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To process streaming signals on graph, some adaptive filtering methods have been extended to the field of graph signal processing in recent years. However, nonlinear adaptive filtering methods have rarely been studied over graph signals. In this work, a nonlinear adaptive filtering model on graph is designed based on functional link network. Then, two distributed diffusion adaptive filtering algorithms on graph are derived to online learning from streaming graph signals. In addition, their normalized versions have also been developed to accelerate the convergence rate. Moreover, The stability and steady-state performance of the proposed algorithms are analyzed. Finally, the simulation results show the effectiveness of the proposed model and algorithms for nonlinear graph signal processing. [ABSTRACT FROM AUTHOR]