Scale-free networks beyond power-law degree distribution

Complex networks across various fields are often considered to be scale free-a statistical property usually solely characterized by a power-law distribution of the nodes' degree k. However, this characterization is incomplete. In real-world networks, the distribution of the degree-degree distance eta, a simple link-based metric of network connectivity similar to k, appears to exhibit a stronger power-law distribution than k. While offering an alternative characterization of scale-freeness, the discovery of eta raises a fundamental question: do the power laws of k and eta represent the same scale-freeness? To address this question, here we investigate the exact asymptotic relationship between the distributions of k and eta, proving that every network with a power-law distribution of k also has a power-law distribution of eta, but not vice versa. This prompts us to introduce two network models as counterexamples that have a power-law distribution of eta but not k, constructed using the preferential attachment and fitness mechanisms, respectively. Both models show promising accuracy by fitting only one model parameter each when modeling real-world networks. Our findings suggest that eta is a more suitable indicator of scale-freeness and can provide a deeper understanding of the universality and underlying mechanisms of scale-free networks.

Automated summary

Complex networks are often considered to be scale-free, characterized by a power-law distribution of the nodes' degree. However, in real-world networks, the distribution of the degree-degree distance, a metric similar to degree, shows a stronger power-law distribution. We investigate the relationship between the two distributions and introduce network models that have a power-law distribution of degree-degree distance but not degree. Our findings suggest that degree-degree distance is a more suitable indicator of scale-freeness.