Mind the Network: Rock Art, Cultural Transmission, and Mutual Information

Abstract

Decorative patterns have long been considered suitable for determining descent, since they are categorized as homologous and adaptively neutral. Rock art, for its part, has often been left aside due to a lack of chronological control. In this paper, we propose a way to treat rock art in order to track Cultural Transmission Paths by means of motif distribution using Northwestern Patagonia as a case study. We present a theoretical and methodological framework for modeling Cultural Transmission Archaeological Paths by constructing a Mutual Information Network between motifs, identifying clusters and defining their associated Site Networks. The results allow us to suggest a hypothetical nuclear region, well known and transited by hunter-gatherers, with few connections to the more distant parts of the study area. This pattern may be related to Patagonia’s population models and fit the suggestion from other fields of inquiry that a sparsely connected and not unnecessarily complex network will be robust enough to sustain information flux.

Appendices

Appendix 1

Before defining Mutual Information, we need to introduce the Shannon Information and the Entropy functions. In his classic paper, Shannon (1948) defined entropy (H) as a measure of uncertainty of a random variable. In a communicational process, a given source emits messages that can be stored in an X variable. Departing from probability distribution of X values, entropy quantifies the level of “surprise” the receiver experiences upon receiving each message. If there is no surprise, there is no Information content (Mitchell 2009), because the message is fully predictable.

As an example, we present a rock art case. Let us introduce variable X, which represents a particular motif, which can take two possible values: 0 means the absence of the motif in a particular site and 1 its presence. Let us suppose that we have assessed 8 sites for this particular motif (see Table A1.1).

Table A1.1 Example of variable X representing the absence (when X takes the value 0) and the presence (when X takes the value 1) of a particular motif in 8 different sites

This motif is present in sites 1, 2, 3, and 4, and it is absent in the rest of the sites. From the data, we can compute the probability that X takes value 0 (which we will call $P\left(X=0\right)$) and the probability that X takes value 1 (which we will call $P\left(X=1\right)$). The probability of X taking a particular value is the frequency of this particular value with respect to the total number of observations. In this example, $P\left(X=1\right)=\frac{4}{8}=\frac{1}{2}$ (because in 4 of the 8 assessed sites the motif is present) and $P\left(X=0\right)=\frac{1}{2}$. Let us note that $P\left(X=1\right)+P\left(X=0\right)=1$ because X can take only two values in this example. The Shannon Information contained in the outcome value 1 of variable X is defined as

$$ h\left(X=1\right)=-{ \log}_2P\left(X=1\right). $$

The entropy of the variable X is defined as the average of the Shannon Information contained in the possible outcomes, thus:

$$ H(X)=P\left(X=1\right)h\left(X=1\right)+P\left(X=0\right)h\left(X=0\right). $$

The entropy of X for the example is $H(X)=1$, because

$$ H\left(X=1\right)=-\frac{1}{2}{ \log}_2\left(\frac{1}{2}\right)-\frac{1}{2}{ \log}_2\left(\frac{1}{2}\right)=\frac{1}{2}+\frac{1}{2}=1 $$

Let us note that:

• The entropy is always greater than or equal to zero. The particular case of entropy zero occurs when the variable X takes a particular value with probability 1 and the rest of the values with probability 0. Then, the uncertainty of the variable is 0, because we are certain that X will take only one possible value.

• Entropy H(X) reaches its maximum value when the probability of the occurrence of the outcomes of X variable is uniform. For the particular case of two possible outcomes, H(X) is maximum when the probability of the two outcomes is the same, $P\left(X=1\right)=P\left(X=0\right)=\frac{1}{2}$, and it reaches the value $H(X)=1$. In this scenario, the uncertainty of the variable X is maximum.

Let us complicate our example a little bit further. We will continue with 4 motifs and 8 assessed sites (Table A1.2).

Table A1.2 Example of variables X ₁, X ₂, X ₃ and X ₄ representing the presence (when variable takes the value 0) and absence (when variable takes the value 1) of four motifs in each of th3 8 assessed sites

In the case of motif 1 and motif 2 (row 1 and row 2 of Table A1.2), the entropy takes the same value ($H\left({X}_1\right)=H\left({X}_2\right)=0$). This occurs because entropy is a function of the probabilities and not of the values of the outcomes. Entropy does not distinguish between two cases which are symmetric (if we change outcomes 1 for 0 and vice versa). The same happens with motifs 3 and 4 (X ₃ and X ₄) that reach the same entropy value, which results $H\left({X}_3\right)=H\left({X}_4\right)=0.81$.

