[1]Jean-Marc Luck

1]Université Paris-Saclay, CNRS & CEA, Institut de Physique Théorique, 91191
Gif-sur-Yvette, France

2]Faculty of Linguistics, Philology and Phonetics, Clarendon Institute, Walton
Street, Oxford OX1 2HG, United Kingdom

# Evolution of grammatical forms: some quantitative approaches

<EMAIL_ADDRESS>Anita Mehta<EMAIL_ADDRESS>[  [

###### Abstract

Grammatical forms are said to evolve via two main mechanisms. These are,
respectively, the ‘descent’ mechanism, where current forms can be seen to have
descended (albeit with occasional modifications) from their roots in ancient
languages, and the ‘contact’ mechanism, where evolution in a given language
occurs via borrowing from other languages with which it is in contact. We use
ideas and concepts from statistical physics to formulate a series of static
and dynamical models which illustrate these issues in general terms. The
static models emphasise the relative numbers of rules and exceptions, while
the dynamical models focus on the emergence of exceptional forms. These
unlikely survivors among various competing grammatical forms are winners
against the odds. Our analysis suggests that they emerge when the influence of
neighbouring languages exceeds the generic tendency towards regularisation
within individual languages.

##  1 Introduction

Historical linguistics is the study of language change over time [1]. It is
principally centred on how linguistic forms evolve in world languages, be
these to do with phonology, morphology, semantics, syntax or core lexicons.
The evolution of languages proceeds in two essentially different ways. The
first route, referred to as phylogeny, describes the ‘vertical’ descent with
modification from more ancient, possibly extinct languages, leading to the
representation of families of languages as branching phylogenetic trees. The
second comprises the ‘horizontal’ borrowing and diffusion between contemporary
languages, brought about via contact between their speakers; this has given
rise to the field of contact linguistics [2, 3].

The mechanisms involved during language contact will be of special importance
in this paper, so we introduce them here. Winford [3] has classified contact
between different linguistic communities under three different heads: he
describes, first, relatively homogeneous communities of monolinguals, most of
whom have little or no contact with speakers of other languages. In these, the
only way that borrowing occurs is via the media, or individual travellers, or
indeed foreign language teaching; the example of Japanese or Russian speakers
borrowing from English is cited as an example. The so-called ‘middle spectrum’
concerns communities which include bilingual or multilingual speakers, an
example of which might be the contact between linguistic minorities and a
dominant host group; the prevalence of French and Flemish in Brussels is cited
as an example. Finally, there are highly heterogeneous communities where
individual multilingualism is high and social and linguistic boundaries are
fluid; an example of this occurs in Northwest New Britain in Papua New Guinea.
The nature of the mutual influence between different languages depends
critically on the extent of contact [3]: slight lexical borrowing is about all
that happens under conditions of casual contact, while structural borrowing
needs the concerned linguistic communities to undergo sustained and intimate
contact. In particular this implies that while the borrowing of content
morphemes like nouns or verbs is common, the borrowing of grammatical features
is relatively rare.

There has been relatively little work done so far from the viewpoint of
statistical physics in modelling linguistic evolution,111In contrast, such
concepts have been used in other areas of language dynamics, such as the
coexistence of two or more languages in a given geographical area [4].
although several quantitative approaches [5, 6, 7, 8, 9, 10, 11] exist. In
this work, we apply statistical physics methodologies to explore the evolution
of morphological features in general, and rules governing verb conjugation in
particular. Our initial motivation for this was a question raised by Ringe and
Yang [12] concerning the evolution of past participles of verbs in English,
via the ‘tolerance principle’ [13]; this states that that there is a maximum
number of exceptions that a rule can tolerate in order to be productive. More
precisely, a rule applied to $N$ items obeys the tolerance principle if the
number $E$ of exceptions (i.e., of items to which it does not apply) is
smaller than the threshold

$E_{N}=\frac{N}{H_{N}}\approx\frac{N}{\ln N}$ (1)

($H_{N}$ is the $N$th harmonic number). From the viewpoint of statistical
physics, this threshold $E_{N}$ is enormously high, since it grows nearly
extensively with the total number $N$ of items. Our own evaluations of a
threshold demarcating rules and exceptions (Section 2) result in smaller and
more realistic values, to which we will draw attention as they occur.

Ringe and Yang claimed that the tolerance principle was able to explain the
prevalence of regular past participles ending in ‘-ed’, but not some more
unlikely irregular forms ending in ‘-uck’, such as ‘stuck’ or ‘struck’ [12].
Our models of competitive dynamics are able to resolve this issue, while also
putting the emergence of unlikely winners in a more general context.

The very formulation of the tolerance principle gives a prominent role to
exceptions. Grammatical rules are an essential feature of linguistic structure
and provide an efficient way of classifying existing forms; there are,
however, always exceptions to these. For instance, the conjugation of the
verbs ‘to be’ or ‘to have’ is rather irregular in most world languages, so
that these verbs constitute marked exceptions to general grammatical rules.

The occurrence of rules and exceptions is not unique to languages. In
mathematics, for example, the classification of semisimple Lie algebras
involves 4 rules (the infinite series $A_{n}$, $B_{n}$, $C_{n}$, and $D_{n}$)
and 5 exceptions ($E_{6}$, $E_{7}$, $E_{8}$, $F_{4}$ and $G_{2}$). Also, the
classification of finite simple groups involves 18 rules (infinite families of
groups) and 26 exceptions (the sporadic groups) (see [14]). Note also that the
numbers of rules and exceptions (4 vs. 5 and 18 vs. 26) are comparable in the
two cases. Rules and exceptions also occur naturally in data clustering, in
the context of computer science and data analysis, where they are termed
clusters and outliers respectively (see e.g. [15, 16]). Clearly, in all of the
above, rules concern either large or infinite series of objects, while
exceptions are isolated.

The plan of this paper is as follows. We begin with a purely static approach
to the interplay between grammar rules and exceptions (Section 2), which in
particular results in sensible values for the threshold that divides them. In
the following sections, we build increasingly sophisticated models of the
dynamical evolution of grammar rules. Successive levels of modelling include
the initial growth of the structured lexicon in a single language (Section 3),
the competition between growth and conversions, e.g. from irregular to regular
forms, in a mature language (Section 4), and finally a network representation
of the comparative evolution of grammar rules in a situation of prolonged
language contact (Section 5). Finally, we summarise and collate our insights
in the Discussion section (Section 6).

##  2 Rules and exceptions: a static approach

This section provides a first approach to the interplay between grammar rules
and exceptions. The arguments used come from a purely static perspective,
being based on optimality: no dynamical evolution is invoked.

We focus here on morphology, and in particular on rules governing verb
conjugation. Consider a language having a total of $N$ verbs, which are
divided into $K$ groups. Each group follows a distinct pattern of conjugation,
and is labelled by an integer $a=1,2,\dots,K$. Group $a$ contains $N_{a}$
verbs, so that

$N=\sum_{a=1}^{K}N_{a}.$ (2)

Groups are assumed to be ranked in order of decreasing size ($N_{1}\geq
N_{2}\geq N_{3}\dots$).

