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Modeling Cultural Transmission of Rituals in Silico: The Advantages and Pitfalls of Agent-Based vs. System Dynamics Models

In: Journal of Cognition and Culture
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  • 1 GEHIR: Generative Historiography of Religion Project, Department for the Study of Religions, Faculty of Arts, Masaryk UniversityREECR: Ritual and the Emergence of Early Christian Religion, Faculty of Theology, University of Helsinki
  • | 2 GEHIR: Generative Historiography of Religion Project, Department for the Study of Religions, Faculty of Arts, Masaryk University
  • | 3 GEHIR: Generative Historiography of Religion Project, Department of Mathematics and Statistics, Faculty of Science, Masaryk University
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Abstract

This article introduces an agent-based and a system-dynamics model investigating the cultural transmission of frequent collective rituals. It focuses on social function and cognitive attraction as independently affecting transmission. The models focus on the historical context of early Christian meals, where various theoretically inspiring trends in cultural transmission of rituals can be observed. The primary purpose of the article is to contribute to theorizing about cultural transmission of rituals by suggesting a clear operationalization of their social function and cognitive attraction. Furthermore, the article challenges recent trends in the field by providing a theoretically feasible model for how, under certain conditions, cognitive attraction can influence the transmission to a relatively greater extent than social function. In the system dynamics model we reproduce the results of our agent-based model while putting some of our basic operational assumptions under scrutiny. We consider approaching social function and cognitive attraction in isolation as a preliminary but necessary step in the process of creating more complex models of the cultural selection of rituals, where the two aspects will be combined to produce ritual forms with greater correspondence to real-world religious rituals.

Abstract

This article introduces an agent-based and a system-dynamics model investigating the cultural transmission of frequent collective rituals. It focuses on social function and cognitive attraction as independently affecting transmission. The models focus on the historical context of early Christian meals, where various theoretically inspiring trends in cultural transmission of rituals can be observed. The primary purpose of the article is to contribute to theorizing about cultural transmission of rituals by suggesting a clear operationalization of their social function and cognitive attraction. Furthermore, the article challenges recent trends in the field by providing a theoretically feasible model for how, under certain conditions, cognitive attraction can influence the transmission to a relatively greater extent than social function. In the system dynamics model we reproduce the results of our agent-based model while putting some of our basic operational assumptions under scrutiny. We consider approaching social function and cognitive attraction in isolation as a preliminary but necessary step in the process of creating more complex models of the cultural selection of rituals, where the two aspects will be combined to produce ritual forms with greater correspondence to real-world religious rituals.

1 Introduction

In this article, we introduce two models: an agent-based model (ABM) and a system dynamics model (SDM), where the latter represents a replication of the former using a different modeling and simulation paradigm. The models focus on two important factors often discussed in the literature as affecting the process of cultural transmission of repetitive collective rituals: social function and cognitive attraction. In our models, we approach these two factors as operating independently, as we assume that they affect ritual’s cultural transmission via different mechanisms.

The primary goal of our article is to contribute to theorizing about cultural transmission of rituals by suggesting a clear operationalization of social function and cognitive attraction. Furthermore, challenging the recently renewed interest in social functions of rituals (e.g., Watson-Jones & Legare, 2016), our models indicate that – in theory – there are conditions under which cognitive attraction appears to have greater importance for cultural transmission of rituals than social function. Finally, we aim to demonstrate some advantages and pitfalls of ABMs versus SDMs for modeling complex socio/cognitive phenomena like religions in general.

Avoiding essentialism when accessing the category of ritual (Snoek, 2006), we pragmatically define rituals as any kind of collective activity involving features of structurally rigid repeated behaviors, that are often causally opaque. Further, we approach it as a phenomenon closely interrelated with magic (Sørensen, 2007), superstition (Vyse, 2013) or fetishism (Frazier, Gelman, Wilson, & Hood, 2009). In terms of mechanistic explanation, we suggest that participation in or observation of a ritual is associated with the activation of cognitive systems responsible for action parsing (Nielbo & Sørensen, 2011), hazard precaution (Liénard & Boyer, 2007), over-imitation (Kapitány & Nielsen, 2017), conceptual blending (Sørensen, 2007), agency (McCauley & Lawson, 2002), as well as intuitive thinking biases such as the law of contagion or the law of similarity (Lindeman & Aario, 2007; Hood, 2009; Vyse, 2013; Subbotsky, 2010; Hutson, 2012). In addition, we assume that rituals, as a form of collective activity, fulfil various social functions by activating cognitive systems associated with identification (Whitehouse & Lanman, 2014), synchrony (Fischer et al., 2013) and shared emotional arousal (Xygalatas et al., 2013).

We understand the two factors (social function and cognitive attraction) as theoretical envelopes for the different families of driving forces leading a population to prefer one ritual form over another over time, either directly or indirectly. In other words, we assume that ritual forms which are either better suited to fulfil particular social functions or evaluated as more cognitively attractive by the participants are also more successful in cultural transmission than their alternatives. However, we approach these two factors as independent and operating via different mechanisms, as there also is an empirical evidence that they sometimes even go against each other (Mitkidis et al., 2014). We see the operationalization of how this can happen and what long-term implications it can have as one of our main contributions.

