  We consider an extension of a binary decision model in which nodes make
decisions based on influence-biased averages of their neighbors' states,
similar to Ising spin glasses with on-site random fields. In the limit where
these influences become very heavy-tailed, the behavior of the model
dramatically changes. On complete graphs, or graphs where nodes with large
influence have large degree, this model is characterized by a new "phase" with
an unpredictable number of macroscopic shocks, with no associated critical
phenomena. On random graphs where the degree of the most influential nodes is
small compared to population size, a predictable glassy phase without phase
transitions emerges. Analytic results about both of these new phases are
obtainable in limiting cases. We use numerical simulations to explore the model
for more general scenarios. The phases associated with very influential
decision makers are easily distinguishable experimentally from a homogeneous
influence phase in many circumstances, in the context of our simple model.
