  We study the behavior of solutions of mutually coupled equations in
heterogeneous random graphs. Heterogeneity means that some equations receive
many inputs whereas most of the equations are given only with a few
connections. Starting from a situation where the isolated equations are
unstable, we prove that a heterogeneous interaction structure leads to the
appearance of stable subspaces of solutions. Moreover, we show that, for
certain classes of heterogeneous networks, increasing the strength of
interaction leads to a cascade of bifurcations in which the dimension of the
stable subspace of solutions increases. We explicitly determine the bifurcation
scenario in terms of the graph structure.
