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In this paper, we investigate the effect of local structures on networkprocesses. We investigate a random graph model that incorporates local cliquestructures to deviate from the locally tree-like behavior of most standardrandom graph models. For the process of bond percolation, we derive analyticalapproximations for large outbreaks and the critical percolation value.Interestingly, these derivations show that when the average degree of a vertexis large, the influence of the deviations from the locally tree-like structureis small. Our simulations show that this insensitivity to local cliquestructures often already kicks in for networks with average degrees as low as6. Furthermore, we show that the different behavior of bond percolation onclustered networks compared to tree-like networks that was found in previousworks can be almost completely attributed to differences in degree sequencesrather than differences in clustering structures. We finally show that theseresults also extend to completely different types of dynamics, by derivingsimilar conclusions and simulations for the Kuramoto model on the same types ofclustered and non-clustered networks.

Network processes on clique-networks with high average degree: the limited effect of higher-order structure