We focus on the "trijunction" between multiscale computations, bifurcation theory and social networks. In particular, we address how the Equation-Free approach, a recently developed computational framework, can be exploited to systematically extract coarse-grained, emergent dynamical information by bridging detailed, agent-based models of social interactions on networks, with macroscopic, systems-level, continuum numerical analysis tools. For our illustrations, we use a simple dynamic agent-based model describing the propagation of information between individuals interacting under mimesis in a social network with private and public information. We describe the rules governing the evolution of the agents' emotional state dynamics and discover, through simulation, multiple stable stationary states as a function of the network topology. Using the Equation-Free approach we track the dependence of these stationary solutions on network parameters and quantify their stability in the form of coarse-grained bifurcation diagrams.