Assumptions

We model the transmission of granny and reef knots within a population through oblique transmission (Cavalli-Sforza and Feldman, Reference Cavalli-Sforza and Feldman1981) and assume a closed system such that when a granny or reef knot is demonstrated, the learned knot is always either a granny or a reef knot. We assume that four parameters can affect the fidelity of social transmission: the learner interprets the demonstrator's knot incorrectly as the knot's mirror image with a probability g (mirroring); the learner copies the perceived trefoil with a probability s (copying fidelity), where the perceived trefoil refers to the learner's interpretation of the demonstrated trefoil, which could either be the demonstrated knot or the mirror image of the demonstrated knot; the learner repeats the trefoil they tied for the first step of the composition of overhand knots with a probability r (repetition); and the learner ties a right-handed trefoil when they do not copy the perceived demonstration with a probability p (handedness).

Using these parameters, we can build a system of recurrence equations to describe relative knot frequencies in the learner generation as a function of their frequencies in the demonstrator generation. We denote the proportion of knot ij tied in the demonstrating generation by f _ij where ij ∈ {RR, LL, RL, LR}, and the knots tied by the learner generation of the population after transmission as $f_{ij}^{\prime}$ where ${f}^{\prime}_{{\rm RR}} + {f}^{\prime}_{{\rm LL}} + {f}^{\prime}_{{\rm RL}} + {f}^{\prime}_{{\rm LR}} = 1$ with each ${f}^{\prime}_{ij}$ taking values in the interval [0,1]. For example, take the granny knot formed by tying two right-handed trefoils and denote it by f _RR. This knot will be transmitted successfully if it is not mirrored and both trefoils that form it are accurately copied by the next generation, denoted by $f_{{\rm RR}} \left ({s^2\lpar {1-g} \rpar } \right)$. However, a right granny could also be formed by mirroring an LL with probability g and accurately copying both trefoils of the perceived knot with probability s ², giving f _LL(s ²g). A right granny could also be formed with no copying fidelity at all (s = 0), if the learner has a bias towards tying right-handed trefoils $f_{{\rm RR}} \left ( {{\lpar {1-s} \rpar }^2p^2} \right)$ or repeating the first knot tied, $f_{{\rm RR}}\left ( {{\lpar {1-s} \rpar }^2\lpar {\,pr} \rpar } \right)$ and so we get the frequency of right granny knots in the population as a function of granny and reef knots already in the population and the probability parameters:

(1)$$\matrix{ {{{\,f}^{\prime}}_{{\rm RR}} = f_{{\rm RR}} \left ({s^2\lpar {1-g} \rpar } \right) + \cdots + f_{{\rm RR}} \left ( {{\lpar {1-s} \rpar }^2p^2} \right) + \ldots} \cr { +\; f_{{\rm RR}}\left ( {{\lpar {1-s} \rpar }^2\lpar {\,pr} \rpar } \right) + \cdots + f_{{\rm LL}}\left( {s^2g} \right) + \ldots} \cr} $$

It is important to think about how the parameters interact with each other. If a learner copies the knot correctly then the learner's likelihood to repeat or tie a right-handed trefoil does not matter. They will do what is shown regardless of their handedness bias or propensity for repetition, and so we can discount repetition and right-hand bias when the knot is accurately copied. In the same way, when the learner simply repeats part of a knot, their right-hand bias does not matter, as they will repeat regardless of this bias. So we can discount right-hand bias when repetition takes place. Figure 2 illustrates how the knot tied may be affected by the observed knot and the four parameters. Note that for each trefoil, the depicted order of parameters on any given branch is arbitrary and does not affect the probability that a particular trefoil form is tied (i.e. the parameters commute; see S5). For instance, the first trefoil can be tied left-handed if the learner both fails to copy the perceived knot and is not subject to a right-handedness bias irrespective of any order by which these processes might take effect. For each trefoil, the model only accounts for combinations of learning processes that lie on the same branch of the probability tree (see Discussion).

Figure 2. Probability tree showing the effect of parameters on the transmission of knot RR. For each trefoil tied, the order by which parameters are shown to take effect along a given branch is arbitrary as the probability that the knot is of form L or R is simply the product of their combined effect.