It is clearly most efficient to remember the conjugations of the $N_{1}$ verbs
in the largest group by means of a single rule. On the other hand, if the
smallest groups ($a$ near $K$) have $N_{a}=1$, it is most efficient to think
of them as exceptions (e.g. verbs such as ‘to be’ or ‘to have’ in most
languages). How, then, can a demarcation line between rules and exceptions be
operationally defined? A natural way of proceeding consists of minimising the
total memorization effort and memory size $I$ needed to learn and store the
conjugations of all verbs comprising both rules and exceptions, or, in
Chomsky’s [17] words, ‘the grammar’ and ‘the lexicon’. If the conjugation in
question has $R$ rules, the latter correspond to the $R$ largest groups. The
remaining $K-R$ groups comprise a total of $E$ exceptions, with

$E=\sum_{a=R+1}^{K}N_{a}.$ (3)

We estimate the requested memory size as

$I=CR+E,$ (4)

where $C$ is the only free parameter of the model. It obeys the inequality
$C>1$, expressing our expectation that it takes more effort and memory size to
remember a full rule than to remember an exception. In particular, the above
inequality ensures that single verbs, belonging to groups with $N_{a}=1$, are
automatically considered as exceptions. Minimising $I$ with respect to $R$
should yield the optimal number of rules.

###  2.1 Exponential size distribution

Consider first the situation where group sizes have an exponential asymptotic
decay of the form

$N_{a}\approx AN{\rm e}^{-\mu a},$ (5)

for some constants $A$ and $\mu$. Since group sizes are obviously integers, we
need to take the integer part of the right-hand side of (5). However,
neglecting this subtlety, we get accurate asymptotic estimates for the
quantities of interest in the realistic situation where the total number $N$
of verbs is large, whereas the parameter $\mu$ is small. Setting $N_{K}=1$
yields an estimate for the total number of groups,

$K\approx\frac{1}{\mu}\ln AN.$ (6)

For a given number $R$ of rules, (3) yields

$E\approx\frac{AN}{{\rm e}^{\mu}-1}{\rm e}^{-\mu R}.$ (7)

The total memory size $I$ is minimal for

$R\approx\frac{1}{\mu}\ln\frac{\mu AN}{C({\rm e}^{\mu}-1)}.$ (8)

This approach predicts that the number of rules grows logarithmically with the
total number of verbs. Considering the relatively small number of conjugation
rules in most world languages, this slow growth of the number of rules makes
very good sense. Our approach also predicts that the number $E$ of exceptions
saturates to the finite limit

$E\approx\frac{C}{\mu}.$ (9)

The actual integer value of $E$ oscillates around the above limit, which, we
note, is much smaller than the threshold (1) predicted by the tolerance
principle.

###  2.2 Power-law size distribution

If group sizes have a power-law decay, namely

$N_{a}\approx\frac{BN}{a^{\theta+1}},$ (10)

with an arbitrary positive exponent $\theta$, we have

$K\approx(BN)^{1/(\theta+1)}.$ (11)

For a given number $R$ of rules, (3) yields

$E\approx\frac{BN}{\theta R^{\theta}}.$ (12)

The memory size $I$ is minimal for

$R\approx\left(\frac{BN}{C}\right)^{1/(\theta+1)}.$ (13)

We have then

$E\approx\frac{C}{\theta}\,R.$ (14)

In this case, the numbers $R$ of rules and $E$ of exceptions grow
proportionally to each other. Their common growth law is subextensive in the
total number $N$ of verbs, and characterised by the growth exponent
$1/(\theta+1)$. This prediction for $E$ is again much smaller than the
threshold (1) predicted by the tolerance principle.

##  3 A dynamical model with growth

This section contains the first of several dynamical approaches to the
evolution of grammar rules and exceptions. We adopt a chronological viewpoint
which assumes that new verbs are added sequentially to the lexicon. Each new
verb typically joins an existing group and follows its conjugation rules,
whereas it rarely, if ever, forms a new group.

This model is freely inspired from the theory of growing networks by
preferential attachment, proposed by Barabasi and Albert [18, 19]. Among
various extensions of the model [20, 21], Bianconi and Barabasi [22, 23] have
shown that the addition of a fitness or attractiveness parameter
characterising each node greatly enriches the model; among other things, it
may induce a condensation phase transition.

Thus, new verbs enter the lexicon sequentially as new nodes in growing
networks. At any given instant, there are $K$ verb groups, indexed
$a=1,\dots,K$. Group $a$ contains $N_{a}$ verbs subject to specific grammar
rules governing their conjugation. The total number of verbs then reads222This
number will be used as an effective measure of ‘time’. In this work, we never
directly compare real (i.e., historical) time to the effective time variables
parametrising the evolution in all our models.

$N=\sum_{a=1}^{K}N_{a}.$ (15)

In this approach, exceptions are not considered explicitly, so that the number
$K$ of groups is identical to the number $R$ of rules.

The new verb number $(N+1)$ joins group $a$ with probability

$p_{a}=\frac{\eta_{a}(N_{a}+c)}{Z(N)},$ (16)

where:

  * •

The first factor $\eta_{a}$ is the intrinsic attractiveness (or fitness)
parameter of group $a$. It is fixed once and for all at the birth of group
$a$, and embodies the Darwinian fit-get-richer effect.

  * •

The second factor $(N_{a}+c)$ is dynamical, in the sense that it grows in the
course of time. It embodies the rich-get-richer, or Matthew, effect. The
constant $c$ is the initial attractiveness of an empty group [20, 21].

  * •

The denominator

$Z(N)=\sum_{a}\eta_{a}(N_{a}+c)$ (17)

ensures the normalisation of the attachment probabilities $p_{a}$ at all
times.

###  3.1 Evolution of number of groups

A new verb group is born, i.e., $K$ is changed to $K+1$, whenever the new verb
starts it, instead of joining an existing one. The geometric picture behind
this is the growth of a forest of $K$ trees, where the joining of a new verb
to an existing group $a$ corresponds to the growth of the $a$th tree, whereas
a newborn tree appears whenever the incoming verb itself starts a new group.

The birth of a new group takes place with probability $c\eta_{K+1}/Z(N)$. In
the late stages of the dynamics, when group sizes are large, the following
deterministic growth law emerges

$\frac{{\rm d}K}{{\rm d}N}\approx\frac{c}{N},$ (18)

yielding

$K\approx c\ln N.$ (19)

The logarithmic growth laws (8) and (19) substantiate our intuition that the
birth of a new grammar rule is a rare event.

###  3.2 Evolution of group sizes

Consider again the late stages, where group sizes are typically large. For a
given draw of the attractiveness parameters $\eta_{a}$ of the groups, the
stochastic rules defining our growth model reduce asymptotically to the
deterministic growth equations

$\frac{{\rm d}N_{a}}{{\rm d}N}\approx\frac{\eta_{a}N_{a}}{Z(N)},$ (20)

with

$Z(N)\approx\sum_{a}\eta_{a}N_{a}.$ (21)

Discarding the rare events where a new verb group is born, so that the number
$K$ of groups remains constant, we rank verb groups according to decreasing
attractiveness ($\eta_{1}>\eta_{2}>\dots>\eta_{K}$). The size of the most
attractive group grows as $N_{1}\approx N$. We have therefore
$Z(N)\approx\eta_{1}N$, so that the sizes of the other groups ($a=2,\dots,K$)
asymptotically obey

$\frac{{\rm d}N_{a}}{{\rm d}N}\approx\frac{\eta_{a}N_{a}}{\eta_{1}N},$ (22)

and hence

$N_{a}\sim N^{\beta_{a}},$ (23)

with

$\beta_{a}=\frac{\eta_{a}}{\eta_{1}}.$ (24)

Our prediction is that, apart from the most favoured one, group sizes grow
with a subextensive power-law, the growth exponents $\beta_{a}<1$ being given
by attractiveness ratios. A similar power-law growth scenario with variable
exponents holds in the Bianconi and Barabasi model of a growing network [22,
23].

