  Most real-world networks display not only a heterogeneous distribution of
degrees, but also a heterogeneous distribution of weights in the strengths of
the connections. Each of these heterogeneities alone has been shown to suppress
synchronization in random networks of dynamical systems. Here we review our
recent findings that complete synchronization is significantly enhanced and
becomes independent of both distributions when the distribution of weights is
suitably combined with the distribution of degrees. We also present new results
addressing the optimality of our findings and extending our analysis to phase
synchronization in networks of non-identical dynamical units.
