  Employing the Kuramoto model as an illustrative example, we show how the use
of the mean field approximation can be applied to large networks of phase
oscillators with assortativity. We then use the ansatz of Ott and Antonsen
[Chaos 19, 037113 (2008)] to reduce the mean field kinetic equations to a
system of ordinary differential equations. The resulting formulation is
illustrated by application to a network Kuramoto problem with degree
assortativity and correlation between the node degrees and the natural
oscillation frequencies. Good agreement is found between the solutions of the
reduced set of ordinary differential equations obtained from our theory and
full simulations of the system. These results highlight the ability of our
method to capture all the phase transitions (bifurcations) and system
attractors. One interesting result is that degree assortativity can induce
transitions from a steady macroscopic state to a temporally oscillating
macroscopic state through both (presumed) Hopf and SNIPER (saddle-node,
infinite period) bifurcations. Possible use of these techniques to a broad
class of phase oscillator network problems is discussed.
