Dense networks with scale-free feature

Abstract: While previous works have shown that an overwhelming number of scale-free networks are sparse, there still exist some real-world networks including social networks, urban networks, information networks, which are by observation dense. In this paper, we propose a novel framework for generating scale-free graphs with dense feature using two simple yet helpful operations, first-order subdivision and Line-operation. From the theoretical point of view, our method can be used not only to produce desired scale-free graphs with density feature, i.e. power-law exponent $\gamma$ falling into the interval $1<\gamma\leq2$, but also to establish many other unexpected networked models, for instance, power-law models having large diameter. In addition, the networked models generated upon our framework show especially assortative structure. That is, their own Pearson correlation coefficients are able to achieve the theoretical upper bound. Last but not the least, we find the sizes of community in the proposed models to follow power-law in form with respect to modularity maximization.