2 The Target Phenomenon

In more narrow terms, in our models we approach the problem of cultural transmission of rituals as a process in which a population of co-religionists goes through a process of selecting rituals with suitable forms from available alternatives. As driven by several interrelated and independent factors, operating partly indirectly, such a selection process is nontrivial phenomenon.

Our cultural selection model is just one variant among many. We choose the design as it seems to be the one most theoretically promising and empirically valid. It is theoretically promising as it enables us to bridge two broader and partially competing perspectives on cultural transmission of rituals and religion more generally: The first perspective is seeing religious beliefs and practices as adaptive tools of human social living and thus viewing rituals as effective social technologies (e.g. Norenzayan, 2016). Drawing on this perspective, the rituals that are successfully transmitted are those which contribute to the greater success of religious groups; for example, by increasing their survival fitness. The second perspective views religious beliefs and practices as a by-product of evolutionary mechanisms evolved for other functions (e.g., Boyer & Baumard, 2016). In that respect, a religious concept or practice might become culturally successful because of the elicitation of particular cognitive mechanisms. From this perspective, concepts or practices become comparatively more widespread in response to the effect of cognitive factors. This happens regardless of whether the concept or practice contributes to the fitness of those who come into contact with it.

It should be evident that adoption of one of these two perspectives influences the amount of attention that scholars from the respective camps pay to these factors. Although it is easy to get rid of the unclear relations between these perspectives on a general level, the issue becomes tricky as we specify exactly how the two factors can affect the cultural transmission of rituals in an empirically valid way. Currently, the empirical evidence for both perspectives suggests that both may be empirically viable.

We suppose that a ritual form relatively better designed in respect to cognitive attraction might become more successful in cultural transmission simply because it will be directly preferred by the ritual participants over the repeated occurrences of the ritual. It will be preferred because it evokes the cognitive systems mentioned above as responsible for making a ritual form more attention grabbing, more memorable, and more motivational for the participants (among other things). On the other hand, we also suppose that the social function of rituals might work differently: rather, in an indirect way. We suggest that as ritual participants go through a ritual more appropriately designed with respect to social function, the participants form stronger social ties among themselves, which motivates them to repeatedly meet again over repeated occurrences of the ritual. Because of a feedback loop (described in detail below), the participants might indirectly begin to prefer a ritual form producing social cohesion in comparison to rituals that do not.

This leads us to ask: under what real-world conditions can we observe these two factors at work simultaneously? While there are many throughout the archaeological, historical, and ethnographic record, here we focus on the historical context of early Christianity in the ancient Mediterranean, toward the end of second century CE. Following an estimate that around 180 CE there were 15,000 Christians in Rome, more than 8,000 in Alexandria, 3,300 in Antioch, and about 1,100 in Thessalonica (Stark, 2006, p. 71), we can expect that in any of these cities there were dozens of semi-autonomous Christian communities. From early on, it was most common for these communities to gather weekly, on Sundays (Llewelyn, 2001), primarily for the purpose of participating in a shared ritual meal (Taussig, 2009). In addition to differences in terms of doctrine, preferences of texts, and references to authoritative figures, the communities appear to differ in how they designed their communal meals, how they call them, etc. (e.g. McGowan, 2010). Some of these differences might be grasped in terms of ritual theory as having implications for the two factors of our interest here (also see Kaše, 2018).

From this foundation, we can imagine the ritual meal practice repeatedly conducted by one or more Christian communities in a city as a market commodity to be “selected” by the Christians living in that city.1 We choose this environment as an ideal setting for approaching the role of our two factors. For instance, in the historical sources (e.g. Justin Martyr, 1 Apol. 67–69 or Didascalia Apostolorum 2.57–58), we have evidence of experimentation with implementing of synchronous activities into the ritual meal structure of some Christian communities; an element generally believed to contribute to the formation of more cohesive social groups (Fischer et al., 2013; Lang et al., 2017; Mogan et al., 2017). In that respect, we can theorize that ritual forms involving this innovation of synchrony would have greater transmissive advantage because of its social function. On the other hand, there is evidence that the ritual meal in many communities was increasingly accompanied by supernatural beliefs concerning the meal elements: the bread and wine as equipped by magical power or used as an amulet or healing substance (e.g. Cyprian, De laps. 26; Novatian [Ps.-Cyp.], De spec. 5). We expect such rituals to be intuitively evaluated as producing such supernatural outcomes (ex. a blessing transforming bread and wine into a magical substance). As such, they should be preferred over those rituals which either 1) do not promise this kind of outcome or 2) are not ritualized enough to be persuasive in that respect. These aspects of ritual could be considered as factors in cognitive attraction, working in complete independence of social function, perhaps sometimes even against it.

Christians were extensively experimenting with their ritual meals until the fourth century and beyond. The hundred-year period from the middle of the second century to the middle of the third century is important because it was a period when many ritual innovations were implemented in the absence of a central authority with the capacity to disturb the process. It is a period involving more than 5000 Sundays, during which an increasing number of Christian communities were negotiating the forms of their ritual meals. Drawing on this, in our model, we zoom to the historically plausible setting of a Roman city from the early Christian period, accommodating several Christian communities differing in the forms of their ritual meals. Specifically, these meals differ by either being better designed in respect to social function or in respect to cognitive attraction. In our models we explore what population level patterns can be generated by slight modifications in how a ritual form is experienced in terms of its social function and cognitive attraction.

