Cultural Consensus Theory: Aggregating Signed Graphs under a Balance Constraint

Abstract

Cultural Consensus Theory (CCT) consists of cognitive models for aggregating the responses of experts to test questions about some domain of their shared cultural knowledge. This paper proposes a new CCT model for a situation where experts judge the ties in a complete signed graph. New to CCT is that the model imposes a side constraint on the aggregation process that requires that the consensus signed graph satisfy the social network property of structural balance. Balanced signed graphs require that the nodes can be partitioned into two sets with positive ties between nodes in the same set and negative ties between nodes in different sets. While the balance constraint is imposed on the consensus aggregation, it is not assumed that each expert's responses satisfy balance because they may be error-prone or biased. The model is presented in terms of signal detection assumptions that allow heterogeneity in expert ability and item difficulty. Bayesian inference of the model is developed using a specially designed Markov Chain Monte Carlo sampler. It is shown that the sampler can recover parameters from simulated data, and then the model is applied to interpret experimental data. Of particular interest is that the model aggregation reveals a single consensus balanced signed graph with a high posteriori probability despite the fact that none of the experts' responses satisfy the balance constraint.