  We study an influence network of voters subjected to correlated disordered
external perturbations, and solve the dynamical equations exactly for fully
connected networks. The model has a critical phase transition between
disordered unimodal and ordered bimodal distribution states, characterized by
an increase in the vote-share variability of the equilibrium distributions. The
random heterogeneities in the external perturbations are shown to affect the
critical behavior of the network relative to networks without disorder. The
size of the shift in the critical behavior essentially depends on the total
fluctuation of the external influence disorder. Furthermore, the external
perturbation disorder also has the surprising effect of amplifying the expected
support of an already biased opinion. We show analytically that the vote-share
variability is directly related to the external influence fluctuations. We
extend our analysis by considering a fat-tailed multivariate lognormal
disorder, and present numerical simulations that confirm our analytical
results. Simulations for other network topologies demonstrate the
generalizability of our findings. Understanding the dynamic response of complex
systems to disordered external perturbations could account for a wide variety
of networked systems, from social networks and financial markets to amorphous
magnetic spins and population genetics.
