles) recount the same battle of David (BHS [2003]). In 2 Samuel 10:18, it is written that seven hundred chariots were destroyed. In 1 Chronicles 19:18, a book generally considered to be later (Römer and Macchi [2009]), seven thousand chariots are destroyed. Even later, in the 1st century CE, Josephus [2013] would speak of seven thousand horsemen. Is the change from seven hundred chariots to seven thousand an accidental error or an intentional one aimed at magnifying David's victory? In any case, the manuscripts of 2 Samuel and 1 Chronicles, although part of the same set (the Hebrew Bible), retain these two contradictory numbers. We are in a case where there was a modification by a copyist at an unknown time due to lack of preserved manuscripts, a modification that then no longer varied, although Josephus's writing may suggest that the contradiction was known and problematic.

### 2.2 Son of God in Mark 1:1

We now give an example of variants on a nominal phrase. The beginning of the Gospel of Mark starts with Ἀρχὴ τοῦ εὐαγγελίου Ἰησοῦ Χριστοῦ [υἱοῦ θεοῦ] (Nestlé et al. [2013]). The presence of the bracket [υἱοῦ θεοῦ] indicates a variant. Indeed, several manuscripts (see Slomp [1977] for a complete list) omit υἱοῦ θεοῦ in Mark 1:1. Let us cite the principal ones:

* Several Greek manuscripts or Latin translations of Irenaeus of Lyons (Irenaeus [1992]), Against Heresies, omit υἱοῦ θεοῦ;

* The Greek manuscripts of Origen's Against Celsus (Chadwick et al. [1980]) omit υἱοῦ θεοῦ;
* The Codex Sinaiticus (4th century, Parker [2010]) omits *υἱοῦ ἴεοῦ* with a later correction: the nomina sacra * Θу* for *υἱοῦ ἴεοῦ* was added between the first two lines;

* The Codex Coridethianus (9th century, Metzger and Ehrman [2005]) omits *υἱοῦ ἴεοῦ*.

Several other manuscripts, in Armenian, Georgian, Greek, or Syriac, omit *υἱοῦ ἴεοῦ*, the latest one, a minor Greek manuscript, dates from the 11th century. These manuscripts are certainly in the minority but too numerous for this omission to be the work of a single copyist. How can we explain the existence of this variant, which eventually disappeared?

A hypothesis was formulated by Ehrman [2011]. Adoptionists were a Christian group attested very early, at least since the 2nd century, which then gradually disappeared, with a local resurgence in Spain in the 8th century. Adoptionists believed that Jesus was born a man and was then adopted by God, generally at his baptism by John the Baptist. From an adoptionist perspective, it is essential to know when, in the Gospels, Jesus is first declared the Son of God. The text of Mark 1:1 *Αρχὴ τοῦ εὐαγγελίου Ἰησοῦ Χριστοῦ υἱοῦ ἴεοῦ* is problematic for Adoptionists. Note that we are here not addressing the question of what the original text of Mark 1:1 is, referring the reader to Wasserman [2011] for that. B. Ehrman's idea is that Adoptionist and Anti-Adoptionist copyists opposed each other and that history is written by the victors: the Adoptionists lost. B. Ehrman's hypothesis requires the existence of a conflict between Adoptionists and Anti-Adoptionists spanning several centuries (at least from the 4th century to the 9th century) and, crucially, a vast geographical area. Could the disappearance of *υἱοῦ ἴεοῦ* not be explained otherwise, for example, by the effect of chance?

