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Randomized neural networks (NNs), such as random vector functional link (RVFL) and extreme learning machine (ELM), have been widely applied in various classification problems owing to their computational efficiency and universal approximation capability. However, such approaches are designed for regular Euclidean data and lack the ability to generalize to complex structured data. Moreover, their randomly generated parameters often lead to a suboptimal decision boundary with a growing requirement of hidden neurons. In this article, we first propose a plain framework, termed randomized graph convolutional networks (RGCNs), to generalize the classic randomized NNs to the non-Euclidean domain. Then, a hybrid framework called evolution-driven RGCN (EvoRGCN) is presented by using adaptive differential evolution with novelty search strategy to seek the globally optimal graph embedding for the plain RGCN. Finally, we recast the classic ELM and RVFL under the proposed frameworks, resulting in four novel semi-supervised models, including the plain models [i.e., graph convolutional extreme learning machines (GCELMs) and graph convolutional RVFL (GCRVFL)] and the optimized models (i.e., O-GCELM and O-GCRVFL). We show that our approaches are the natural generalization of the traditional randomized NNs in the non-Euclidean domain. Furthermore, our approaches not only retain the advantages of the classic approaches but also enable them to handle graph data. We compare our approaches against many existing methods across regular datasets and graph benchmarks, demonstrating that the proposed approaches dramatically outperform the compared methods with better generalization ability and robustness. Particularly, we quantitatively show the performance ranking of different randomized NNs, i.e., O-GCRVFL $> $ O-GCELM $\approx $ GCRVFL $> $ GCELM $\approx $ RVFL $> $ ELM. [ABSTRACT FROM AUTHOR]