This subextensive growth law (23) with continuously variable exponents
$\beta_{a}$ suggests a smooth crossover between rules and exceptions, instead
of a sharp line of demarcation dividing the two (see the static approach of
Section 2). Notice once again that the predicted sizes of all unfavoured
groups are much smaller than the threshold (1) involved in the tolerance
principle.

##  4 A dynamical model with growth and conversions

This second dynamical approach describes a later stage in the evolution of a
mature language. Verbs might spontaneously change groups, converting, for
instance, from an irregular to a regular form. This conversion mechanism
competes with the growth mechanism introduced in Section 3; it may result in
the enrichment and eventual dominance of a verb group which is not per se the
most attractive. This is a manifestation of the phenomenon of winning against
the odds, which we have explored in various contexts [24, 25, 26].

Again, we discard the rare events where a new group of verbs is born, so that
the number $K$ of groups is constant, and the group label $a$ runs from 1 to
$K$. The total rate of conversions from group $b$ to group $a$ is assumed to
be proportional to the sizes of both groups. It therefore reads
$C_{ab}N_{a}N_{b}$, where the individual conversion rates $C_{ab}$ are the
entries of a constant skew-symmetric conversion matrix of size $K\times K$.

In the presence of conversions, the evolution equations (20) therefore read

$\frac{{\rm d}N_{a}}{{\rm
d}N}=N_{a}\left(\frac{\eta_{a}}{Z(N)}+\frac{1}{N^{2}}\sum_{b}C_{ab}N_{b}\right),$
(25)

with

$N=\sum_{a}N_{a},\quad Z(N)=\sum_{a}\eta_{a}N_{a}.$ (26)

The first and second terms in the parentheses in (25) respectively describe
the competing growth and conversion mechanisms. The second term has been
rescaled by $1/N^{2}$, in order to ensure that the strengths of both competing
mechanisms remain comparable in the regime of large $N$, i.e., for very mature
languages. The description of this conversion mechanism is similar to that
used in earlier work on the coexistence of two or more languages in
competition [27].

The relative sizes of the various verb groups, defined as the ratios

$x_{a}=\frac{N_{a}}{N}\quad(a=1,\dots,K),$ (27)

obey the reduced evolution equations

$N\frac{{\rm d}x_{a}}{{\rm
d}N}=x_{a}\left(\frac{\eta_{a}}{z}-1+\sum_{b}C_{ab}x_{b}\right),$ (28)

with the sum rules

$\sum_{a}x_{a}=1,\quad\sum_{a}\eta_{a}x_{a}=z.$ (29)

The dynamical system (28) shares a remarkable property with that encountered
in earlier work [27]. For generic values of the attractiveness parameters
$\eta_{a}$ and the conversion rates $C_{ab}$, the coupled evolution equations
(28) have a single attractor, which consists of an attractive fixed point
${\bm{x}^{\star}}=\\{x_{a}^{\star}\\}$. In general, some components
$x_{a}^{\star}$ are positive, whereas others vanish.

  * •

If $x_{a}^{\star}>0$, group $a$ is said to be a survivor. Its size grows
asymptotically in proportion to the total number of verbs, as

$N_{a}\approx Nx_{a}.$ (30)

  * •

If $x_{a}^{\star}=0$, group $a$ is said to be extinct. According to the
deterministic equations (28), $x_{a}$ falls off exponentially in $N$. In the
microscopic stochastic context, this means that the group concerned goes
extinct in a finite time.

The unique attractor of the dynamical system (28) defines the pattern of
survivors. Its principal characteristic is the number $M$ of survivors. Some
specific sets of model parameters can be chosen so as to have either $M=1$
(corresponding to the emergence of a grammatical consensus, in the sense that
only one group survives), or $M=K$ (all groups survive), or an arbitrary
number of groups in the range $M=1,\dots,K$, survive. We emphasise that in the
present setting, the number $M$ of surviving groups is identical to the number
$R$ of rules.

From here on, we consider the specific situation where verbs convert from less
to more regular forms. This is motivated by the observed general tendency for
irregular verbs in most languages to ‘regularise’ with time (see e.g. [7]),
even when the former are well established (see, for example, the gradual
regularisation of the past participle ‘wed’ to an increasingly accepted
‘wedded’ [7]).

For simplicity, we now rank verb groups according to decreasing regularity,
$a=1$ being the most regular, and $a=K$ the most irregular group. The
conversion mechanism then takes the form of a simple descent at some constant
rate $g$. Within this framework, the entries of the conversion matrix read

$C_{ab}=\left\\{\matrix{+}g\quad&(a<b),\cr
0\hfill(a=b),\cr-g\hfill(a>b).\right.$ (31)

###  4.1 The case of two verb groups in competition

We consider first the case of two verb groups with attractiveness parameters
$\eta_{1}$ and $\eta_{2}$. For $K=2$, the reduced evolution equations (28)
read

$\displaystyle N\frac{{\rm d}x_{1}}{{\rm
d}N}=x_{1}\left(\frac{\eta_{1}}{z}-1+gx_{2}\right),$ $\displaystyle
N\frac{{\rm d}x_{2}}{{\rm d}N}=x_{2}\left(\frac{\eta_{2}}{z}-1-gx_{1}\right),$
(32)

with

$x_{1}+x_{2}=1,\quad\eta_{1}x_{1}+\eta_{2}x_{2}=z.$ (33)

The two control parameters are the attractiveness ratio

$q=\frac{\eta_{1}}{\eta_{2}}$ (34)

and the conversion rate $g$.

Figure 1 shows the phase diagram of the model in the $q$–$g$ plane. In the
absence of conversions $(g=0$), only the most attractive verb group survives,
namely group 1 for $q>1$ and group 2 for $q<1$. The presence of conversions
triggers several novel phenomena. First, both groups survive simultaneously in
an intermediate range

$1-q<g<\frac{1-q}{q},$ (35)

whose boundaries are respectively shown in blue and green. In this coexistence
range we have

$x_{1}^{\star}=\frac{g+q-1}{(1-q)g},\quad
x_{2}^{\star}=\frac{1-q(1+g)}{(1-q)g}.$ (36)

The first fraction $x_{1}^{\star}$ (resp. the second fraction $x_{2}^{\star}$)
vanishes continuously as the blue (resp. green) boundary is approached.
Second, the competition between the growth and conversion mechanisms allows
for the emergence of survivors against the odds; thus, in the domain between
the green curve and the vertical dashed line (labelled in red), group 1
survives despite being unfavoured.

Figure 1:  Phase diagram of the model of two verb groups with growth and
conversions in the $q$–$g$ plane. Blue and green curves: boundaries of
coexistence range. Vertical dashed line: neutral line where
$\eta_{1}=\eta_{2}$. Numbers between parentheses: labels of surviving
group(s). Red label: survivor against the odds.

###  4.2 The general case of $K$ verb groups

For a higher number $K\geq 3$ of competing verb groups, the determination of
the full multidimensional phase diagram of the model is intractable.

We consider instead a statistical ensemble, where the intrinsic attractiveness
parameters $\eta_{a}$ of verb groups are modelled as independent quenched
random variables drawn from some probability distribution. We choose for
definiteness the exponential distribution with unit width:

$f(\eta)={\rm e}^{-\eta}.$ (37)

The main features of the model, including the logarithmic growth law (39),
would remain qualitatively unchanged for any bounded or rapidly decaying
attractiveness probability distribution.