In our models, we simulate an environment in which the ritual forms are tied with particular places, which we can imagine as meeting sites of the Christian communities in a city. One half of the places have the capacity to be parameterized to produce an experience associated with higher social function of the ritual, the second a capacity to produce an experience associated with higher cognitive attraction of the ritual. As we manipulate the strength of these experience by varying the parameters of the models, we affect what type of ritual places the simulated ritual participants prefer over time. Thus, by parametrization of our models, we can produce three outcomes: (1) The ritual sites associated with both types of experiences are equally preferred over time; (2) ritual sites associated with social function experience are preferred over time; (3) ritual sites associated with cognitive attraction experience are preferred over time. Investigating the conditions under which each of these three scenarios can occur is one of the main contributions of this article. In what follows, we describe an ABM and the results of testing the model. Afterwards, we turn to the system dynamics model and the results from testing it.

3 Agent-Based Model Description

To describe our ABM, we use an extension of the standard ODD protocol consisting of the model’s overview, design concepts, and details (Grimm et. al., 2006; Grimm et al. 2010; Grimm et al., 2013). To better describe the human-like behavior of our model, the extension adds a section on decision making – therefore ODD + D protocol (Müller et al., 2013). As decision making processes are the most complex component of our model, we postpone its description until the details section. The ODD + D can be found in the Appendix to the article. However, much of this information will be presented here in a less technical format. Therefore, some information from the previous section and ODD + D is repeated here.

3.1 Overview of the Model

The model serves as a theory-building conceptualization of the transmission dynamics of collective rituals inside a population of interacting individuals as driven by two non-deliberative decision-making factors: cognitive attraction and social function. It creates an idealized environment for a bottom-up cultural selection of two ritual forms reflecting the two factors. The model aims to show specific dynamics associated with the two factors, revealing ratcheting effect towards the ritual form characterized by cognitive attraction.

The model consists of an environment of discrete ritual places divided into two even groups: social function groups (SFg) and cognitive attraction groups (CAg). In these places, a population of human agents holds gatherings (encounters) at each turn. At initialization, a population of 200 agents is randomly and evenly distributed over 10 ritual places and each agent forms their first encounter memory with the other agents at their ritual place. Each agent has constrained memory for ritual experiences gained from such encounters. Each group of ritual places affects the experience differently. The agent is bound to use one from three ritual experience seeking strategies (random, conservative and adaptive). The usage ratio of strategies is fixed in the population (10% random, 50% conservative, 40% adaptive) and for individual agents changes each turn randomly. Time is modelled in discrete time-steps and the same process repeats each time-step.

At the beginning of each time-step, each agent adopts one experience seeking strategy. This strategy is randomly allocated along a fixed distribution, as discussed above. The behavior driven by the strategy results in choosing one ritual place and moving there. For the agent, this place becomes its ritual encounter place for that time step.

The process of agent allocation differs according to the experience seeking strategy.

Random and conservative strategies follow simple heuristics. When adopting the random strategy, the agent chooses randomly from all possible places. In a conservative strategy, the agents move to the same ritual place that they occupied in the previous time-step.

In the adaptive strategy, the agents decide by considering each ritual place in terms of its prospective encounter value, which consists of a combination of the social value (familiarity with agents located in the ritual places based on past encounters) and affinity for cognitive attraction value (see further). All ritual places are marked with the prospective ritual encounter value and the place with the highest mark is chosen and the agent moves there.

The marks of the ritual places are computed by measuring a fit between the content of agent’s memory (i.e., visited places and encountered agents over the last 5 time-steps) and places visited by a proportion of agents using random or conservative strategies. Here the main two drivers (SFc and CAc) influence the computation in a different way.

The score of a ritual place is ascribed in several steps. First, the agent examines all ritual places and checks their visitors while comparing them with those which the agent holds in memory from previous encounters. Whenever the agent recognizes another agent on the examined ritual place as someone in its memory, the agent increases the score of the ritual place by one. In a null model (i.e., when the effect of SFc and CAc are set up to zero) the agent goes into a place scoring highest in that respect. When the SFc is higher than zero, the agent takes into consideration previous encounters with other agents at the ritual space. For each agent that was previously encountered on SF place, the score is increased by 1 and multiplied by (1 + SFc value). Thus, the agent pays higher attention to encounters formed on the SF places than to encounters formed on the CA places.

Second, when the CAc parameter is higher than zero, the agent modifies the marks of all ritual places on the basis of what type of places the agent visited over the past 5 time-steps. The most recent memory has the strongest impact again. Thus, if the agent visited a CA place one week back, the agent multiplies the overall mark of all current CA places by (1 + [CAc value * 5]), where 5 reflects the fact that it is the most recent memory from the 5 slots. Further, if the agent visited a CA place two weeks back as well, the agent multiplies the overall mark of all current CA places again, now by (1 + [CAc value * 4]) etc. Thus, in agreement with the theoretical framework, past visits of CA places make the agent cumulatively more motivated to visit this type of places again and again over time, regardless with whom the agent can meet there. There are two main exogenous drivers: social-function-coefficient or SFc (value 0–1) and cognitive-attraction-coefficient or CAc (value 0–1). They represent simple abstractions of targeted factors used to compute the outcome of the adaptive strategy while shifting the tendency to gather to at a ritual place from the respective group. In the model, each time-step represents a time unit of a week. The model does not end up with any equilibrium state, due to the hardwired variation resulting from the constant presence of random experience seeking strategy. However, the configuration of the coefficient associated with the two factors can create a significant preference for one or other ritual form over time.

