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Networks are useful representations for analyzing and modeling real-world complex systems. They are often both scale-free and dense: their degree distribution follows a power-law and their average degree grows over time. So far, it has been argued that producing such networks is difficult without externally imposing a suitable cutoff for the scale-free regime. Here, we propose a new growing network model that produces dense scale-free networks with dynamically generated cutoffs. The link formation rule is based on a weak form of preferential attachment depending only on order relations between the degrees of nodes. By this mechanism, our model yields scale-free networks whose scaling exponents can take arbitrary values greater than 1. In particular, the resulting networks are dense when scaling exponents are 2 or less. We analytically study network properties such as the degree distribution, the degree correlation function, and the local clustering coefficient. All analytical calculations are in good agreement with numerical simulations. These results show that both sparse and dense scale-free networks can emerge through the same self-organizing process.

Ordinal Preferential Attachment: A Self-Organizing Principle Generating Dense Scale-Free Networks