The survivors thus form a random pattern, whose statistics depend only on the
number $K$ of groups and the conversion rate $g$. This pattern can be easily
predicted at small and large $g$. If $g$ is either zero or very small, the
growth mechanism dominates and only the most attractive group survives. If $g$
is very large, the conversion mechanism wins; now the only survivor is the
most regular group. In the intermediate regime where the conversion rate $g$
is moderate, so that growth and conversion are comparable, several survivors
may coexist.

We first focus on the mean number $\langle M\rangle$ of survivors, where the
mean value is taken over the distribution (37) of attractiveness parameters.
In the case of two verb groups ($K=2$), the exactly known phase diagram of the
model (see Figure 1) yields

$\langle M\rangle=\left\\{\matrix{}{4}{4-g^{2}}\quad&(g\leq
1),\cr\cr\frac{\displaystyle g+3}{\displaystyle g+2}\quad(g\geq 1).\right.$
(38)

This expression goes to unity at small and large $g$, as expected. It takes
its maximal value, $\langle M\rangle=4/3$, at $g=1$; its first derivative is
discontinuous at this point, as indicated by the cusp in the black curve in
Figure 2. This singularity is due to the endpoint at $(q=0,g=1)$ of the blue
line in Figure 1.

Figure 2:  Mean number $\langle M\rangle$ of survivors against conversion rate
$g$ for several numbers $K$ of verb groups (see legend). Black curve: exact
analytical result (38) for $K=2$. Coloured curves: numerical data for $K=3$ to
6.

We have investigated the behaviour of larger systems ($K\geq 3$) by means of
numerical simulations, determining the value of the fixed point
${\bm{x}^{\star}}=\\{x_{a}^{\star}\\}$ for many independent draws ($10^{7}$
for each parameter set) of the attractiveness parameters $\eta_{a}$. Figure 2
shows the mean number $\langle M\rangle$ of survivors plotted against the
conversion rate $g$ for several $K$ (see legend). The black curve shows the
exact analytical result (38) for $K=2$. The other curves show the outcome of
numerical simulations. The qualitative form of the dependence of $\langle
M\rangle$ on $g$ is independent of $K$, with $\langle M\rangle$ departing
quadratically from unity at small $g$, reaching its maximum with a cusp at
$g=1$, and slowly converging back to unity at large $g$.

Figure 3 shows the maximal mean number $\langle M\rangle_{\rm max}$ of
survivors, corresponding to the conversion rate $g=1$, plotted against $\ln
K$, for $K$ up to 12. The excellent agreement with the regression line
demonstrates that this quantity grows logarithmically with the number $K$ of
groups, as

$\langle M\rangle_{\rm max}\approx A\ln K,$ (39)

with a prefactor $A\approx 0.5$. The appearance of this logarithmic law again
emphasises the conformity of our model with the general principles laid out in
earlier sections (see (8), (19)). In the present context, the clear
implication of this law is that only very few verb groups survive from an
initial panoply of possibilities.

Figure 3:  Maximal mean number $\langle M\rangle_{\rm max}$ of survivors,
corresponding to $g=1$, plotted against $\ln K$, for a number $K$ of verb
groups up to 12. Symbols: numerical data. Circled symbol: exact value $\langle
M\rangle_{\rm max}=4/3$ for $K=2$. Full line: least-square fit with slope
0.498.

In addition to the number $M$ of survivors, the whole pattern of survivors is
also of interest. In the absence of conversions, only the most attractive
group survives. As the conversion rate $g$ increases, survivors against the
odds [24, 25, 26] – i.e., those which do not belong to the most attractive
groups – gradually become more and more frequent. We define the variable
$P_{\rm para}$, the ‘paradoxical’ probability that the most attractive group
does not belong to the pattern of survivors, to explore this issue further.

In the case of two verb groups ($K=2$), the paradoxical probability $P_{\rm
para}$ is nothing but the statistical weight of the region lying between the
green curve and the vertical dashed line in the phase diagram of the model
(see Figure 1), which is evaluated as

$P_{\rm para}=\frac{g}{2g+4}.$ (40)

The relevant region does not touch the endpoint $(q=0,g=1$). Hence, and at
variance with the mean number of survivors, $P_{\rm para}$ has a smooth
dependence on $g$.

Figure 4 shows the paradoxical probability $P_{\rm para}$ plotted against the
conversion rate $g$ for several values of $K$ (see legend). The black curve
shows the exact analytical result (40) for $K=2$. The other curves show the
outcome of numerical simulations. As expected, $P_{\rm para}$ increases
steadily as a function of the conversion rate $g$, departing linearly from
zero at $g=0$, exhibiting a shoulder for $g$ slightly below unity, and slowly
saturating to the limit value

$P_{\infty}=\frac{K-1}{K}$ (41)

at very large $g$. For infinitely large $g$, there is indeed only one
survivor, namely the most regular group ($a=1$), whose probability of also
being the most attractive one is $1/K$.

Figure 4:  Paradoxical probability $P_{\rm para}$ against conversion rate $g$
for several numbers $K$ of verb groups (see legend). Black curve: exact
analytical result (40) for $K=2$. Coloured curves: numerical data for $K=3$ to
6.

##  5 Languages in contact

In the previous section, we showed that conversions provided a mechanism for
gradual regularisation of initially irregular grammatical forms. This occurs
with increasing usage within the same language [7], i.e. it is an intra-
language mechanism.

On the other hand, our initial motivation for this study was the unlikely
survival of irregular past participles such as ‘stuck”, when the favourite to
win at the time was clearly ‘sticked’ [12]. We suggest that this emergence is
due to contact with other languages, i.e., that it is attributable to an
inter-language mechanism.

As mentioned in the Introduction, the field of contact linguistics concerns
such linguistic influence. We cite below a couple of instances of languages in
prolonged contact, which have led to their deep modification [3]. The first
concerns the contact of Old English with Norse, and then that of Middle
English with Norman French, which were the precursors of English in its
present form. The second concerns the Balkan Sprachbund,333This term, meaning
‘union of languages’ refers to a situation where there is prolonged contact
across geographically contiguous language communities [3]. where speakers of
Greek, Romanian, and various Slavic languages were in contact for almost a
millennium. In both cases, there has been an appreciable amount of convergence
in the morphology and syntax of the languages in contact, despite the sizeable
differences between them originally.

In the following, we model the effect of such contact among a given family of
languages. We represent all verb groups of this linguistic family as the nodes
of a graph, and the couplings between them as bonds connecting these nodes.
Bonds which connect nodes pertaining to the same language represent the
conversion mechanism introduced in Section 4, whereas those joining nodes
pertaining to different languages represent the new ingredient of linguistic
contact. This description of language contacts is static, in the sense that
the topology of the network does not change over the course of time. It
therefore describes e.g. the influence of Norse on Old English, or that of
French on Middle English, but not both together.

For the sake of simplicity, we write the corresponding evolution equations in
the following linear form:

$\frac{{\rm d}N_{a}}{{\rm
d}N}=\frac{\eta_{a}}{Z(N)}\left(N_{a}+\sum_{b}g_{ab}N_{b}\right).$ (42)

These equations present analogies and differences with the evolution equations
(25). In both cases $\eta_{a}$ is the intrinsic attractiveness parameter of
group $a$. Within the present linear setting, the couplings $g_{ab}$ represent
the strength of all conversion and contact effects described above, whereas in
(25) the conversion mechanism involves a more traditional bilinear form. In
general, the matrix ${\bm{g}}=\\{g_{ab}\\}$ is not symmetric. More
importantly, it is expected to be sparse, as only similar grammar rules
pertaining to different languages will influence each other. Furthermore, the
couplings $g_{ab}$ connecting nodes pertaining to different languages, i.e.,
representing contact between distinct languages, must be positive, whereas
those between nodes pertaining to the same language may take both signs.
Finally, the denominator $Z(N)$ is there to ensure that the sum rule

$\sum_{a}N_{a}=N$ (43)

holds, where $N$ is the total number of verbs in all languages of the family
under consideration.