## 2.3 The Pericope Adulterae

One of the most well-known episodes in the Gospels is the account of the adulterous woman, which appears in the Gospel of John 7:53-8:11 (Nestlé et al. [2013]). This pericope, however, is not attested in all manuscripts, including some major witnesses such as the 4th-century Sinaiticus and Vaticanus Codices (Keith [2008]). But the Pericope Adulterae is shortly attested by Papias, as Eusebius of Caesarea reports in Book III of his Ecclesiastical History, and is present in the Didascalia Apostolorum, a 3rd-century text whose Greek original is lost but whose Syriac and Latin translations are known to us, possibly sharing a source with the Gospel of Luke (Hughes [2013]). Moreover the Pericope Adulterae is present in the late 4th/early 5th-century Latin/Greek Codex Bezae Cantabrigensis (Parker [1992]). In fact, we cannot say that one of the sets of manuscripts including or excluding the Pericope Adulterae is truly a minority (Wasserman and Knust [2020]): including minuscules, manuscripts that do not include the Pericope Adulterae are found until the 14th century. Starting from the 15th century, all manuscripts include the Pericope Adulterae.
This raises the question: is the Pericope Adulterae a later invention, the insertion of an oral or Lukan tradition about Jesus, or the insertion of a tradition spuriously attributed to Jesus? For what reasons might a copyist have decided to insert or, conversely, to omit this pericope? A moral argument is put forward by Augustine of Hippo in his early 5th century book De Adulterinis Conjugiis (Augustine [1996]): the Pericope Adulterae can be interpreted as a de facto authorization of adultery. There might have been a conflict between copyists over moral issues. But this idea of Augustine does not seem to have had much legacy. Whatever the case, we have here a striking passage whose presence or absence varied across manuscripts for centuries before becoming widely but belatedly accepted.

## 2.4 Extinct manuscripts

Let us give two examples of extinction or non-extinction of two texts with equal status. The Codex Sinaiticus (4th century, Parker [2010]) contains the Christian Bible in Greek, including the oldest and most complete copy of the New Testament. The books contained within the Codex Sinaiticus have been recopied thousands of times. However, the Codex Sinaiticus also contains books later considered non-canonical, specifically the Epistle of Barnabas and the Shepherd of Hermas (Holmes [2007]). The Epistle of Barnabas was hardly recopied, and it took the 19th-century discovery of the Codices Sinaiticus and Hierosolymitanus (Milavec [2003]) for its text to become known. The Shepherd of Hermas, for its part, was recopied and transmitted, at least in its Latin version.

Several factors account for these possible disappearances: physical destruction (e.g. fires, warfare), material decay of the writing substrates, evolving literary canons, or the suppression of texts deemed heretical. However, can we posit the loss of works purely due to chance occurrences?

## 2.5 Textual stability amidst manuscript growth

There was a significant surge in the number of copied biblical manuscripts during the period spanning the 10th to the 14th century (Aland and Aland [2023]) with thousands of witnesses. This era saw an exponential increase in the production and dissemination of texts. While the sheer number of extant manuscripts grew tremendously, leading to a corresponding explosion in the total count of non-significant variants, the core text itself has become rather stable. The number of comparisons required to check every pair of manuscripts grows quadratically with the number of manuscripts. A core challenge posed by this massive increase in manuscript witnesses is the method of control required for scholarly analysis.

# 3 Textual drift

For the sake of completeness, we summarize Istas [2025] in this section. We assume that there is a constant number of manuscripts in each generation. We
will discuss this assumption in the next section. We assume that there are two
sources of randomness in the copyist model:

* The first source is due to contamination between manuscripts. The idea is that this copyist is the opposite of the theologically trained and conflict-driven copyist envisioned by Ehrman. On the contrary, the random copyist is indecisive: if he/she has access to the same variants, he/she copies them; if he/she has access to two different variants, he/she chooses one to copy at random.

* The second source concerns the choice of manuscripts. A binomial draw with replacement is performed so as not to favor any variant.

In this model, the number of variants is a finite-state Markov chain (e.g. Douc et al. [2018]); once a variant has disappeared, it cannot reappear. The Markov chain will converge toward the domination of one variant. This domination is solely due to chance; there is no need to assume the victory of one group of copyists over another. Of course, the majority variant has the greatest chance of prevailing. This model accounts for the variants seen in Section 2.2: a minority variant eventually gets eliminated by chance without needing to assume proactive human intervention. The case in Section 2.1 (two different variants that survive in two different texts) is also well-modeled by this approach. By analogy with genetic drift, we propose to name this phenomenon textual drift.