Now let us compare two other motifs (X and Y) in the same 8 sites. Let us suppose that the observed values result in Table A1.3.

Table A1.3 Example of two variables X and Y representing the presence (when variable takes the value 0) and absence (when variable takes the value 1) of two motifs in 8 sites

We are interested in detecting if there is any type of correlation between these two motifs. Does information about motif 5 give information about motif 6, or are they independent variables? Mutual Information helps to answer this question by quantifying the information gain that we obtain from one variable when we know the other variable and vice versa. The Mutual Information is defined as:

$$ I\left(X,Y\right)=H(X)-H\left(X\Big|Y\right) $$

where $H\left(X\Big|Y\right)$ is the conditional Entropy of variable X given that we know the variable Y (Cover and Thomas 1991). As we just mentioned, Mutual Information measures the difference in the uncertainty of X variable when we know Y variable and vice versa. When variables X and Y are independent, then the fact of knowing Y does not reduce the uncertainty of X. Formally, $H\left(X\Big|Y\right)=H(X)$, because the uncertainty of X is the same (regardless of whether Y is known or not). Then $I\left(X,\ Y\right)=H(X)-H\left(X\Big|Y\right)=0$, reflecting that the knowledge of one of the variables (X or Y) does not say anything about the other. In the other extreme case, the uncertainty of variable X is fully reduced when we know Y variable, $H\left(X\Big|Y\right)=0$ (if we know the value of Y, then the certainty of X variable is complete, because we are sure of the value which X takes), then the Mutual Information is maximum $I\left(X,\ Y\right)=H(X)-H\left(X\Big|Y\right)=H(X)$.

Returning to the example of Table A1.3, we compute $I\left(X,\ Y\right)=0.548$. Thus, X and Y are not independent. We can observe that every time motif 5 is present (X = 1), then motif 6 is present too (Y = 1) (notice that the inverse case is not met: in site 6 although motif 6 is present, motif 5 is absent).

Finally, it is important to remark that Mutual Information does not say anything about the sense of the information gain. Hence, in Table A1.4 we present other example which leads to the same value of Mutual Information as the previous example ($I\left(X,\ Y\right)=0.548$). But in this case, we can note that each time motif 7 is present (X = 1), then motif 8 is absent (Y = 0). Then, motif 7 gives us information about motif 8, but they are negatively correlated.

Table A1.4 Example of two variables X and Y representing the presence (when variable takes the value 0) and absence (when variable takes the value 1) of two motifs in 8 assessed sites

Appendix 2

List of motifs and character states (Taken from Scheinsohn et al. 2009).

Appendix 3: Simulation

In order to discard the possibility of obtaining the values of Mutual Information of Table 2 by chance, we performed a simulation in which we randomly assigned the same number of 1s and 0s (the amount of “presences” of motifs in the sites) from the database in the X_i variable. We generated 1000 random assignments. The obtained distribution of Mutual Information is shown in Fig. A3.1, where the threshold value u corresponds to a p-value (the probability of finding a case greater than the observed 0.093) of 1.56 % in the distribution of the Mutual Information obtained by random assignment. This means that, for the extreme case of less correlated pairs (on threshold value u 0.093), it is possible to obtain this correlation by random assignment with less than 1.56 (a low probability). Then, we performed another statistical test in which for each pair of motifs, the random assignment was made considering the same value as in our Table 2. For instance, comparing motif 50 with 15, we assigned 1 and 0 taking into account that motif 50 was present in 3 sites and motif 15 was present in 5 sites (see Table 2). We obtained a p-value less than 2 % for the three first cases corresponding to the threshold value u (the first three pairs of Table 2) and less than 0.2 % for the rest of the cases. Notice that these three cases, which are the ones with the greater possibility of being obtained by random, are also the ones with fewer information (given the small sample). Nevertheless we decided to include it on the Mutual Information Network, since in archaeology it is usual to deal with absence of information. In any case, with this exception, this test allows us to sustain that the probability of obtaining these values of Mutual Information correlation with the rest of the pair of motifs is low. These sets of correlated motifs above the threshold value u will be used to construct the MIN, in which nodes represent motifs and links represent the Mutual Information between them.

Fig. A3.1

Histogram of Mutual Information values obtained by assigning the amount of 1s and 0s observed in the database randomly in a matrix of 49 columns and 43 rows. The p-value of 0.156 for u means that the probability of obtaining a value of Mutual Information greater than the threshold value is 1.56 %