In the model, agent memory is a fixed 5-positional stack filled with information about past encounters, where the freshest memory is at the top of the stack and where each position represents one of the past time-steps. When the stack exceeds 5, the most time distant memory is deleted. The memory consists of two parts, the ritual quality memory and social encounter memory. The first is responsible for remembering the ritual quality of the visited place, that is to which group of places the place belongs. The second is responsible for memorizing all individual agents which were present in the place during the encounter. Both parts are crucial for computing the values on which is based the adaptive experience seeking strategy. The social encounter memory is constrained to limited capacity of 150 slots for memory of encounter with individual agents. More recent encounter memories hold more space in this memory than older onward. Formally represented, the count of agents from the most recent encounter is multiplied by 3; in the case of the second most recent encounter, the count of agents is multiplied by 2. In the case of older encounters, the count is represented in the memory as one by one. As a result of this, the agents met over older encounters are pushed out of the memory as the memory capacity is depleted by those met more recently.

4 Agent-Based Model Experiment

To investigate our model, the model was run under seven conditions for 500 time-steps (see Table 1). Data was collected at the end of each simulation run (at time-step 500). Each condition consists of specific combination of parameter values and was run 100 times in order to work with the average outcomes. Due to model stochasticity, the results of any one simulation run vary, and therefore using the results from only one run may bias the results. The model was implemented in NetLogo 5.3.1, the statistical evaluation was done in Python 2.7 using Pandas v0.21 package, graphs were plotted with matplotlib.

Table 1
Table 1

Outcomes in seven conditions as mean of total visits in factor group in 100 runs

Citation: Journal of Cognition and Culture 18, 5 (2018) ; 10.1163/15685373-12340041

Results allowed us to discern three different outcome categories (See Table 1 and Figure 1 below). The first is null model, with SFc and CAc both equal to 0. In this condition, the total visits to either SF or CA ritual places remained about the same – neither target factor drives the behavior. The second and third outcome study the conditions under which target factor drives changes towards its respective group. Each of the second and third outcome category shows different strengths in dependency to the other factor, where a) the target factor acts alone, b) the target factor slightly shifts the visits in its own direction, despite the presence of the other factor, or c) the target factor does it decisively.

Figure 1
Figure 1

Boxplot of total visits under the 7 conditions

Citation: Journal of Cognition and Culture 18, 5 (2018) ; 10.1163/15685373-12340041

To investigate internal dynamics, we also include the model behavior as recorded at the end of a simulation for conditions 2c (Figure 2) and 3c (Figure 3). The first five places with yellow numbers represent the group of ritual places where ritual form favoring SF factor acts (SFg); the places with red numbers represent the group of ritual places where ritual form favoring CA factor acts (CAg).

Figure 2
Figure 2

NetLogo visualization of condition 2c – SF factor overcomes CA

Citation: Journal of Cognition and Culture 18, 5 (2018) ; 10.1163/15685373-12340041

Figure 3
Figure 3

NetLogo visualization of condition 3c – CA factor overcomes SF

Citation: Journal of Cognition and Culture 18, 5 (2018) ; 10.1163/15685373-12340041

Both visualizations show that even in conditions which favor one group of ritual places due to the high force of the driving factor, there can exist a strong group in the non-favored set of ritual places (Fig. 2 – The sixth place with 23 agents, Fig. 3 – the fourth place with 34 agents). These places of resistance represent a group of agents acting over a long period of time with a conservative or adaptive strategy, leading to strong mutual memory ties, which forms a resistance to change against agents acting with the random strategy.

The most interesting outcome of the model are the diverse conditions under which there emerge the situations (2) and (3), revealing that the CAc parameter has a significantly stronger impact upon the behavior of the model than the SFc parameter.

5 System-Dynamics Model Description

The results of testing the agent-based model above clearly demonstrated that, following our operationalization of the key factors and the parametric setting of the model, the factor of cognitive attraction can much more easily dominate the process of cultural transmission of rituals. This is a valuable insight, as it indicates that seemingly small role of cognitive attraction on the individual level can have substantial implications for the behavior of the system at the population level. To explore this observation further we decided to investigate our model using a different kind of model, a system-dynamics model (SDM), that focuses on the population level attributes of the system (Sterman, 2000).

In this model, the population consists of items (individuals) that are characterized by two attributes: (1) presence in a ritual place of one of the two groups (SF-social function or CA-cognitive attraction) and (2) strategy for choosing a ritual place in the next time instant (random, conservative, or adaptive). The strategies are the same as those described in the previous section. The decision process for each strategy will be imitated by mathematical rules operating on a stock of individuals divided into definite parts, since the SDM deals neither with individuals nor with their attributes. In order to promote a driving force of social function, we divide the process of decision making into two steps, where the result of the first one is completely driven by the SF. We assume that a certain proportion of adaptive participants replies a commitment to a place with SF quality independently on remembered fellow-participants. This proportion represents the strength of the social experience associated with the ritual in consideration. Alternatively, we can think of it in the way that the social function causes increment of conservative behavior. The rest of adaptive participants chooses a type of ritual site according to evaluation of remembered encounters; i.e., on “an experienced sociality” or “a shared cognitivity”. The attraction of cognitive aspects of a ritual is expressed by higher sensitivity to remembered encounters in places with the CA quality.

