Comparison of Simulations with a Mean-Field Approach vs. Synthetic Correlated Networks

It is well known that dynamical processes on complex networks are influenced by the degree correlations. A common way to take these into account in a mean-field approach is to consider the function k(nn)(k) (average nearest neighbors degree). We re-examine the standard choices of knn for scale-free networks and a new family of functions which is independent from the simple ansatz k(nn)proportional to k(alpha) but still displays a remarkable scale invariance. A rewiring procedure is then used to explicitely construct synthetic networks using the full correlation P(h divide k) from which k(nn) is derived. We consistently find that the k(nn) functions of concrete synthetic networks deviate from ideal assortativity or disassortativity at large k. The consequences of this deviation on a diffusion process (the network Bass diffusion and its peak time) are numerically computed and discussed for some low-dimensional samples. Finally, we check that although the k(nn) functions of the new family have an asymptotic behavior for large networks different from previous estimates, they satisfy the general criterium for the absence of an epidemic threshold.