Evolutionary dynamics

Each set of parameter values 0 ≤ (s, g, r, p) ≤ 1, determines the evolutionary trajectory and a single equilibrium point, where $f_{ij} = {f}^{\prime}_{ij} = \hat{f}_{ij}$, (expressions for equilibrium states are given in Section S6). If s = 0, the system jumps to a stable equilibrium point determined by the p and r and is unaffected by starting values of f _ij. In contrast, if copying is always accurate, s = 1, and mirroring never occurs, g = 0 (0 ≤ r ≤ 1), the population does not evolve from starting frequencies, so if a small perturbation in frequencies is induced, the population remains at the new frequencies. If there is some copying, 0 <s <1, the population evolves to a stable equilibrium, such that the population returns to the original equilibrium state following a small perturbation in frequencies.

Figure 3 illustrates the effect of copying and mirroring on equilibrium frequencies. In Figure 3a, the value of s is set lower than in Figure 3b, resulting in a relatively small change in the values of $\hat{f}_{{\rm RR}}$, $\hat{f}_{{\rm LL}}$ and $\hat{f}_{{\rm RL}}$ and $\hat{f}_{{\rm LR}}$. This shows that copying needs to be highly probable for mirroring to affect the proportion of knots tied in the population. We notice that the two reef knot frequencies, f _LR and f _RL, are always equal at equilibria. This is consistent with the fact that LR and RL represent the same knot mathematically (see Section S1).

Figure 3. Parts (a) and (b) show the proportion of knots at equilibria as a function of the probability of mirroring when copying fidelity of the perceived knot is low and high, respectively. The values of $\hat{f}_{{\rm LR}}$ and $\hat{f}_{{\rm RL}}$ are equal so these are represented by the same line on the graph, while $\hat{f}_{{\rm RR}}$ and $\hat{f}_{{\rm LL}}$ are represented by separate lines. Parts (c) and (d) show evolutionary trajectories when the probability of mirroring is low and high, respectively. Each arrow represents the change in relative frequency of each type of knot in the population, starting from sole existence in each corner to a mixture of different knots in the interior of the tetrahedron. The solid disc is the equilibrium state which is evolved towards no matter the starting frequencies. Frequencies are plotted in tetrahedral space using Barycentric coordinates (see Section S8).

Prior to reaching an equilibrium state, evolutionary dynamics typically follow a smooth trajectory (assuming 0 < s < 1), but a high probability of mirroring can cause oscillations in the trajectory when copying fidelity is high. When mirroring is low (Figure 3c), we see the system evolve in a smooth curve to a point strongly affected by the handedness, pand repetition, r. The high value of p causes the point to be slightly closer to the corner f _RR = 1 than f _LL = 1 but the low value of r does not cause the point to be as close to the f _RL + f _LR = 1 boundary as we may expect. In Figure 3d, mirroring is likely to occur. Coupled with the high copying fidelity, the system evolves to a similar equilibrium point as shown in Figure 3c, but the high probability of mirroring causes the path to oscillate to the point rather than evolve in a smooth trajectory.

Humans are likely to copy a perceived demonstration with some success but to make some mistakes. In this circumstance (0 < s < 1), there are some conditions where s does not affect equilibrium state frequencies.

If p = 1/2, we have

$$\matrix{ {{\hat{\,f}}_{{\rm RR}}} \hfill & { = {\hat{\,f}}_{{\rm LL}} = \displaystyle{{\lpar {1 + r} \rpar } \over 4}\qquad} \hfill & {} \hfill & \qquad \hfill \cr {{\hat{\,f}}_{{\rm RL}}} \hfill & { = {\hat{\,f}}_{{\rm LR}} = \displaystyle{{\lpar {1-r} \rpar } \over 4}\qquad} \hfill & {} \hfill & \qquad \hfill \cr}$$

while if g = 0, we have

$$\matrix{ {{\hat{\,f}}_{{\rm RR}}} \hfill & { = p\lpar {\,p + r-pr} \rpar \qquad} \hfill & {} \hfill & \qquad \hfill \cr {{\hat{\,f}}_{{\rm LL}}} \hfill & { = \lpar {1-p} \rpar \lpar {1-p + pr} \rpar \qquad} \hfill & {} \hfill & \qquad \hfill \cr {{\hat{\,f}}_{{\rm RL}}} \hfill & { = {\hat{\,f}}_{{\rm LR}} = \lpar {\,p-1} \rpar \lpar {\,pr-p} \rpar \qquad} \hfill & {} \hfill & \qquad \hfill \cr}$$

Most combinations of parameter values result in an excess of granny knots over reef knots at equilibrium. As noted above, any repetition, r, will favour the granny knot, but even when repetition never occurs, the population is still more likely to tie granny knots than reef knots if there is any handedness bias, p ≠ 1/2. Mirroring typically has little influence on the relative equilibrium frequency of granny to reef knots when 0 < s < 1 and has no influence when either s = 0 or s = 1. Figure 4a illustrates the predominance of granny knots at equilibrium, taking the case where there is no repetition in the absence of guidance, r = 0, and intermediate mirroring, g = 1/2. The bias towards granny knots is strongest when handedness bias, p, is either high or low and the copying coefficient, s, is low; in other words, when individuals consistently tie with the same handedness rather than copying a different knot.