The SDM encapsulates all individuals into a stock, which assumes all individuals to be homogeneous. Hence, there is collective memory (not individual memory) and we are not able to deal with a recollection of a certain agent met by another agent at a particular site. Instead, we assume that the probability of a remembrance of average individual to another one is proportional to amount “stored” in the “collective memory”.

In the model, we distinguish two parts of the “collective memory”: social and cognitive. We imagine that an agent who would have committed to a place of SF and CA type in the ABM can be recorded to the “social” and “cognitive” storage respectively. A memory evaluation for the decision to commit one of the two types of ritual sites consists in appraisal amounts of the dual memory storage. The greater the amount of “cognitive” storage, the more likely an individual is to choose a CA site; the same assumption holds for the “social” storage. The decision reacts more elastically to the “cognitive” memory storage than to the “social” one, i.e., a unity change of “cognitive” storage amount produces (slightly) greater change in decision probability than a corresponding change of “social” storage.

Further, we need to specify the process of filling the memory. In the ABM above, if the number of “average” ritual site attendants is huge then not all of them can be recorded in an individual memory; the same we assume of the abstract collective memory in the SDM. That is, a proportion of remembered individuals committing to a ritual site may be determined by ritual site’s average size; the greater the site (the more committed) implies the less proportion of remembered individuals. The both parts of memory are cleared with a time progress, the encounters are forgotten. The rate of this oblivion may differ in the distinct parts of the memory. Another element of the model is what we call social satisfaction. We suppose that it is greater after visit of a SF site than of a CA site. A measure of this “social benefit of the SF site” is characterized by a “social function parameter”.

The presented considerations and assumptions were implemented into an SDM (for the flow chart see figure 4). The model consists of two “basic” and two “ancillary” stocks. The basic stocks stand for the occupancy of the two ritual sites. In this model we do not consider any absolute population size, the stocks only express a relative proportion of the population committed the particular group of ritual sites. Since we assume a “closed population”, the stocks are cyclically connected by flows – SF runs to CA and CA runs to SF. The ancillary stocks represent amounts of the two parts of memory. The inflow to the stock representing the memory on occupation of SF sites is influenced by the corresponding basic stock; the outflow from it, expressing the oblivion process, depends on its contents. The similar statement holds for the second ancillary stock representing the remembered occupancy of CA sites. The whole process is directed by seven parameters. Two of them characterize the strategies, other two specify the adaptive decision and the remaining three the remembrance and oblivion of encounters in ritual sites. Namely, let us denote:

Figure 4
Figure 4

The SDM of evolution of the ritual sites occupancy created in PowerSim Studio software. See the text for explanation.

Citation: Journal of Cognition and Culture 18, 5 (2018) ; 10.1163/15685373-12340041

p, q

proportions of random and conservative strategies, respectively,

r

proportion of adaptive individuals committing SF site immediately,

CAS

sensitivity to cognitive attraction,

γ

rate of remembered individuals,

w, v

rates of oblivion of encounters in SF and CA sites, respectively.

All of the parameters are dimensionless and ranging from 0 to 1. The parameter r can be considered as strength or intensity of the social function, the parameter CAS represents an extra contribution of cognitive attraction to the decision to commit a particular type of ritual site. The parameters r and CAS are counterparts of SFc and CAc coefficients utilized in the AB model described in the previous section. The CtoSrnd and StoCrnd express the rate of random transition from CA site to SF site and from SF site to CA site, respectively. They depend on the parameter p. Technically, the variables CtoSrnd and StoCrnd are random variables with the expected value one half (for the “random individual”, the probability of changing the site equals to the probability of remaining in it). Hence, it is possible to simulate expectations with constant values CtoSrnd = StoCrnd = ½, or to generate (pseudo)random transitions.

The Adaptive variable represents the number of adaptive individuals. It depends naturally on amounts of randomly flowing and of remaining individuals; i.e., on variables CtoSrnd and StoCrnd and on the parameter q. The variable is composed from two components, To_SF and Memory_dep. The first one expresses the rate of decision to commit the SF site on the basis of social satisfaction, this component depends on the parameter r. The second component computes the available number of individuals making decision on the basis of remembered encounters. The rates of adaptive change of the site group are described by the variables gS (flow from CA site to SF site) and gC (flow from SF site to CA site). Both depend on the Memory_dep component of the Adaptive variable and on the amounts of the two memories. The dependence of the rates gS and gC on the SF memory is immediate, the dependence on the CA memory is modified by the sensitivity parameter CAS through the variable sensitCA. The remaining variables are rates of the flows.

6 System-Dynamics Model Experiment

In order to investigate the SDM, we investigated the results of the model under a range of values for key parameters. A primary question asked was whether it permits an invasion of CA rituals to environment occupied by SF sites. Hence, the initial values used for simulation was 1 for SF stock and 0 for CA stock.

