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In our previous work [Chiba, Medvedev, arXiv:1612.06493], we initiated amathematical investigation of the onset of synchronization in the Kuramotomodel (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 thecoupling 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 ofsynchronization in the KM on graphs is realized via a pitchfork bifurcation.The formula for the stable branch of the bifurcating equilibria involves theprincipal eigenvalue and the corresponding eigenfunctions of the kerneloperator defined by the limit of the graph sequence used in the model. Thisestablishes an explicit link between the network structure and the onset ofsynchronization in the KM on graphs. The results of this work are illustratedwith 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.

The mean field analysis of the Kuramoto model on graphs II. Asymptotic  stability of the incoherent state, center manifold reduction, and  bifurcations