Seeking for Simplicity in Complex Networks

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.