Figure 4. (a) A density plot showing the proportion of granny knots at equilibrium, denoted by $x = \hat{f}_{{\rm RR}} + \hat{f}_{{\rm LL}}$, as a function of handedness bias, p and copying fidelity, s, where g = 1/2 and r = 0. (b) A probability tree showing knots tied in the absence of biases in handedness (p = 1/2; top two layers affecting first and second trefoil) and repetition biases (r = 1/2; bottom layer, affecting second trefoil).

Also note that the absence of non-random copying error does not lead to equal knot frequencies; granny knots are expected in higher frequency than reef knots (Figure 4b; also see that in Figure 5b the blue disc is not in the centre of the tetrahedron). This occurs because of the way the parameters interact to affect the knot forms: consider for instance the case in the absence of handedness bias and repetition bias p = 1/2, r = 1/2 (under these conditions, mirroring and copying fidelity do not affect equilibrium frequencies). Figure 4b shows that the probability of tying each knot is P(LL) = 3/8, P(RR) = 3/8, P(RL) = 1/8 and P(LR) = 1/8.

Figure 5. (a) Histograms of parameter values simulated from the experiment, with acceptance interval d(O,S) ≤ 0.0075. Red lines indicate unbiased parameter values, p = 1/2 and r = 1/2, giving equal probability of tying right- and left-handed trefoils and equal probability of repeating the previous knot as not, respectively. (b) Evolutionary trajectories of the four knot forms, where f _ij = 1 in each corner and frequencies are equal at the centre of the tetrahedron. Trajectories using the mean posterior parameter values $\lpar {\bar{s},\bar{p},\bar{g},\bar{r}} \rpar$ are shown by the grey arrows and black disc, $\hat{f}_{{\rm LL}} = \hat{f}_{{\rm RR}} = 0.415 $, $ \hat{f}_{{\rm LR}} = \hat{f}_{{\rm RL}}\, {=}$ 0.085. The blue arrows and disc, $\hat{f}_{{\rm LL}} = 0.375 $, $\hat{f}_{{\rm RR}} =0.375 $, $\hat{f}_{{\rm LR}} = \hat{f}_{{\rm RL}} = 0.125$ show the trajectories in the absence of handedness bias and repetition bias (p = 1/2, r = 1/2) assuming no mirroring, g = 0, and the mean posterior parameter value for copying fidelity, $\bar{s} = 0.81$ (note that mirroring and copying fidelity do not affect the equilibrium state here).

There are only two cases where the equilibrium proportion of granny and reef knots is equal $\lpar\,\, {{\hat{f}}_{{\rm RR}} + {\hat{f}}_{{\rm LL}} = {\hat{f}}_{{\rm RL}} + {\hat{f}}_{{\rm LR}}} \rpar$. The first case is when copying is not perfect, 0 ≤ s < 1, the first knot is never repeated, r = 0, and there is no handedness bias, p = 1/2, where 0 ≤ g ≤ 1. The absence of repetition prevents predominance of granny knots, and the lack of handedness bias prevents the prevalence of either granny knot. The second case is when copying is perfect, s = 1, and there is some mirroring 0 < g ≤ 1, where 0 ≤ p ≤ 1 and 0 ≤ r ≤ 1. Here copying the perceived knot form is always perfect, but mirroring causes tying of the opposite handedness to that demonstrated. Both these cases are illustrated in Figure 4a. Finally, we note that reef knots can only be more prevalent than granny knots if this is exhibited in their starting frequencies and when the system does not evolve (s = 1 and g = 0; discussed above).

Fitting the social transmission model to experimental data

Using ABC (Sunnåker et al. Reference Sunnåker, Busetto, Numminen, Corander, Foll and Dessimoz2013), we can use our model to estimate parameter values that predict the experimental data. ABC works on the same premise as Bayes’ theorem, relating conditional probability of parameters θ, to data D by the rule

(2)$$p\lpar {\theta \vert D} \rpar = \displaystyle{{\,p\lpar {D \vert \theta} \rpar p\lpar \theta \rpar } \over {\,p\lpar D \rpar }},$$

where p(θ|D) is referred to as the posterior, p(θ) represents the prior beliefs before any data is available, p(D|θ) the likelihood of data D occurring given the prior and p(D) the evidence (Gelman et al., Reference Gelman, Carlin, Stern and Rubin2003). With this rule, we can calculate the posterior by taking the product of prior beliefs with the likelihood of data occurring, divided by the evidence observed. To obtain the probability of data D given parameter θ, we use our model to simulate data for a given parameter set and decide whether it fits the observed data. We construct a metric to describe our observed data such that we can accept or reject the simulated parameter set depending on whether it generated data within a tolerated proximity from the observed. The retained parameter distributions give us p(θ|D).