The provided experiments with the SDM shows that a positive but small sensitivity of decision probability on “cognitive memory” storage (the parameter CAS) has a little impact on the behavior of the model output – the occupancy of the SF sites dominates but it does not exclude the CA sites. But increment of the parameter CAS causes the occupancy of CA sites to stabilize at a higher level than SF sites even if the immediate decision to remain in the SF site (the parameter r) is high. The rates of oblivion (the parameters w and v) influence rather rate of evolution than qualitative results of it, but high rate of oblivion may suppress the impact of high parameter CAS. The typical results of simulations are depicted in figure 5.

Figure 5
Figure 5

Results of the SDM experiment. The fixed parameters are p = 0.1, q = 0.5, γ = 0.5, varying parameters are indicated below particular figures. The results of simulation are plotted by broken lines, the smooth lines represent theoretic expectations (see description of the random variables CtoSrnd and StoCrnd for explanation).

Citation: Journal of Cognition and Culture 18, 5 (2018) ; 10.1163/15685373-12340041

7 Discussion

The two models demonstrate that we are able to produce a range of outcomes relevant to the study of ritual behaviors using multiple computer modeling and simulation paradigms. In the most modest sense, this means that our modeling of the two factors may be successful at simulating preferences for certain ritual forms over time. The ABM results revealed that it is easier for cognitive attraction to dominate ritual selection than social function. Our SDM enabled us to investigate the whole system from another abstract perspective and examine this behavior of the model in a more mathematically rigid way. Since ABMs may be more easily presented using natural language, they also represent suitable starting point for first formalized exploration of target phenomena. However, the ABM paradigm allows us to implement previously hidden and unexplored assumptions. On the contrary, an SDM that has been built “bottom-up” (i.e., from small number of parameters through related auxiliary functions to stocks with definite rates of change) does not allow such relaxed approach and instead require detailed explication of all implemented variables since the beginning. Moreover, such a model can be more easily interpreted into the language of mathematics (using differential or difference equations) that enables more clear analysis of the role of individual parameters.2 As we were able to observe similar outputs from both our models, we consider our results to suggest that researchers should be encouraged to take the cognitive attraction of rituals as theoretically seriously as they do the social functionality of a ritual.

Our models represent only one variant among many concerning how to approach the question of cultural transmission of collective rituals. One aspect for future consideration is that we hold social function and cognitive attraction separately as independently operating factors. Without a doubt, there exist real ritual forms which would score very well in respect to both social function and cognitive attraction, thus representing best candidates for success in any cultural transmission competition. We do not consider these variants, as this would be more meaningful if we could also manipulate ritual forms in respect to their constitutive features, such as rigidity, internal repetitiveness, goal-demotion, or involvement of synchronous movements, etc. and consider how their combinations interact. For instance, while goal-demotion is attention grabbing aspect of an activity and thus can produce higher cognitive attraction of a ritual (Nielbo & Sørensen, 2011), it is also an aspect making an activity worse in promoting cooperation than its goal-directed alternative (Mitkidis et al., 2014). Thus, to approach social function and cognitive attraction as variables to be manipulated simultaneously in respect to one ritual form would require us to consider that some ritual features increase social function while decrease cognitive attraction and vice versa. We can imagine this as one possible direction for our future research.

Further, since the rituals in our models are connected with the ritual sites, the ritual forms are stable elements within our modeling environment. The only aspect that is measured and changes over time is the proportion of visits of the ritual sites. In that sense, we directly model a preference of certain types of ritual. We expect that in a historical setting these preferences would have crucial impact upon cultural selection of appropriate ritual forms by communities as the communities will adopt the most popular ritual forms. To address this aspect, we can imagine an extended version of our model, where proportionally less visited sites will be abandoned or change their ritual practice over time. However, by producing more complex data, we expect that such version of the model would be more difficult to analyze in respect to operationalization of our two factors, which was our primary interest here.

This is associated with another limitation of our models, which is the way how we implement social function. We focus only on what can be called an internal social function. This means that our operationalization grasps only how social function affects social behavior of ritual participants in respect to their ritual preferences. We understand this is an extremely limited understanding of social function, as it ignores external social function such as how it contributes to long-term survival of the groups which adopt it. We have two reasons for this limitation. First, we have chosen such an operationalization of social function which would allow us to directly compare its effects to the effects of cognitive attraction. Therefore, we opted for a modeling design that allows us to investigate social function and cognitive attraction as working next to each other, instead of seeing them as two mutually energizing aspects characterizing a ritual form and differing only in that they operate on a different level. Although we agree that this alternative approach could be both theoretically promising and empirically reliable under some conditions, it would be less suitable for the theoretical and analytical purposes we address here. Second, although it is reasonable to assume that there are important external social functions of rituals, it is difficult to grasp them in terms of the bottom-up approach, in which we focus on how such phenomena as religious communities arise from interactions of individuals instead of approaching groups as stable units of analysis (cf. Wildman & Sosis, 2011). In that, we remain faithful to our anti-essentialist position in respect to formation of religious groups and religion in general. Further, despite how intuitive it sounds, it is not so easy to untangle the complex causal chain by which can certain ritual forms contribute to the long-term survival of a group. For modeling purposes, it would require the implementation of several other factors, like variables representing survival challenges for the ritually interacting individuals.