# 4 Pólya Urns and Textual Dynamics

In the previous section 3, there were two sources of randomness: the randomness associated with contamination and the randomness associated with the selection of manuscripts. What happens if we remove the contamination and keep only the random selection of the manuscript? We will follow the classical Pólya urn model (Eggenberger and G. Pólya [1923], Pólya [1930]). As before, we have two variants, A and B, of the same manuscript. Initially, there are N manuscripts, $X_0$ manuscripts with variant A and $N - X_0$ with variant B. A manuscript is randomly selected from the N manuscripts, it is copied once without modifications. These two manuscripts are returned to the set of manuscripts: we therefore have $N + 1$ manuscripts, with $X_1$ variants A and $N + 1 - X_1$ variants B. The operation is repeated as many times as desired. At generation n, what is the behavior of the frequency $f_n = X_n/(N + n)$ of variant A and the frequency $1 - f_n$ of variant B? Is there one of the two frequencies that tends towards zero? We have the following proposition (e.g. Mahmoud [2008]).

**Proposition 1.**

*The frequency $f_n$ converges in distribution to a beta distribution with parameters $X_0$ and $N - X_0$.*

The important point for us is that the beta distribution is a distribution
with a density. Consequently, the frequency $f_n$ does not converge towards 0
nor towards 1, it continues to fluctuate randomly with a limit distribution, the beta distribution, which depends on the initial parameters. This model without contamination does not lead to the disappearance of one of the two variants. This model well illuminates the case of the Pericope Adulterae, where for centuries we see a significant number of the two variants of the manuscripts (i.e., with or without the Pericope Adulterae) coexist.

However, we will see that a simple and realistic modification of the hypotheses changes the conclusion. A drawback of the previous model is that it assumes manuscripts live forever. Let us modify the process as follows: at each iteration, in addition to copying a manuscript, we remove one by making it "die". There is now a constant number N of manuscripts at each iteration. Let us denote $X_n$ the number of variants A at each generation, so there are $N - X_n$ variants B. The sequence $X_n$ is a Markov chain with a finite state space; the states 0 and N are absorbing, the others transient. This Markov chain will converge in finite time to one of the two states A or B. There is disappearance of one of the two variants. This example shows us that the conclusions, even qualitative ones, of the model are unstable. This model of random copies is unstable with respect to a change of hypotheses and therefore has less validity.

# 5 Modeling Manuscript Extinction

To study the possible extinction of manuscripts, two natural models are available to us: the birth-death processes and the Galton-Watson processes. The former are in continuous time, the latter in discrete time. One is not superior to the other, and this section can be seen as the natural complement to Guidi et al. [2025], Godreau et al. [2025].

It is not surprising that a manuscript copied on average less than once disappears. But can a manuscript copied on average more than once disappear? Galton-Watson processes (e.g. Istas [2005]) are fundamental models of population dynamics. These random processes have their origins in the study of family name extinction (Galton and Watson [1875]). Galton-Watson processes describe the evolution of a population over discrete time periods, where each individual in the current generation independently produces a random number of offspring for the next generation. Key characteristics of Galton-Watson processes are:

* Discrete Time: The process evolves in distinct, non-overlapping generations.

* Independence: Each individual reproduces independently of others.

* Identical Distribution: The number of offspring for each individual follows the same probability distribution.

Let $Z_n$ denote the number of manuscripts at generation n. We will assume the existence of an original text: $Z_0 = 1$. Let us denote by $p_k, k \ge 0$ the offspring distribution:

$$p_k = \mathbb{P}(Z_1 = k), k \ge 0.$$
One assume $p_0 \neq 0$. The expected offspring rate is:

$$m = \sum_{k \geq 0} k p_k.$$

We only consider here the case where the expected offspring rate $m$ is greater than 1. An essential tool for the study of Galton-Watson processes is the generating function:

$$\phi(s) = \sum_{k \geq 0} p_k s^k, s \in \mathbb{R}.$$

We are interested in the extinction of the Galton-Watson process, denoted as Ext:

$$Ext = \{\exists n > 0, Z_n = 0\}.$$

**Proposition 2.**

* The probability of the event Ext is the smallest positive root of the fixed-point equation $\phi(s) = s$.

* There exists a random variable $W$, with expectation 1, such that $Z_n/m^n$ converges (a.s.) to $W$. Moreover, conditionally on the non-extinction of the Galton-Watson process, $W$ is non-zero.

