  We propose a simple model for a binary decision making process on a graph,
motivated by modeling social decision making with cooperative individuals. The
model is similar to a random field Ising model or fiber bundle model, but with
key differences on heterogeneous networks. For many types of disorder and
interactions between the nodes, we predict discontinuous phase transitions with
mean field theory which are largely independent of network structure. We show
how these phase transitions can also be understood by studying microscopic
avalanches, and describe how network structure enhances fluctuations in the
distribution of avalanches. We suggest theoretically the existence of a
"glassy" spectrum of equilibria associated with a typical phase, even on
infinite graphs, so long as the first moment of the degree distribution is
finite. This behavior implies that the model is robust against noise below a
certain scale, and also that phase transitions can switch from discontinuous to
continuous on networks with too few edges. Numerical simulations suggest that
our theory is accurate.