This second point is associated with one important aspect of our modeling ontology. The behavior of our agents might be described, analyzed and visualized by means of various tools from social network analysis and network science and further interpreted in the light of combination of this paradigm with the social brain hypothesis (Dunbar, 1998; Sutcliffe et al., 2012). The changing content of agents’ social memory might be grasped as their dynamic ego-network and the social memory capacity as an evolutionary constraint upon the number of maintained social relationship. From that standpoint, an impact of social function might be associated with how it modifies content and structure of ego-networks. As far as we know, this perspective has not been elaborated in respect to social function of rituals. However, as the computational generation of “Dunbarian” ego-networks has been a subject of several recent studies (e.g. Conti et al., 2011), it also promises to be a good starting point for future more empirically reliable agent-based models targeting associated phenomena.

8 Conclusion

The main goal of this article has been to promote theorizing concerning the role of the two factors discussed in the literature as influencing the process of cultural transmission of rituals. Following our own progress, we introduced the agent-based model followed by the system-dynamics model. We hope that this order is meaningful from a reader’s perspective, as it also demonstrates our own research progress, starting with theoretically stimulating real world phenomenon observation (early Christian ritual meals), continuing with translation of this observation into an artificial environment, and culminating in mathematical simplification of this former translation. Generally, we verified our theoretical assumptions by showing them as feasible in the formalized models. In the future, we would like to continue with this research logic, focusing on how social function and cognitive attraction interplay in forming complex ritual experiences and how this can contribute to preference and survival of particular ritual forms.

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Appendix: ODD Kaše et al.

Agent-Based Model Description: ODD Protocol

To describe our ABM, we use an extension of the standard ODD protocol consisting of the model’s overview, design concepts, and details (Grimm et. al., 2006; Grimm et al. 2010; Grimm et al., 2013). To better describe the human-like behavior of our model, the extension adds a section on decision making – therefore ODD + D protocol (Müller et al., 2013). As decision making processes are the most complex component of our model, we postpone its description until the details section. The goal of the protocol is to make the model understandable. Therefore, some information from the previous section will be repeated here, but in a more technical form and order.

Overview

Purpose

The model serves as a theory-building conceptualization of the transmission dynamics of collective rituals inside a population of interacting individuals as driven by two non-deliberative decision-making factors: cognitive attraction and social function. It creates an idealized environment for a bottom-up cultural selection of two ritual forms reflecting the two factors. The model aims to show specific dynamics associated with the two factors, revealing ratcheting effect towards the ritual form characterized by cognitive attraction.

Entities, State Variables, and Scales

The model consists of an environment of discrete ritual places divided into two even groups: social function groups (SFg) and cognitive attraction groups (CAg). In these places, a population of human agents holds gatherings (encounters) at each turn. Each agent has constrained memory for ritual experiences gained from such encounters. Each group of ritual places affects the experience differently. The agent is bound to use one from three ritual experience seeking strategies (random, conservative and adaptive). The usage ratio of strategies is fixed in the population (10% random, 50% conservative, 40% adaptive) and for individual agents changes each turn randomly.

There are two main exogenous drivers: social-function-coefficient or SFc (value 0–1) and cognitive-attraction-coefficient or CAc (value 0–0.1). They represent simple abstractions of targeted factors used to compute the outcome of the adaptive strategy while shifting the tendency to gather to at a ritual place from the respective group. Time moves in discrete time-steps. At each time-step a ritual encounter is held in one of the ritual places. In the model, each time-step represents a time unit of a week. The model does not end up with any equilibrium state, due to the hardwired variation resulting from the constant presence of random experience seeking strategy. However, the configuration of the coefficient associated with the two factors can create a significant preference for one or other ritual form over time.

Process Overview and Scheduling

Time is modelled in discrete time-steps and the same process repeats each time-step. At the beginning of each time-step, each agent adopts one experience seeking strategy. This strategy is randomly allocated along a fixed distribution, as discussed above. The behavior driven by the strategy results in choosing one ritual place and moving there. For the agent, this place becomes its ritual encounter place for that time step. The process of agent allocation differs according to the experience seeking strategy. Randomly acting agents choose a random place from all possible places. Conservatively acting agents choose the same place as its last encounter. Agents with the “adaptive” strategy compute their choice on the basis of its encounter memory and actual situation in ritual places by aiming to maximize its ritual experience; i.e., to meet the “best” acquaintances, which are the agents that are most familiar from previous encounters. In this computation the two main driving factors come into play and influence decision making. At the end of the turn, every agent adds the actual encounter to their memory, where previous encounters are already stored.

Design Concepts

Individual Decision Making – Each agent makes one decision per time-step; i.e., the agent decides which to visit, from the all ritual places. For this the agent uses one of the experience seeking strategies discussed above (random, conservative, adaptive) (see below sections on “Agent memory” and “Experience seeking strategies computation”).

Learning – The agents form social relationships based on the ritual experience. This experience further influences the decision making during the experience seeking strategy with adaptive computation.

Sensing – While located in a place during the ritual encounter, the agents sense and remember the present agents and the type of the respective ritual place. Further, when deciding according to the adaptive experience seeking strategy, the agents have an unrestricted view of the world in the sense that they have full access to the information about allocations of the already located agents (those who behave according to the random and conservative strategy during the given time-step).

