Figure 1.
View of the extensive ceremonial center of Teotihuacan as seen from the top of the Pyramid of the Moon.
In front is the ceremonial plaza of the Pyramid of the Moon with a small ritual platform in the middle. The Street of the Dead can be seen stretching into the distance, its sides lined with pyramidal platforms that would have originally supported temple structures. Mimicking the mountainous horizon on the left is the Pyramid of the Sun. (Photo courtesy of Iliana Mendoza).
Figure 2.
Architecture of a typical Three-Temple Complex (TTC).
Central part of a mockup of the Street of the Dead (located in the Museo de Sitio de la Cultura Teotihuacana). The beginning of the plaza in front of the Pyramid of the Moon can be seen at the very top, while the Pyramid of the Sun is on the right. A typical TTC can be seen on the bottom right: three temples face each other around a square plaza, with the two equally sized temples flanking a larger temple in their middle. Another TTC can be seen in the center of the photo, with its plaza opening to the Street of the Dead. (Photo courtesy of Iliana Mendoza).
Figure 3.
Spatial distribution of 22 Three-Temple Complexes (TTCs) of Teotihuacan.
Only TTCs of an intermediate scale were selected as neighborhood centers. The Pyramid of the Sun, Pyramid of the Moon and known examples of smaller TTCs contained within excavated apartment compounds were excluded because they seem to belong to different spatial and temporal contexts. The shortest paths between any two TTCs are approximated as a rectangle because much of the city is densely occupied and roughly conforms to a grid-like pattern. Distances between TTCs were measured by hand on Millon’s [3] map. Minor measurement errors are visible, since the nodes do not overlap 100%. The effect of these distances on social constraints was incorporated into a pilot study, but not in the final model (see Methods section for details).
Figure 4.
Ten examples of self-organized constraint satisfaction before self-optimization.
During each behavior update a representative in the network is randomly selected, and allowed to adjust its behavior (to +1 or −1) if the new choice satisfies more constraints posed by its connections with all the others. Because a representative’s received utility is calculated as the weighted sum of its satisfied constraints, priority is given to satisfying more important (more weighty) connections. As representatives repeatedly optimize their choices, the network’s sum of utilities U increases. However, as shown by these 10 independent trajectories starting from arbitrary initial conditions, this weak form of self-organization typically becomes trapped in one of several suboptimal behavioral configurations (U = 173.58 for global optima). Only first 400 behavior updates shown; after that U stays the same until the end of the run in all cases.
Figure 5.
Sum of utilities reached after convergence during a series of 200 ritual interruptions.
In contrast to Figure 4, convergences happen in sequence rather than independently, and only the final sum of utility is plotted (behavioral updates leading to convergence are not shown). During a ritual interruption all behavioral states are reset to arbitrary states, as before, but agents’ changes to their social connection weights are retained. To allow comparison with Figure 4, we show the sum of utilities U based on the original network’s constraints only. Self-optimization is clearly visible: the network is quickly and spontaneously transformed such that updates of behavior, despite being selfish and rational, consistently converge on a globally optimal configuration (U = 173.58).
Figure 6.
Comparison of the extent of cooperation in four variations of the Teotihuacan network model.
We compared cooperation in two respects: As a function of self-optimization, i.e. whether connection changes were preserved over the preceding 200 ritualized perturbations or not (red versus blue bars). And as a function of social organization, i.e. whether the network is divided only into 22 neighborhoods or also additionally into four larger-scale districts (Neighborhoods versus Districts). For each case we assessed the outcomes of 20,000 independent convergences. The bars show the average number of agreements reached by the social network given the constraints of its original organization (4290 agreements equal complete consensus). Error bars represent one standard deviation.
Figure 7.
Mural painting located in the Tepantitla apartment compound.
This is a representative section of a much more extensive scene showing a large number of modestly dressed people who are engaged in a variety of unusual activities. Most of them do not seem to be engaged in the mundane activities imposed by the needs of daily life. Some of the people’s interactions with plants and flowers have been interpreted in terms of the consumption of psychoactive substances (see, e.g., on the bottom left where a person is eating a plant while rainbow-like streams are flowing from his head). Scenes such as these are indicative of large-scale communal rituals or events at Teotihuacan. (Photo courtesy of Iliana Mendoza).