  The Kuramoto model of coupled second order damped oscillators on convergent
sequences of graphs is analyzed in this work. The oscillators in this model
have random intrinsic frequencies and interact with each other via nonlinear
coupling. The connectivity of the coupled system is assigned by a graph which
may be random as well. In the thermodynamic limit the behavior of the system is
captured by the Vlasov equation, a hyperbolic partial differential equation for
the probability distribution of the oscillators in the phase space. We study
stability of mixing, a steady state solution of the Vlasov equation,
corresponding to the uniform distribution of phases. Specifically, we identify
a critical value of the strength of coupling, at which the system undergoes a
pitchfork bifurcation. It corresponds to the loss of stability of mixing and
marks the onset of synchronization. As for the classical Kuramoto model, the
presence of the continuous spectrum on the imaginary axis poses the main
difficulty for the stability analysis. To overcome this problem, we use the
methods from the generalized spectral theory developed for the original
Kuramoto model. The analytical results are illustrated with numerical
bifurcation diagrams computed for the Kuramoto model on Erd\H{o}s--R\'enyi and
small-world graphs. Applications of the second-order Kuramoto model include
power networks, coupled pendula, and various biological networks. The analysis
in this paper provides a mathematical description of the onset of
synchronization in these systems.
