Title: Scale-free networks are rare,
Abstract: A central claim in modern network science is that real-world networks are
typically "scale free," meaning that the fraction of nodes with degree $k$
follows a power law, decaying like $k^{-\alpha}$, often with $2 < \alpha < 3$.
However, empirical evidence for this belief derives from a relatively small
number of real-world networks. We test the universality of scale-free structure
by applying state-of-the-art statistical tools to a large corpus of nearly 1000
network data sets drawn from social, biological, technological, and
informational sources. We fit the power-law model to each degree distribution,
test its statistical plausibility, and compare it via a likelihood ratio test
to alternative, non-scale-free models, e.g., the log-normal. Across domains, we
find that scale-free networks are rare, with only 4% exhibiting the
strongest-possible evidence of scale-free structure and 52% exhibiting the
weakest-possible evidence. Furthermore, evidence of scale-free structure is not
uniformly distributed across sources: social networks are at best weakly scale
free, while a handful of technological and biological networks can be called
strongly scale free. These results undermine the universality of scale-free
networks and reveal that real-world networks exhibit a rich structural
diversity that will likely require new ideas and mechanisms to explain.