Topological Properties and Transition Features Generated by a New Hybrid Preferential Model

A new hybrid preferential model (HPM) is proposed for generating both scale-free and small world properties. The topological transition features in the HPM from random preferential attachment to deterministic preferential attachment are investigated. It is found that the exponents γ of the power law are very sensitive to the hybrid ratio (d/r) of determination to random attachment, and γ increases as the ratio d/r increases. It is also found that there exists a threshold at d/r = 1/1, beyond which γ increases rapidly and can tend to infinity if there is no random preferential attachment (r=0), which implies that the power law scaling disappears completely. Moreover, it is also found that when the ratio d/r increases, the average path length L is decreased, while the average clustering coefficient C is increased. Compared to the BA model and random graph, the new HPM has both the smallest L and the biggest C, which is consistent with most real-world growing networks.