Introducing the reduced effective time

$s=\int_{1}^{N}\frac{{\rm d}N^{\prime}}{Z(N^{\prime})}$ (44)

brings the evolution equations (42) to the form

$\frac{{\rm d}N_{a}}{{\rm
d}s}=\eta_{a}\left(N_{a}+\sum_{b}g_{ab}N_{b}\right).$ (45)

These equations are autonomous, in the sense that they no longer involve any
explicit time dependence. They can be recast in matrix form:

$\frac{{\rm d}N_{a}}{{\rm d}s}=\sum_{b}M_{ab}N_{b},$ (46)

with

$M_{ab}=\eta_{a}(\delta_{ab}+g_{ab}),$ (47)

where $\delta_{ab}$ is the Kronecker symbol.

We are chiefly interested in the late-time regime of the evolution described
by the contact network under consideration. There, all group sizes grow
asymptotically as

$N_{a}\approx V_{a}\,{\rm e}^{\lambda s},$ (48)

where $\lambda$ is the largest eigenvalue of the constant dynamical matrix
${\bm{M}}=\\{M_{ab}\\}$, whereas the amplitudes $V_{a}$ are proportional to
the components of the associated right eigenvector, such that

$\lambda V_{a}=\sum_{b}M_{ab}V_{b}.$ (49)

The total number of verbs thus grows as

$N\approx V\,{\rm e}^{\lambda s},\quad V=\sum_{a}V_{a}.$ (50)

We expect that $\lambda$ and the $V_{a}$ are positive in realistic
circumstances, even when the dynamical matrix ${\bm{M}}$ is not symmetric.
These expectations have been confirmed by a range of numerical explorations.

The relative sizes of the groups in the late-time regime are therefore
dictated by the components $V_{a}$ of the leading eigenvector of the dynamical
matrix ${\bm{M}}$. The spectral problem at hand presents a formal analogy with
Anderson localisation within the tight-binding formalism [28, 29, 30]. More
precisely, the dynamical matrix ${\bm{M}}$ is analogous to the tight-binding
Hamiltonian ${\cal H}$ describing the motion of a single electron in a random
potential. The largest eigenvalue $\lambda$ is analogous to the ground-state
energy $E_{0}$ of the electron. Finally, the components $V_{a}$ of the
associated right eigenvector are analogous to the components of the ground-
state wavefunction of this one-body problem. From a very general viewpoint,
the geometry of the underlying network and the distribution of the couplings
determine the nature (extended, localised, fractal, etc.) of the wavefunction.
The analogy of our problem with that of Anderson localisation implies that
differing network geometries and model parameters will lead to a rich
diversity of behaviour in the asymptotic distribution of verb group sizes.

###  5.1 The linear chain

We first investigate the idealised situation where nodes form an infinite
linear chain, with asymmetric couplings between nearest neighbours. In this
context, (49) reads

$\lambda V_{n}=\eta_{n}(V_{n}+g_{n,n-1}V_{n-1}+g_{n,n+1}V_{n+1}),$ (51)

with obvious notations. For specificity, we consider the case where the node
at the origin is favoured, in the sense that its attractiveness parameter is
$\eta_{0}=1$, whereas all other nodes have $\eta_{n}=q<1$.

Consider the pristine case where all couplings are the same
$(g_{n,n-1}=g_{n,n+1}=g$). In this simple situation, the analogy with the
tight-binding model is as follows. The favoured node at the origin acts as an
attractive impurity, where the wavefunction is expected to be largest. It can
be checked that the largest eigenvalue $\lambda$ corresponds to a localised
impurity state of the form

$V_{n}=z^{|n|},$ (52)

which falls off exponentially with the distance $|n|$ to the origin. The
eigenvalue $\lambda$ and the decay constant $z$ are determined by the two
equations

$\lambda=1+2gz=q\left(1+g\left(z+\frac{1}{z}\right)\right),$ (53)

hence

$\displaystyle z$ $\displaystyle=$ $\displaystyle\frac{2qg}{w+1-q},$
$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle
1+\frac{4qg^{2}}{w+1-q},$ (54)

with the notation

$w=\sqrt{(1-q)^{2}+4q(2-q)g^{2}}.$ (55)

In the more general case, where the couplings $g_{n,n-1}$ and $g_{n,n+1}$ are
arbitrary and modelled as, say, quenched random variables, the above picture
of a localised impurity state around the favoured node, with an exponentially
decaying wavefunction, remains qualitatively correct. To get more quantitative
results, we look at the theory of fluctuations in one-dimensional Anderson
localisation (see [31] and the references therein). The latter theory predicts
that the amplitude $V_{n}$ is distributed log-normally in the asymptotic
limit. More precisely, for a large distance $n$, the logarithmic ratio

$L_{n}=-\ln\frac{V_{n}}{V_{0}}$ (56)

is approximately distributed according to the Gaussian law

$f(L_{n})\approx\frac{1}{\sqrt{2\pi\gamma_{2}n}}\,\exp\left(-\frac{(L_{n}-\gamma_{1}n)^{2}}{2\gamma_{2}n}\right),$
(57)

where $\gamma_{1}$ and $\gamma_{2}$ are the first two Lyapunov exponents of
the problem, such that444The impurity state (52) of the pristine case
described above fits within this scheme, with $\gamma_{1}=-\ln z$ and
$\gamma_{2}\to 0$.

$\langle L_{n}\rangle\approx\gamma_{1}n,\quad\mathop{\rm
var}L_{n}\approx\gamma_{2}n.$ (58)

The probability of winning against the odds for a node at a distance $n$ from
the favoured one sitting at the origin is defined as

$P_{n}={\rm Prob}(V_{n}>V_{0})={\rm Prob}(L_{n}<0).$ (59)

Using (57), this becomes

$P_{n}\approx\frac{1}{2}\mathop{\rm
erfc}\left(\gamma_{1}\sqrt{\frac{n}{2\gamma_{2}}}\right).$ (60)

This expression falls off exponentially with distance $n$, according to

$P_{n}\sim{\rm e}^{-\mu n},$ (61)

with

$\mu=\frac{\gamma_{1}^{2}}{2\gamma_{2}}.$ (62)

In our context, the above analysis suggests that the probability of finding an
unlikely winner (e.g. an irregular grammatical form) decays exponentially with
the graph distance $n$ between that form and the closest most regular (or
otherwise favoured) form. The illustration of our formalism in the idealised
geometry of an infinite chain will serve as a template for the analysis in the
next subsections, where we will formulate our problem in more realistic
settings.

###  5.2 A two-dimensional ‘toy’ network

We now look for winners against the odds in the setting of a $3\times 3$
network involving three related languages, denoted A, B and C (see Figure 5).
Individual nodes in a row correspond to verb groups in a given language; we
consider thus a total of three verb groups in each of the three languages.
This geometry is far more realistic than the previous one of an infinite
linear chain, even though it is not motivated by a specific example. Three is
indeed the right order of magnitude for the number of verb groups, and more
generally for competing grammatical forms. It is also expected to be a good
estimate of the number of closely related languages with significant borrowing
exchanges. One may think of English, Dutch and German.

