Dissemination of Opinions And Ideas Via Complex Contagion on Social Networks Alex Stivala, Yoshihisa Kashima, Garry Robins, Michael Kirley The University of Melbourne, Australia AOARD FA

1.Approach to cultural dynamics 2.The Axelrod model of cultural dissemination 3.Cultural space 4.Social space 1.diversity and community 5.Culture and Cooperation 6.Ongoing work: correlated cultural vectors Outline

How can individuals’ particular meaning- making activities with other individuals in situ can collectively form, maintain, or transform what looks like a context-general system of meaning? How does what looks like a context-general system of meaning constitute the local information environment, so as to shape the individuals’ particular meaning-making activities? Cultural Dynamics

Approach to Cultural Dynamics Neo-diffusionist approaches (e.g., Campbell, 1975; Dawkins, 1976; Cavalli-Sforza & Feldman, 1981; Boyd & Richerson, 1985) to cultural dynamics: Social transmission (or diffusion) of cultural information in a population as central to cultural dynamics. Adaptive cultural information tends to be retained and become widespread in a population, and prevalence of cultural information in a population (i.e., how often is it available, accessible, and applicable in situations) determines its population characteristics.

Sender Presentation Interpersonal Process Acceptance (giving evidence of comprehension) t Receiver Joint Intention Instrumental goal Social goals Coordination Mechanisms Common t Common t + Δ t t + Δ cultural information Encoded cultural information Grounding Model of Cultural Transmission e.g., Kashima, et al. (2007)

The Axelrod (1997) model of cultural dissemination It captures some of the essential aspects of the grounding model. “Culture” is represented by a vector of F features each with q possible values. Cultural similarity is the number of matching traits. Agents on a lattice, only neighbors can interact. At each step a random agent and one of its neighbors is chosen

Axelrod’s Diffusion Model Pfau et al. (2012) Physica A

The Axelrod model (cont.) With probability proportional to their similarity they interact: – A random trait on one agent is changed to become equal to that trait on the other Keep going until convergence, At which point all neighboring agents have either identical or completely distinct culture vectors

What the Axelrod model represents Homophily: individuals prefer to interact with similar others Social influence: interactions typically cause others to become more similar What is surprising: this does not necessarily result in all agents converging to the same culture!

Absorbing states of the Axelrod model There are two possible absorbing (frozen) states: – Monocultural: All agents have the same culture vector. – Multicultural: Any two agents in the same region have the same culture. Any two agents that are neighbors but in different regions have completely distinct cultures. A phase transition between the two states occurs at the critical value of q (number of traits).

Example of an Axelrod model run with number of features F = 5, number of traits q = 15, on a 10 x 10 lattice. The darkness of the lines between sites indicates the cultural similarity (white is identical, black is completely different). Y. Kashima, M. Kirley, A. Stivala, and G. Robins, Modelling cultural dynamics, in Computational Social Psychology, Frontiers of Social Psychology (Psychology Press, New York, 2016, in press).

Co-evolution in Physical, Social, and Cultural Spaces Pfau et al. (2012) Physica A

Initial research questions What is the structure of “cultural space”? How do culture, ideas, and opinions spread, and how is this affected by the structure of cultural and social space? How then do we best spread innovations or compelling ideas through a social network?

Cultural Space: Ultrametric structure of real opinion vectors increases long-term diversity Valori, L., Picciolo, F., Allansdottir, A., & Garlaschelli, D. (2012). Reconciling long-term cultural diversity and short-term collective social behavior. Proceedings of the National Academy of Sciences, 109(4), Correlations between empirical culture (opinion) vectors induce an ultrametric (hierarchical) structure that facilitates short- term cooperation based on similarity, but also long-term diversity compared to random vectors. Initial connected cultural components Final number of cultures

Ultrametricity + variance in cultural distance allows greater diversity Stivala, A., Robins, G., Kashima, Y., & Kirley, M. (2014). Ultrametric distribution of culture vectors in an extended Axelrod model of cultural dissemination. Scientific Reports, 4:4870. Culture vectors simulated by evolution from a small number of “prototype” cultures are ultrametrically distributed, as are the other two schemes (neutral evolution and trivial ultrametric), but have greater variance in intervector distance and result in even greater diversity surviving at the absorbing state of the Axelrod model.

Social Space: Residential segregation and social networks The Schelling (1969) model of residential segregation shows that even a small preference to live near similar others leads to high degrees of segregation. And if social network ties are formed based on homophily and proximity, it would seem then that highly clustered social networks are not compatible with diversity (low segregation) [Neal & Neal 2014].

Diversity and sense of community are incompatible? Neal, Z. P., & Neal, J. W. (2014). The (in)compatibility of diversity and sense of community. American Journal of Community Psychology, 53(1-2), 1-12.

Cultural evolution shows a way out But if we consider the possibility of culture vectors that contain both immutable (Schelling) and mutable (Axelrod) features, the situation changes. If cultural diversity is sufficiently large, then under some conditions diversity is no longer negatively correlated with social network clustering coefficient (Stivala et al., 2016).

Diversity and community can coexist Stivala, A., Robins, G., Kashima, Y., & Kirley, M. (2016). Diversity and Community Can Coexist. American Journal of Community Psychology, 57(1-2),

Evolution of cooperation Public goods game: prisoner’s dilemma with more than two players Each player in a lattice region either contributes (cooperates) or not (defects) a fixed amount to a pool. The pool is multiplied by a fixed factor, and the proceeds equally distributed among the participants. Under what conditions does the dominant strategy (defection) not drive out cooperation?

Culture and cooperation There is a huge amount of work on iterated games and evolution of cooperation. And an extensive literature on variations and extensions of the Axelrod model of cultural dissemination. But now we combine these models and investigate the coevolution of culture and cooperation.

Culture and conditional cooperation An Axelrod model, but with noise and multilateral influence (more than two participants). An interaction in the model consists of playing multiple public goods games. So also a coevolutionary spatial public goods game. Probability of participation and cooperation (for cooperators) is proportional to cultural similarity. Both a cultural trait and the cooperate/defect trait are updated probabilistically according to the total payoff in the public goods games.

Coevolution of culture and cooperation example

Cooperation only survives in multicultural states with no noise Stivala, A., Kashima, Y., & Kirley, M. (2016). Culture and cooperation in a spatial public goods game. Manuscript submitted to Physical Review E.

Ongoing work So far it has always been assumed that the features in the culture vectors are independent. (The correlation structure in the empirical / ultrametric case is between agents, not features within agents). But what happens if there are correlations between features? – And further, what if these correlations persist; there is some network of relationships between features?

Connections between features For every pair of features i, j we have a conditional probability (learned from empirical or simulated data) of feature i having trait value x given that feature j has trait value y (for all trait values 1 <= x, y <= q). Then when updating a feature, with this probability we also update a “connected” (correlated) feature.

Publications arising from the grant Stivala, A., Robins, G., Kashima, Y., & Kirley, M. (2016). Diversity and Community Can Coexist. American Journal of Community Psychology, 57(1-2), Stivala, A., Kashima, Y., & Kirley, M. (2016). Culture and cooperation in a spatial public goods game. Manuscript submitted to Physical Review E.

Acknowledgments Asian Office of Aerospace Research and Development (AOARD) grant number FA Australian Research Council (ARC) grant number DP Victorian Life Sciences Computation Initiative (VLSCI) grant number VR0261