Interaction – Agents are limited to sensing each other during the ritual encounter. In that sensing, the most basic social relationship (familiarity) is established and held in individual memory.

Stochasticity – The significant stochastic mechanisms in the model are the 1) initialization, where agents get their first experience and 2) the random experience seeking strategy, which is followed by 10% agents at any given time-step.

Collective – During the simulation, agents become part of emergent collectives (networks of interconnected agents) based on their encounter experiences (memory). The analysis of these collectives is not part of this article.

Heterogeneity – The agents have the same probabilities to use one of the three strategies. But what differs is their ritual experience changing between time-steps.

Observation – The measured output of the simulation is the total difference of visits between the two ritual group places. The outcome can have three different states: (1) The random variation beats the underlying forces of ritual activity and none of the group of ritual places representing the two ritual forms becomes significantly preferred, i.e., none of the group has more visits than the other; (2) the ritual places with SF quality have more visits; (3) the ritual places with CA quality have more visits.

Details

Initialization

At initialization, a population of 200 agents is randomly and evenly distributed over 10 ritual places and each agent forms their first encounter memory with the other agents at their ritual place.

Ritual activity

At each time-step each agent choses a place of their ritual encounter using their experience seeking strategy, which is randomly distributed among the agents at the beginning of each time-step. The assigned experience seeking strategy leads the agent to find an appropriate ritual place and to move there. The agents assigned with random and conservative strategies proceed first, the agents with adaptive strategy follow. The agents behaving according to the adaptive strategy base their decision upon their access to information about locations of the previously located agents behaving according to the random and conservative strategies. At the end of the time-step the memory of agents is updated by the type of the ritual place and those encountered there.

Agent memory

The agent memory is a fixed 5-positional stack filled with information about past encounters, where the freshest memory is at the top of the stack and where each position represents one of the past time-steps. When the stack exceeds 5, the most time distant memory is deleted.

The memory consists of two parts, the ritual quality memory and social encounter memory. The first is responsible for remembering the ritual quality of the visited place, that is to which group of places the place belongs. The second is responsible for memorizing all individual agents which were present in the place during the encounter. Both parts are crucial for computing the values on which is based the adaptive experience seeking strategy.

The social encounter memory is constrained to limited capacity of 150 slots for memory of encounter with individual agents. More recent encounter memories hold more space in this memory than older onward. Formally represented, the count of agents from the most recent encounter is multiplied by 3; in the case of the second most recent encounter, the count of agents is multiplied by 2. In the case of older encounters, the count is represented in the memory as one by one. As a result of this, the agents met over older encounters are pushed out of the memory as the memory capacity is depleted by those met more recently.

Experience seeking strategies computation

Random and conservative strategies follow simple heuristics. When adopting the random strategy, the agent chooses randomly from all possible places. In a conservative strategy, the agents move to the same ritual place that they occupied in the previous time-step.

In the adaptive strategy, the agents decide by considering each ritual place in terms of its prospective encounter value, which consists of a combination of the social value (familiarity with agents located in the ritual places based on past encounters) and affinity for cognitive attraction value (see further). All ritual places are marked with the prospective ritual encounter value and the place with the highest mark is chosen and the agent moves there.

The marks of the ritual places are computed by measuring a fit between the content of agent’s memory (i.e., visited places and encountered agents over the last 5 time-steps) and places visited by a proportion of agents using random or conservative strategies. Here the main two drivers (SFc and CAc) influence the computation in a different way.

The score of a ritual place is ascribed in several steps. First, the agent examines all ritual places and checks their visitors while comparing them with those which the agent holds in memory from previous encounters. Whenever the agent recognizes another agent on the examined ritual place as someone in in its memory, the agent increases the score of the ritual place by one. In a null model (i.e., when the effect of SFc and CAc are set up to zero) the agent goes into a place scoring highest in that respect. When the SFc is higher than zero, the agent takes into consideration previous encounters with other agents at the ritual space. For each agent that was previously encountered on SF place, the score is increased by 1 and multiplied by (1 + SFc value). Thus, the agent pays higher attention to encounters formed on the SF places than to encounters formed on the CA places.

Second, when the CAc parameter is higher than zero, the agent modifies the marks of all ritual places on the basis of what type of places the agent visited over the past 5 time-steps. The most recent memory has the strongest impact again. Thus, if the agent visited a CA place one week back, the agent multiplies the overall mark of all current CA places by (1 + [CAc value * 5]), where 5 reflects the fact that it is the most recent memory from the 5 slots. Further, if the agent visited a CA place two weeks back as well, the agent multiplies the overall mark of all current CA places again, now by (1 + [CAc value * 4]) etc. Thus, in agreement with the theoretical framework, past visits of CA places make the agent cumulatively more motivated to visit this type of places again and again over time, regardless with whom the agent can meet there.

1

We are well aware of the fact that ancient Mediterranean world was much less individualistic than the modern world. However, there still was a substantial proportion of those who could behave deliberately in religious matters, openly expressing their changing preferences associated with particular Christian or non-Christian practices.

2

It is fair to note here that this is not a general statement on comparison of ABMs and SDMs; one can imagine that it is possible to model one phenomenon using ABM with few parameters and using SDM consisting of many stocks linked by complicated feedbacks with flows controlled by a large number of random variables.

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