  We focus on the role played by the node degree distribution on the way
collective phenomena emerge on complex networks. To address this question, we
focus analytically on a typical model for cooperative behaviour, the Majority
Rule, applied to dichotomous networks. The latter are composed of two kinds of
nodes, each kind $i$ being characterized by a degree $k_i$. Dichotomous
networks are therefore a simple instance of heterogeneous networks, especially
adapted in order to reveal the effect of degree heterogeneity. Our main result
are that degree heterogeneity affects the location of the order-disorder
transition and that the system exhibits non-equipartition of the average
opinion between the two kinds of nodes. This effect is observed in the ordered
phase and in the disordered phase.
