Deciphering the global organization of clustering in real complex networks

We uncover the global organization of clustering in real complex networks. To this end, we ask whether triangles in real networks organize as in maximally random graphs with given degree and clustering distributions, or as in maximally ordered graph models where triangles are forced into modules. The answer comes by way of exploring m-core landscapes, where the m-core is defined, akin to the k-core, as the maximal subgraph with edges participating in at least m triangles. This property defines a set of nested subgraphs that, contrarily to k-cores, is able to distinguish between hierarchical and modular architectures. We find that the clustering organization in real networks is neither completely random nor ordered although, surprisingly, it is more random than modular. This supports the idea that the structure of real networks may in fact be the outcome of self-organized processes based on local optimization rules, in contrast to global optimization principles.