Epistasis between cultural traits causes paradigm shifts in cultural evolution

Every now and then the cultural paradigm of a society changes. While current models of cultural shifts usually require a major exogenous or endogenous change, we propose that the mechanism underlying many paradigm shifts may just be an emergent feature of the inherent congruence among different cultural traits. We implement this idea through a population dynamics model in which individuals are defined by a vector of cultural traits that changes mainly through cultural contagion, biased by a ‘cultural fitness’ landscape, between contemporary individuals. Cultural traits reinforce or hinder each other (through a form of cultural epistasis) to prevent cognitive dissonance. Our main result is that abrupt paradigm shifts occur, in response to weak changes in the landscape, only in the presence of epistasis between cultural traits, and regardless of whether horizontal transmission is biased by homophily. A relevant consequence of this dynamics is the irreversible nature of paradigm shifts: the old paradigm cannot be restored even if the external changes are undone. Our model puts the phenomenon of paradigm shifts in cultural evolution in the same category as catastrophic shifts in ecology or phase transitions in physics, where minute causes lead to major collective changes.

2) It could be a bit uncomfortable reading about "fitness" without a proper definition of the concept until much later. I understand that the organization of the manuscript might required the fitness to be introduced later, but perhaps the authors could find a solution not to delay the presentation of the concept so much.
3) There seems to be an inconsistency in the definition of dH. The first definition states that it is the usual Hamming distance, i.e. the number of differences in the values of the cultural traits between two nodes. But later the authors say that the maximum value of dH is reached for a given value of alpha. How is that the Hamming distance depends on alpha if it only measures the number of coincidences? At least this not clear considering the role assigned to alpha 4) The nodes are numbered and ordered accordingly. If we locate the nodes in the ring, the epistasis, when present, links each node with its neighbours in only one direction and always in order. Is this restriction something that could affect the results? It is hard to accept that there is no reciprocity in the epistasis unless the value of k is close to the value of N.
5) The fitness, that is governing the evolution of the cultural profile of the population, is not affected by homophily so why should homophily be important? Is not that a trivial results?

Review form: Reviewer 2
Is the manuscript scientifically sound in its present form? Yes

Are the interpretations and conclusions justified by the results? Yes
Is the language acceptable? Yes Do you have any ethical concerns with this paper? No

Recommendation?
Accept with minor revision (please list in comments)

Comments to the Author(s)
In the manuscript entitles "Epistasis between cultural traits cause paradigm shifts in cultural evolution" by Ignacio Pascual et al. the authors develop a simple model for cultural dissemination showing that paradigm shifts can occur as emergent phenomena of the underlying dynamics.
In this model, the individuals are characterized by a vector of cultural traits that change mainly through cultural contagion biased by "cultural fitness" lanscape. The authors show that this model produces paradigm shifts in response to weak changes in the landscape only in the presence of cultural epistasis.
The model is perfectly framed and its equations are rigourosly derived. The presented results are clear and informative and the discussion highlight the virtues of the model, but also its limitations and possible future works.
The manuscript is well written and, from my point of view, it deserve to be published in RSOS, after the authors can clarify some minor comments and questions: The reviewers and handling editors have recommended publication, but also suggest some minor revisions to your manuscript. Therefore, I invite you to respond to the comments and revise your manuscript.
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Once again, thank you for submitting your manuscript to Royal Society Open Science and I look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch. In this work the authors present a model for paradigms shift along cultural evolution, based on Axelrod' s original proposal for cultural evolution. The main conclusion in this work is that epistasis, properly defined in the context of the present model, is responsible for these shifts.
The work explores original ideas and present a novel formal model, that although based on previous works, contains interesting new ideas.
The presentation of the model, the mathematical approach and results are presented in a clear way but I have some observation listed below i would ask the authors to address 1) Fig 1 is not clear. In the text the authors say that figs. a and c show the fitness and b and d the cultural state, while the legend in the vertical axis reads "fitness" in all cases. The number of nodes seems not to be the same, not even when considering the pairs and b, c and d.
By the definition, the cultural states can be only 0 or 1, while the plot shows something different, perhaps due to an issue with the graphing soft.
2) It could be a bit uncomfortable reading about "fitness" without a proper definition of the concept until much later. I understand that the organization of the manuscript might required the fitness to be introduced later, but perhaps the authors could find a solution not to delay the presentation of the concept so much.
3) There seems to be an inconsistency in the definition of dH. The first definition states that it is the usual Hamming distance, i.e. the number of differences in the values of the cultural traits between two nodes. But later the authors say that the maximum value of dH is reached for a given value of alpha. How is that the Hamming distance depends on alpha if it only measures the number of coincidences? At least this not clear considering the role assigned to alpha 4) The nodes are numbered and ordered accordingly. If we locate the nodes in the ring, the epistasis, when present, links each node with its neighbours in only one direction and always in order. Is this restriction something that could affect the results? It is hard to accept that there is no reciprocity in the epistasis unless the value of k is close to the value of N.
5) The fitness, that is governing the evolution of the cultural profile of the population, is not affected by homophily so why should homophily be important? Is not that a trivial results?
Reviewer: 2 Comments to the Author(s) In the manuscript entitles "Epistasis between cultural traits cause paradigm shifts in cultural evolution" by Ignacio Pascual et al. the authors develop a simple model for cultural dissemination showing that paradigm shifts can occur as emergent phenomena of the underlying dynamics.
In this model, the individuals are characterized by a vector of cultural traits that change mainly through cultural contagion biased by "cultural fitness" lanscape. The authors show that this model produces paradigm shifts in response to weak changes in the landscape only in the presence of cultural epistasis.
The model is perfectly framed and its equations are rigourosly derived. The presented results are clear and informative and the discussion highlight the virtues of the model, but also its limitations and possible future works.
The manuscript is well written and, from my point of view, it deserve to be published in RSOS, after the authors can clarify some minor comments and questions: 1 -In line 14 of page 5, the sentence reads "The last term in this expression" referring to equation (2.5). However, this equation have only one term.
2 -The description of the Kauffman's NK landscape is a little bit confusing for me. Could the authors improve this section? Here, in line 27 which describe equation (2.10) they refer to an index |j| which seems to be absent in the mentioned equation. Decision letter (RSOS-191813.R1) 30-Jan-2020 Dear Dr Cuesta, It is a pleasure to accept your manuscript entitled "Epistasis between cultural traits causes paradigm shifts in cultural evolution" in its current form for publication in Royal Society Open Science. The comments of the reviewer(s) who reviewed your manuscript are included at the foot of this letter.
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Thank you for your fine contribution. On behalf of the Editors of Royal Society Open Science, we look forward to your continued contributions to the Journal. Thank you very much for your accepting our manuscript for publication in Royal Society Open Science. We have gone through all minor comments that the referees made and either fixed them or provided a response to them all. In what follows you will find the detailed responses. The main conclusion in this work is that epistasis, properly defined in the context of the present model, is responsible for these shifts.

