A fractal and scale-free model of complex networks with hub attraction behaviors

It is widely believed that fractality of complex networks originate from hub repulsion behaviors (anticorrelation or disassortativity), which means that large degree nodes tend to connect with small degree nodes. This hypothesis was demonstrated by a dynamical growth model, which evolves as the inverse renormalization procedure, proposed by Song et al. Now we find that the dynamical growth model is based on the assumption that all the cross-box links have the same probability e to link to the most connected nodes inside each box. Therefore, we modify the growth model by adopting the flexible probability e, which makes hubs to have higher probability to connect with hubs than non-hubs. With this model, we find that some fractal and scale-free networks have hub attraction behaviors (correlation or assortativity). The results are the counter-examples of former beliefs. Actually, the real-world collaboration network of movie actors also is fractal and shows assortative mixing.