  In our previous work [Chiba, Medvedev, arXiv:1612.06493], we initiated a
mathematical investigation of the onset of synchronization in the Kuramoto
model (KM) of coupled phase oscillators on convergent graph sequences. There,
we derived and rigorously justified the mean field limit for the KM on graphs.
Using linear stability analysis, we identified the critical values of the
coupling strength, at which the incoherent state looses stability, thus,
determining the onset of synchronization in this model.
  In the present paper, we study the corresponding bifurcation. Specifically,
we show that similar to the original KM with all-to-all coupling, the onset of
synchronization in the KM on graphs is realized via a pitchfork bifurcation.
The formula for the stable branch of the bifurcating equilibria involves the
principal eigenvalue and the corresponding eigenfunctions of the kernel
operator defined by the limit of the graph sequence used in the model. This
establishes an explicit link between the network structure and the onset of
synchronization in the KM on graphs. The results of this work are illustrated
with the bifurcation analysis of the KM on Erd\H{o}s-R{\' e}nyi, small-world,
as well as certain weighted graphs on a circle.