In practice, the question arises as to whether a Galton-Watson process can actually become extinct after several generations. Since $\mathbb{P}(Z_n = 0/Z_{n-1}) = p_0^{Z_{n-1}}$ and $Z_n/m^n \sim W$, it follows that the Galton-Watson process can only stop after several generations and with a non-negligible probability if $m$ is not too large and if $p_0$ is not too small.

Let us give a numerical example. Suppose the manuscript has a probability $p$ of being copied twice and a probability $1-p$ of not being copied. Since $m > 1$, $p$ must be between $1/2$ and 1. The extinction probability $q$, solution of $1-p+pq^2 = q$, is $(1-p)/p$. For example, for $p = 0.55$, $q$ is 0.82.

Let us return to the second part of Proposition 2. When there is no extinction, the number of manuscripts grows exponentially. This constitutes a limitation on the use of Galton-Watson processes for modeling the number of copied manuscripts: there is either extinction or exponential growth. Specifically, this model cannot be used to study the disappearance of variants when the number of manuscripts is limited or increases slowly but is highly relevant to the examples of section 2.4.

# 6 The small world of manuscripts

With the growth in the number of manuscripts copied through the centuries, such as biblical texts, one might intuitively expect an exponential increase in
the number of significant variants. Interestingly, this has not been the case. In the preceding sections, we examined models that naturally lead to the extinction of variants. However, these models generally assume a bounded number of manuscripts over time, with the Pólya Urn Model (discussed in Section 4) serving as a counter-example. Here, we shift our focus to the paradoxical behavior observed within Erdős-Rényi random graphs as we investigate the mechanisms that constrain variant proliferation despite continuous growth in the underlying textual support. The theory of random graphs is quite technical to explain. We provide a summary here, referring to Curien [2025] for a detailed exposition.

The Erdős-Rényi random graph is constructed from a set of $n$ points and each possible edge between any two distinct points is included independently with a probability $p$. $p$ is the probability that an edge exists between any two points. A given point is therefore connected on average to $(n-1)p$ points. We assume $p$ fixed and $n$ grows to $+\infty$. We get three phases of connectivity.

1. Subcritical phase (Isolated components) as $p \ll 1/n$

The graph is sparse. Almost all points are in small, isolated connected components. The graph is highly fragmented.

2. Critical phase (Emergence of a giant component) as $p \sim 1/n$

This is the critical threshold. Once $np$ crosses 1, a giant connected component suddenly appears.

3. Supercritical phase (Connected graph) as $p \gg \log(n)/n$

The graph is dense. The giant component rapidly absorbs all other components. The graph becomes connected with high probability. This means that almost all points are connected and there is a path between virtually any pair of vertices. The threshold $p_c = \log(n)/n$ is often called the connectivity threshold.

Let us take a numerical example with $p = 0.05$. If $n \ll 20$, we are in the subcritical phase; if $n \sim 20$, in the critical phase; if $n \gg 30$ in the supercritical phase. Let $n = 40$. There are $40 \times 39/2 = 780$ pairs of points. On average, in the Erdős–Rényi random graph, there are $40 \times 39/2 \times 0.05 = 39$ connected pairs of points. And yet, almost all 40 points are connected to each other.

Now, let us consider $n$ manuscripts. Each pair of manuscripts has a probability $p$ of being related: either by belonging to the same group of copyists, or to groups that maintain ties or links. We consider $p$ as given. This is where the importance of the a priori paradoxical behavior of the Erdős-Rényi random graph emerges. While $p$ is fixed, we observe three phases:

* $n$ is small: the manuscripts are isolated from one another.

* $n$ increases and is approximately $1/p$: a giant connected component appears.

* $n$ continues to increase: almost all the manuscripts are indeed connected.
The interconnected yet random nature of manuscript comparison (the Erdős–Rényi random graph) created a self-correcting, decentralized network strong enough to eliminate significant deviations and stabilize the core text, while simultaneously being too weak and localized to eradicate every trivial spelling or grammatical difference.

## Acknowledgements

This work was supported by the ANR SHERBET (CE38-2023) project.

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