The conjugation rules of the verb groups (in different languages) which are
aligned vertically in a column are assumed to be very similar to each other. A
single verb group is favoured in each language (large symbols), i.e., its
attractiveness reads $\eta_{a}=1$, whereas the other two groups have
$\eta_{a}=q<1$ (small symbols). The couplings (blue lines) are limited to
nearest neighbours. It is when the most favoured groups are chosen to be
different in the three languages that non-trivial behaviour emerges.

Figure 5:  Schema of the toy network with 3 languages, denoted A, B and C, and
3 verb groups in each language. Large symbols: favoured nodes. Small symbols:
unfavoured nodes. Blue lines: bonds carrying either symmetric or asymmetric
couplings.

We study both symmetric and asymmetric isotropic random couplings. In the
symmetric case, the couplings $g_{ab}=g_{ba}$ along the 12 bonds are
independently drawn from the exponential law of parameter $\Delta$:

$f_{\Delta}(g)=\frac{{\rm e}^{-g/\Delta}}{\Delta}.$ (63)

In the asymmetric case, both $g_{ab}$ and $g_{ba}$ are two independent
positive random variables drawn from the above distribution, so that there are
altogether 24 random couplings.

We consider the fates of nodes 1 and 9, which are furthest from the favoured
nodes, and so are least likely to win. For this purpose, we examine the
logarithmic ratio (see (56))

$L=-\ln\frac{V_{1}}{V_{3}}.$ (64)

The high symmetry of the network implies that considering the alternative
ratios $V_{1}/V_{7}$, $V_{9}/V_{3}$ and $V_{9}/V_{7}$ would yield
statistically identical results. A negative value of $L$ implies that node 1
wins against the odds; in other words, we have $V_{1}>V_{3}$, despite mode 3
being the most attractive in language A (Note that there is no direct coupling
between nodes 1 and 3). The probability that node 1 wins against the odds
therefore reads

$P_{\rm win}={\rm Prob}(L<0).$ (65)

Figure 6:  Probability distribution of the logarithmic ratio $L$ (see (64)),
for a fixed small coupling width $\Delta=0.1$ and several attractiveness
ratios $q$ (see legend). Vertical dashed lines: neutral value ($L=0$). Upper
panel: symmetric couplings. Lower panel: asymmetric couplings.

Figure 6 shows plots of the probability distribution $f(L)$ of the logarithmic
ratio $L$ for symmetric (top) and asymmetric (bottom) couplings, with a fixed
small coupling width $\Delta=0.1$ and several attractiveness ratios $q$ (see
legend). Data have been obtained by numerically solving the $9\times 9$
eigenvalue equation (49) for many independent draws of the random couplings.
Even though the network size is small, the overall shape of the plotted
distributions is close to the asymptotic Gaussian profile (57) obtained on the
infinite chain. The asymmetry of the couplings appears to play a rather minor
role, in the sense that both series of curves are rather similar to one
another. When $q=1$ (black curves), so that the attractiveness is the same
throughout the network, the distributions are symmetric around the neutral
value $L=0$ (vertical dashed lines), and $P_{\rm win}=1/2$, as expected. As
$q$ is decreased, the distributions shift progressively to the right, with
$\langle L\rangle$ growing steadily with the difference $1-q$, whereas their
shapes remain roughly unchanged. The portion of the curves corresponding to
negative values of $L$ shrinks accordingly, indicating that winning against
the odds becomes increasingly difficult as the attractiveness contrast $1-q$
is increased. This observation is made quantitative in Figure 7, showing that
the probability $P_{\rm win}$ of winning against the odds (see (65)) falls off
more rapidly than exponentially with the attractiveness contrast $1-q$.

Figure 7:  Plot of $-\ln(2P_{\rm win})$ against the attractiveness contrast
$1-q$, where $P_{\rm win}$ is the probability that node 1 wins against the
odds (see (65)). Symbols: data extracted from those shown in Figure 6 for
symmetric and asymmetric couplings. Blue curves: quadratic fits suggesting
that $P_{\rm win}$ falls off faster than exponentially as a function of the
attractiveness contrast $1-q$.

The present network embedding, with its explicit depiction of contact with
neighbouring languages, is more apposite than the setting of the infinite
chain for the problem of unlikely winners among grammatical forms. However,
the striking resemblance between the distributions presented in Figure 6 for
our rather small toy network and the Gaussian profile (57) for the infinite
chain is a strong indicator of the relevance of the theory of one-dimensional
Anderson localisation to the present problem. This analogy enables us to
reduce the problem of finding unlikely winners in a dynamical system, where
many agents are in simultaneous competition, to one involving the exponential
decay of $P_{\rm win}$ with distance from the nearest favoured node. We will
make good use of this simplification in Section 5.3, when the collective
competition intrinsic to our problem is embedded on complex networks.

###  5.3 Complex networks

A ‘map’ of linguistic influences can be expected to have a complex topography,
comprising regions of strong linguistic contact (i.e., high connectivity), as
well as relatively isolated regions with weak or no linguistic contact. It is
the former that are relevant in the context of the question we ask: can the
emergence of irregular grammatical forms which survive against the odds be
attributed to the influence of ‘neighbouring’ languages? We therefore home in
on what will be the most sophisticated, as well as the most abstract, version
of our model; here, all grammar rules (governing verb conjugation, in this
instance) in a family of related languages are embedded in a complex network
(see e.g. [32, 33]).

We choose for definiteness the geometry of random regular graphs (see e.g.
[34, 35]). These are randomly connected networks where each node has the same
prescribed degree $k\geq 3$, i.e., each node is connected to exactly $k$ other
nodes. The main qualitative features of the model, to be described below,
would remain essentially the same for other structureless models of complex
networks, where degrees $k$ have mild fluctuations around the mean degree
$\langle k\rangle$. Scale-free networks, where node degrees $k$ obey a broad
power-law distribution, might however give rise to a different phenomenology.

Some of the connections involve two nodes in the same language, while others
embody contact with neighbouring languages; as before, the bonds between nodes
in the same language represent conversion, while those connecting nodes
belonging to different languages represent interlingual contact. We do not –
unlike for the toy network of the previous subsection – specify which is
which; we focus instead only on the nodes which win against the odds. The
distribution of attractiveness parameters is again assumed to be bimodal. Some
fraction $\rho$ of the nodes are favoured, and have attractiveness parameters
$\eta_{a}=1$, whereas all other nodes have the same smaller attractiveness
$\eta_{a}=q$. The microscopic distribution of couplings $g_{ab}$ entering the
dynamical matrix is left unspecified. We indeed rely on the key outcome of the
analysis of the one-dimensional setting, i.e., the exponential falloff of the
probability of winning against the odds with distance $n$ to the nearest
favoured node (see (61)). For specificity, we assume a purely exponential
decay law of the form

$P_{n}={\rm e}^{-\mu n}.$ (66)

The parameter $\mu$ increases with the attractiveness contrast $1-q$, in a way
which depends on the distribution of the couplings $g_{ab}$, as suggested by
(62). The greater the attractiveness contrast, therefore, the less will be the
likelihood of finding winners against the odds.

The central question we wish to address concerns the probability $P_{\rm win}$
that any given node (e.g., the origin O) wins against the odds, as a function
of the model parameters $\rho$ and $\mu$. This can be written as

$P_{\rm win}=\sum_{n\geq 1}f_{n}P_{n},$ (67)

where $f_{n}$ is the distribution of the distance $n$ of the origin O to the
nearest favoured node. Note that the term $f_{0}=\rho$ does not enter the sum
in (67), since this corresponds to the current node being favoured, and
therefore does not contribute to winners against the odds.