The presentation of the model, the mathematical approach and results are presented in a clear way but I have some observation listed below i would ask the authors to address
We thank the referee for the very positive overall judgment and for the careful revision. There y axis in b and d was incorrectly labeled, thank you for spotting the error: it has to be population density. All simulations have 64 nodes (2 6 ). The figure has been substituted by its raw, unedited version for clarity. Here, any impression of a different number of nodes is due to the 2D projection.
By the definition, the cultural states can be only 0 or 1, while the plot shows something different, perhaps due to an issue with the graphing soft.
The reviewer is right that each trait takes values 0 or 1. But the cultural state is defined by a vector of such traits which is assigned a real fitness value between 0 and 1. We have revised the description of the model and of the variables involved and have found no mistake that could lead to that confusion. Therefore, we have not changed the ms in regard to that comment.
2) It could be a bit uncomfortable reading about "fitness" without a proper definition of the concept until much later.
I understand that the organization of the manuscript might required the fitness to be introduced later, but perhaps the authors could find a solution not to delay the presentation of the concept so much.
We have moved previous subsections introducing the definition of fitness landscapes and Kauffman's NK model; they precede now the definition of the system dynamics. The last paragraph in the latter section has been also moved for consistency and an introductory sentence (highlighted in red) has been added. We are unable to spot the inconsistency stated by the referee. Indeed, dH is the usual Hamming distance. The maximum we talk about corresponds to a complex function of different parameters and variables, among them dH and the degree of homophily. And this function has a maximum when certain combination of dH and alpha occurs. This does not imply that dH depends on alpha since, as the referee states, it just measures the number of coincidences. This nonetheless, we have rephrased the sentence to clarify that it is not the increase in alpha that directly modifies the distance between individuals.

3) There seems to be an inconsistency in the definition of dH. The first definition states that it is the usual
4) The nodes are numbered and ordered accordingly. If we locate the nodes in the ring, the epistasis, when present, links each node with its neighbours in only one direction and always in order. Is this restriction something that could affect the results? It is hard to accept that there is no reciprocity in the epistasis unless the value of k is close to the value of N.
Kauffman's NK model is admittedly a cartoon model for real epistasis. However, we use it because its qualitative properties correctly capture the gross features of epistasis. In a cultural context, data are insufficient to describe even the most generic properties of the landscape. The reciprocity the reviewer mentions holds in general in molecular sequences, but perhaps this is not so in cultural traits. For example, being vegetarian implies that a person does not eat meat, but not eating meat can depend on other personal choices. Despite the lack of reciprocity implicit in the definition of the landscape, the fact that the NK model is able to quantitatively mimic the landscape of RNA secondary structure, for example, strongly supports that this restriction does not affect the results. (Note that nodes are linked to the same number of neighbors in both directions, as explained right after Eq. (2.2) -(2.10) in the previous version. Whether symmetrically linked, as done, or always in one direction and in order, is however irrelevant for the results.) 5) The fitness, that is governing the evolution of the cultural profile of the population, is not affected by homophily so why should homophily be important? Is not that a trivial results?
Homophily affects the distribution of the population in the fitness landscape, and it is in this sense that it might be important: it changes the cultural diversity and often clusters the population in separated groups. Still, it does not affect the occurrence of paradigm shifts. In our view, this result is not trivial. On the contrary, since homophily is an important factor in cultural spread, it is important to show that its presence does not condition the occurrence of such shifts.

Reviewer: 2
Comments to the Author(s)