  We propose and analyze a nonlinear dynamic model of continuous-time
multi-dimensional belief formation over signed social networks. Our model
accounts for the effects of a structured belief system, self-appraisal,
internal biases, and various sources of cognitive dissonance posited by recent
theories in social psychology. We prove that agents become opinionated as a
consequence of a bifurcation. We analyze how the balance of social network
effects in the model controls the nature of the bifurcation and, therefore, the
belief-forming limit-set solutions. Our analysis provides constructive
conditions on how multi-stable network belief equilibria and belief
oscillations emerging at a belief-forming bifurcation depend on the
communication network graph and belief system network graph. Our model and
analysis provide new theoretical insights on the dynamics of social systems and
a new principled framework for designing decentralized decision-making on
engineered networks in the presence of structured relationships among
alternatives.
