  In this paper, we investigate the effect of local structures on network
processes. We investigate a random graph model that incorporates local clique
structures to deviate from the locally tree-like behavior of most standard
random graph models. For the process of bond percolation, we derive analytical
approximations for large outbreaks and the critical percolation value.
Interestingly, these derivations show that when the average degree of a vertex
is large, the influence of the deviations from the locally tree-like structure
is small. Our simulations show that this insensitivity to local clique
structures often already kicks in for networks with average degrees as low as
6. Furthermore, we show that the different behavior of bond percolation on
clustered networks compared to tree-like networks that was found in previous
works can be almost completely attributed to differences in degree sequences
rather than differences in clustering structures. We finally show that these
results also extend to completely different types of dynamics, by deriving
similar conclusions and simulations for the Kuramoto model on the same types of
clustered and non-clustered networks.

Physics and Society;Probability