The distribution $f_{n}$ is evaluated as follows. The number $M_{n}$ of nodes
situated at distance at most $n$ from the origin O reads

$\displaystyle M_{n}$ $\displaystyle=$ $\displaystyle
1+k+k(k-1)+\cdots+k(k-1)^{n-1}$ (68) $\displaystyle=$
$\displaystyle\frac{k(k-1)^{n}-2}{k-2}.$

This result relies on the property that a random regular graph is locally
treelike, so that cycles are rare, and therefore negligible. In other words,
local properties of random regular graphs coincide with those of the infinite
Cayley tree of degree $k$, also known as the Bethe lattice in the physics
literature. The exponential growth law (68) of the number of nodes with
distance is an expression of the fact that the fractal dimension of the
network is formally infinite. The diameter $n$ of a large network indeed grows
logarithmically with its total mass $M$, as

$n\approx\frac{\ln M}{\ln(k-1)}.$ (69)

This is a manifestation of the so-called small-world effect [32, 33].

We now use the result (68) to evaluate the probability $F_{n}$ that the
distance between O and the nearest favoured node is larger than $n$. This is
identical to the probability that all the $M_{n}$ nodes in the first $n$
shells around O are unfavoured, so that $F_{n}=(1-\rho)^{M_{n}}$. The
distribution that we seek to evaluate is then nothing but the difference

$f_{n}=F_{n-1}-F_{n}\qquad(n\geq 1),$ (70)

i.e.,

$f_{n}=(1-\rho)^{M_{n-1}}\left(1-(1-\rho)^{k(k-1)^{n-1}}\right).$ (71)

The probability of winning against the odds is now readily obtained by
inserting (66) and (71) into (67).

An interesting scaling regime takes place when the density $\rho$ of favoured
nodes is small. There, the typical distance to the nearest favoured node is
large. The distribution $f_{n}$ is peaked around a well-defined mean distance,
which grows logarithmically as

$n^{\star}\approx\frac{|\ln\rho|}{\ln(k-1)},$ (72)

with a bounded variance around this mean value. Two distinct regimes emerge,
according to whether the sum entering (67) is dominated by the first few
values of $n$ or by $n\approx n^{\star}$. These regimes are defined by
comparing two inverse lengths, viz. $\mu$, characterising the exponential
decay law (66) of the winning probability $P_{n}$, and $\ln(k-1)$,
characterising the exponential proliferation (68) of nodes around a given
node.

  * •

For $\mu>\ln(k-1)$, the distant-dependent probabilities $P_{n}$ fall off fast
enough that the sum in (67) is dominated by finite values of $n$, i.e., $n\ll
n^{\star}$. Winners against the odds are actually not too far from being
likely winners, since the nearest favoured node is nearby. For small $\rho$,
we have

$f_{n}\approx k(k-1)^{n-1}\rho,$ (73)

and therefore

$\displaystyle P_{\rm win}$ $\displaystyle\approx$ $\displaystyle\sum_{n\geq
1}k(k-1)^{n-1}{\rm e}^{-\mu n}\rho$ (74) $\displaystyle\approx$
$\displaystyle\frac{k\rho}{{\rm e}^{\mu}-(k-1)}$

starts increasing linearly in $\rho$.

  * •

For $\mu<\ln(k-1)$, the distance-dependent probabilities $P_{n}$ fall off
slowly enough that the sum in (67) is dominated by $n\approx n^{\star}$.
Winners in this case are genuinely against the odds, since the nearest
favoured node is quite distant. We have

$P_{\rm win}\sim P_{n^{\star}}\sim\rho^{\alpha},$ (75)

where the growth exponent $\alpha$ depends linearly on $\mu$, according to

$\alpha=\frac{\mu}{\ln(k-1)}.$ (76)

  * •

In the borderline case where $\mu=\ln(k-1)$, the summand

$f_{n}P_{n}\approx\frac{k\rho}{k-1}$ (77)

is nearly flat up to $n^{\star}$. We thus obtain

$P_{\rm win}\approx\frac{k\rho|\ln\rho|}{(k-1)\ln(k-1)}.$ (78)

We pause briefly to consider the significance of the parameter $\alpha$ (see
(76)). A similar ratio characterised survivors in an earlier model of
competitive dynamics on networks [25], where the probability of survival of a
node depended on the ratio of its mass to its average degree distribution; the
‘heavier’ the node, and the less connected it was to others, the better its
chances of survival. In the present case, the role of the mass in [25] is
played by the the attractiveness contrast $(1-q)$ (recall that
${\mu}\sim(1-q)$), with the parameter $\ln(k-1)$ representing the effect of
the (constant) degree distribution. This analogy puts our work into a more
general context: the more attractive a grammar rule is, and the less connected
it is to direct competitors, the more it is likely to persist.555The important
difference is that all survivors were considered in [25], whereas here only
the subset of survivors against the odds is considered. The same reasoning
though clearly applies to both.

Figure 8 shows plots of the probability $P_{\rm win}$ of winning against the
odds against the density $\rho$ of favoured nodes, for $k=3$ and several
values of the parameter $\mu$ (see legend). The initial rise of $P_{\rm win}$
at small $\rho$ is faster than linear for the two upper curves ($\mu<\ln 2$)
(see (75)) and linear for the two lower curves ($\mu>\ln 2$) (see (74)). The
borderline case ($\mu=\ln 2$) (see (78)) is also shown (thick black curve). In
the other limiting situation $(\rho\to 1)$, only

$f_{1}=(1-\rho)(1-(1-\rho)^{k})$ (79)

scales linearly with the difference $1-\rho$, so that

$P_{\rm win}\approx{\rm e}^{-\mu}(1-\rho).$ (80)

Figure 8:  Probability $P_{\rm win}$ of winning against the odds for a typical
node of a random regular graph, against the density $\rho$ of favoured nodes,
for $k=3$ and several values of $\mu$ (see legend).

For any value of the parameter $\mu$, the probability $P_{\rm win}$ vanishes
both for $\rho\to 0$ and for $\rho\to 1$. This must clearly be the case: the
complete absence of favoured sites will not cause winners against the odds to
be generated at all, while if nearly all sites are favoured, winners will be
very much with the odds, and not against them. We notice also that $P_{\rm
win}$ is overall larger for $\mu<\ln(k-1)$ than for $\mu>\ln(k-1)$. This ties
up with the arguments we were making above: we would expect that winners would
be more numerous when there are many unfavoured nodes under the umbrella, so
to speak, of a favoured site (see (75)) than in the opposite situation (see
(74)).

The plots in Figure 8 also clearly manifest the presence of an optimal density
$\rho^{\star}$ of favoured nodes, where $P_{\rm win}$ reaches its maximal
value $P_{\rm win}^{\star}$. Figure 9 shows a plot of this optimal density
against $\mu$ for $k=3$. At small $\mu$, $\rho^{\star}$ starts growing
linearly. At large $\mu$, we have $P_{\rm win}\approx{\rm e}^{-\mu}f_{1}$ (see
(79)), so that $\rho^{\star}$ saturates to the limiting value

$\rho_{\infty}=1-(k+1)^{-1/k}.$ (81)

(The precise values of $P_{\rm win}^{\star}$ are however less informative, as
they depend on the assumed exact exponential form with unit amplitude (66) of
$P_{n}$.) The interpretation of these plots is as follows: when $\mu$ is
small, one does not need a high density of favoured sites to achieve $P_{\rm
win}^{\star}$; the unfavoured sites themselves are close enough to being
winners. When, however, $\mu$ is large, we need a far higher density of
favoured sites to have a sphere of influence strong enough to attain $P_{\rm
win}^{\star}$. The optimal density $\rho^{\star}$ is therefore an increasing
function of $\mu$ until it saturates to $\rho_{\infty}$.

