Structural constraints in complex networks

We present a link–rewiring mechanism to produce surrogates of a network where both the degree distribution and the rich–club connectivity are preserved. We consider three real networks, the AS–Internet, the protein interaction and the scientific collaboration. We show that for a given degree distribution, the rich–club connectivity is sensitive to the degree–degree correlation, and on the other hand the degree–degree correlation is constrained by the rich–club connectivity. In particular, in the case of the Internet, the assortative coefficient is always negative and a minor change in its value can reverse the network’s rich–club structure completely; while fixing the degree distribution and the rich–club connectivity restricts the assortative coefficient to such a narrow range, that a reasonable model of the Internet can be produced by considering mainly the degree distribution and the rich–club connectivity. We also comment on the suitability of using the maximal random network as a null model to assess the rich–club connectivity in real networks. PACS numbers: 89.75.-k, 89.75.Da, 89.75.Fb, 89.20.Hh, 82.39.Rt, 87.23.Ge, 05.70.Ln