Taking our observed data from Table 1 as a 4 × 4 matrix O and simulating data of the same form using our model to give a 4 × 4 matrix s, we compare these two sets of data using the metric;

(3)$$d\left( {O,S} \right) = \sum\limits_{i,j} a_{ij}^2,$$

where a _ij are the entries of the matrix O − S. This metric is proportional to finding the Euclidean distance between the points in the two matrices.

Describing the process in more detail, we ran a Monte Carlo simulation (Hastings, Reference Hastings1970) where the number of simulated learners exposed to each of the four demonstrated knots matched the number of participants in each experimental condition (see Table 1). A value for each of the four parameters (p, g, r, s) was sampled from a uniform distribution between 0 and 1. The knot tied by each simulated learner was derived by walking through the relevant probability tree with a Bernoulli trial at each internal node (e.g. Figure 2 for a learner that observes demonstration of knot RR). This simulation procedure was repeated many times to build up parameter distributions for all the simulations that resulted in a metric value, $d\lpar {O,S} \rpar = \sum\nolimits _{i,j}a_{ij}^2 \le 0.0075$, coming from fewer than $0.5\%$ of the simulations.

Figure 5a shows uncertainty in handedness bias, with a broad distribution around a mean of $\bar{p} = 0.5$ (sd = 0.28) which is where handedness bias is absent. The model predicts that individuals mirror fairly frequently ($\bar{g} = 0.39,sd = 0.07$) but that knots are mirrored less often than they are correctly interpreted. There is uncertainty in the posterior estimate of the repetition bias, but with a trend to be more likely to repeat the handedness of the first trefoil tied than not ($\bar{r} = 0.66,sd = 0.24$). Finally, there is relatively high copying fidelity of the perceived knot ($\bar{s} = 0.81,sd = 0.08$).

We can establish what effect our parameter estimates would have on the cultural evolution of granny and reef knots by plugging the central tendency values into the model. For illustration, we use the mean from each posterior parameter distribution, but note that sampling from the posterior distributions each generation gives similar results (see Section S11.)

Figure 5b illustrates how the population evolves towards a single polymorphic equilibrium state, no matter the starting distribution (grey arrows leading to black disc). Compared with the case where handedness and repetition errors are random (p = 1/2, r = 1/2; blue arrows leading to blue disc), the mean posterior estimate of repetition bias results in a higher equilibrium frequency of granny over reef knots. The mean posterior estimate for the handedness coefficient is unbiased, $\bar{p} = 1/2$, resulting in the equal equilibrium frequency of left and right forms of granny knot. As established in the social transmission model analysis, the posterior copying fidelity value, $\bar{s} = 0.81$ actually has no effect on the equilibrium frequencies when there is no handedness bias. The posterior mirroring value is not large enough to cause the characteristic oscillating dynamics shown in Figure 3d. We see that the equilibrium frequency results in a prevalence of granny knots over reef which can also be explored by sampling from the posterior distribution of parameter values, allowing us to see the relative frequency of granny knots over reef knots for one generation (see Section S12).

As an aside, in Section S9 we compare our social transmission model against a non-parametric estimate of equilibrium frequencies which can be found simply by iterating this proportional change without specifying the effects of specified learning processes (Claidière et al., Reference Claidière, Scott-Phillips and Sperber2014). Our parametric social transmission model results in similar equilibrium frequencies to this non-parametric prediction, indicating that the close match in the proportional change in knot frequencies over one social transmission episode caused by implementing ABC is preserved over multiple generations.

A comprehensive out-of-sample test of the model is for a future study, although we note that, for composite knots within the Ashley corpus, the proportion of granny to reef knots exactly matches the 3 to 1 ratio predicted by the equillibrium state of our model when handedness and repetition errors are random (p = 0.5, r = 0.5; blue disc in Figure 5b) and is fairly similar to that predicted using the posterior parameter estimates (mean posterior estimates give $83\%$ granny knots; black disc in Figure 5b; Ashley, Reference Ashley1993; Scanlon, Reference Scanlon2016).