Scale-free networks with positive integer exponent

Scale-free feature is a popularly observed characteristic in various kinds of complex systems. This feature is often determined by verifying that degree distribution follows power-law distribution when using complex network to depict the underlying structure of complex systems under consideration. Previous works have proven that power-law exponent γ plays a key role in the analysis of topological structure of networks of this type. In this work, we propose a principled framework by cycle-based renewing operation to construct scale-free networks that allow for an arbitrary positive integer to be power-law exponent candidate. That is to say, exponent γ now belongs to N∗ (positive integers set). Next, we study the relationship between exponent γ and other fundamental structural parameters of the proposed scale-free networks, and verify that some pre-existing statements or declarations associated with topological structure of scale-free network can be further improved. For instance, dense scale-free networks can have a larger diameter, and thus have no small-world property. In addition, some new findings are also obtained. For example, given an arbitrary exponent γ∈N∗, there exists scale-free network that turns out to have a Pearson correlation coefficient nearly close to the theoretical upper bound in the limit of large graph size. The results obtained may shed lights on more comprehensive understanding of scale-free networks.