Figure 9:  Optimal density $\rho^{\star}$ of favoured nodes, such that the
probability $P_{\rm win}$ is maximal, against the parameter $\mu$ for $k=3$.
Dashed horizontal line: limiting value $\rho_{\infty}=1-4^{-1/3}\approx
0.370039$.

It turns out that the overall dependence of $P_{\rm win}$ on the density
$\rho$ of favoured nodes is always rather accurately represented by the
phenomenological formula

$P_{\rm win}\approx\rho^{\alpha_{\rm eff}}-\rho.$ (82)

Figure 10 shows a plot of the effective exponent $\alpha_{\rm eff}$, obtained
by means of a nonlinear fit, against $\mu$ for $k=3$. This effective exponent
(green curve) interpolates smoothly between the two exact growth exponents in
the low-density scaling regime (red and blue lines).

Figure 10:  Green curve: effective exponent $\alpha_{\rm eff}$ entering the
approximate formula (82), against $\mu$ for $k=3$. Red line: exact growth
exponent (76) for $\mu<\ln 2$. Blue horizontal line: exact unit exponent of
the linear law (74) for $\mu>\ln 2$. Vertical dashed line: borderline value
$\mu=\ln 2$.

We now summarise and interpret the above results. The linguistic map in a
region of strong linguistic contact, alluded to at the beginning of this
section, is constructed by embedding our model of competing grammar rules on a
sparse complex network, here chosen to be the Bethe lattice.666We mention for
completeness that the present work has only little to do with the theory of
Anderson localisation on the Bethe lattice [36], which plays a major role in
recent work on many-body localisation (see e.g. [37, 38, 39]). There, the
Bethe lattice arises as a template for the Fock space of a quantum many-body
problem. The mechanisms of competition involve both linguistic contact between
different languages and regularisation within a given language, which act
against each other. Our analysis is based on an effective description, via
analogies with Anderson localisation, of the eigenvector associated with the
largest eigenvalue $\lambda$ of the dynamical matrix ${\bm{M}}$ on this
lattice, and involves only its exponential decay with distance from the set of
favoured nodes. We have discovered the existence of a low-density scaling
regime, where irregular forms vastly outnumber regular ones and two kinds of
unlikely winners emerge: the first correspond to grammatical forms which are
linguistically close to a favoured regular form, while the second correspond
to those which are linguistically rather distant from it.

These results provide a natural framework within which the emergence of
‘stuck’ can be explained. According to Ringe [12], the competing past
participles of the verb ‘stick’ were ‘sticked’, ‘stuck’, ‘stoke’ and ‘stoked’,
and the application of the tolerance principle, which takes a one-body view of
the problem, did not predict the winner. In our formalism, all these
coexisting grammatical forms compete with each other in parallel in a
dynamical model. In the relevant dynamical regime, where irregular forms far
outnumber the only ‘regular’ competitor ‘sticked’ ($\rho$ small), the scenario
of (75) applies, and a deeply irregular form such as ‘stuck’, linguistically
very distant from the regular form ‘sticked’, emerges as a winner against the
odds.

##  6 Discussion

We have, in the above, used statistical physics methodologies to model the
evolution of grammatical rules. Starting with a very general approach, we have
provided a useful way of classifying rules and exceptions in typical
situations. A major result of this static approach is that exceptions are so
called for a good reason: i.e., that they occur rarely, their number growing
either logarithmically or as a subextensive power law in the number of items.
Our work quantifies a well-known example from linguistics, viz. the paucity of
verb groups in most world languages, by demonstrating that the birth of a new
grammatical rule (or a new verb group) is a very rare event (see the
logarithmic laws (8), (19), (39)). All of the above is in stark contrast to
the high threshold predicted by the tolerance principle, which has a nearly
extensive (i.e., nearly maximal) growth law for the number of allowed
exceptions.

The dynamical models we have presented later in the paper have corroborated
these insights; we have there focused on morphological issues in verb
conjugation, and their evolution. The two main mechanisms we have included in
our models are first, the conversion of forms from irregular to regular within
a given language, and second, the influence of ‘neighbouring’ languages with
which it is in prolonged contact. We have built up our models progressively,
adding these ingredients one at a time to observe their full consequences. Our
first dynamical model of Section 3 involves only the growth of the lexicon and
of grammar rules in a single language in the first stage of its evolution. In
Section 4, we have added a conversion mechanism, which describes the rather
universal tendency towards regularisation observed by linguists [7]. Finally,
in Section 5, we have completed the picture by adding linguistic change via
prolonged contact between similar languages, and argued that this mechanism
might well be responsible for the introduction of novel irregular forms into a
given language.

We have strongly emphasised the appearance and persistence of winners or
survivors against the odds in all our dynamical models of linguistic
evolution; these unlikely winners are precisely the irregular linguistic forms
which persist in the face of several competitors, irregular as well as
regular. All the models we present include this essential ingredient of
collective and simultaneous competition. In Sections 4 and 5, we have
quantified the probability of occurrence of these winners against the odds, as
a function of model parameters in a variety of situations. Finally, we have
constructed a model linguistic map in Section 5.3 incorporating both intra-
language regularisation and inter-language contact, and shown that forms which
are linguistically very far from favoured can indeed emerge, and persist; in
particular, this scenario allows us to explain the unlikely persistence of the
deeply irregular grammatical forms mentioned in [12]. It seems very likely
that such persistence occurs when the influence of interlingual contact
exceeds that of intralingual regularisation.

The quantitative proof of the above contention would entail the formulation by
quantitative linguists of realistic network models of language neighbourhoods.
These would require a full knowledge of linguistically appropriate
distributions of conversion and contact interactions for closely linked
language groups (see Section 5.3). Such distributions might, for example, be
obtained from numerical experiments on the phylogenesis of selected
grammatical forms (past participles in our case) in the languages concerned.
It would be most interesting to see if our model predictions are verified,
i.e., if, in the case where contact exceeds conversion, unlikely irregular
forms survive.

In summary, the main conclusions of our work are as follows. First, attractive
grammar rules survive best in the absence of strong competitors within or
outside the language concerned; second, there is an optimal density of
favoured rules which maximises the probability of less attractive rules
winning against the odds; third, this optimal density decreases in proportion
to the difference in attractiveness between favoured and unfavoured rules; and
finally, despite overall tendencies towards regularisation, irregular forms
may persist in a given language because of their strong similarities with
sufficiently attractive forms in other, closely related, languages.

#### Acknowledgments

We acknowledge with thanks discussions and exchanges with Aurélia Elalouf,
Guillaume Jacques and Mattis List, which introduced us to the rich fields of
historical linguistics and contact linguistics. AM warmly thanks the
Leverhulme Trust for the Visiting Professorship that funded part of this
research, as well as the Faculty of Linguistics, Philosophy and Phonetics,
Oxford and the Institut de Physique Théorique, Saclay, for their hospitality.

#### Author contribution statement

Both authors contributed equally to the present work, were equally involved in
the preparation of the manuscript, and have read and approved the final
manuscript.

#### Data availability statement

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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