  Complex networks can be understood as graphs whose connectivity deviates from
those of regular or near-regular graphs, which are understood as being
`simple'. While a great deal of the attention so far dedicated to complex
networks has been duly driven by the `complex' nature of these structures, in
this work we address the identification of simplicity, in the sense of
regularity, in complex networks. The basic idea is to seek for subgraphs
exhibiting small dispersion (e.g. standard deviation or entropy) of local
measurements such as the node degree and clustering coefficient. This approach
paves the way for the identification of subgraphs (patches) with nearly uniform
connectivity, therefore complementing the characterization of the complexity of
networks. We also performed analysis of cascade failures, revealing that the
removal of vertices in `simple' regions results in smaller damage to the
network structure than the removal of vertices in the heterogeneous regions. We
illustrate the potential of the proposed methodology with respect to four
theoretical models as well as protein-protein interaction networks of three
different species. Our results suggest that the simplicity of protein
interaction grows as the result of natural selection. This increase in
simplicity makes these networks more robust to cascade failures.
