Structural properties of extended pseudo-fractal scale-free network with higher network efficiency

Abstract Complex networks, as an important model for studying complex systems, have been gradually studied in various extension models. Firstly, the network is proved to be sparse by calculating the density of the extending pseudo-fractal network. Secondly, it is proved that the network nodes conform the power-law distribution by enumerate the degree sequence of nodes with power law index $ 2 \lt \gamma \leq 2.585 $ shows that the network has scale-free feature. Thirdly, it is also demonstrated that the average clustering coefficient $ \overline{C_{g}} $ can reach a theoretical lower limit value 0.8182 and the increase of extension coefficient m makes the $ \overline{C_{g}} $ higher than traditional fractal networks. Fourthly, we derive the analytical expression and limit expression of the average path length, and conclude that it has small-world effect. Meanwhile, it is shown that the average path length $ \overline{D_{g}} $ is logarithmically related to Ng growth relationship in Fg. Finally, it is concluded that this extended pseudo-fractal network becomes more effective under the effect of m.