Scale-free networks are rare (1801.03400v1)

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.

• The paper rigorously tests the scale-free hypothesis by statistically analyzing nearly 1000 diverse networks, revealing that only 4% meet stringent power-law criteria.
• The analysis shows significant domain-specific variation, with biological and social networks exhibiting minimal evidence of strong scale-free structures compared to technological networks.
• Alternative models like exponential and log-normal distributions often outperform power-law fits, suggesting a need to revise traditional network formation theories.

Scale-Free Networks are Rare: A Comprehensive Analysis

The analysis presented explores the empirical prevalence of scale-free networks by examining a vast array of nearly 1000 network data sets from diverse domains including social, biological, technological, and informational sources. The paper rigorously tests the scale-free hypothesis which claims that most real-world networks follow a power-law degree distribution defined by $k^{-\alpha}$, typically with $2 < \alpha < 3$. This research aims to quantitatively assess the validity of this hypothesis using state-of-the-art statistical tools.

The authors, Anna D. Broido and Aaron Clauset, apply advanced statistical methodologies to fit the power-law model to each degree distribution and evaluate its plausibility using goodness-of-fit tests. They further compare these models to alternative distributions using a likelihood ratio test. The scope of this paper is significant given the breadth of the network data sets examined, obtained from the Index of Complex Networks (ICON), which ensures a diverse representation across multiple scientific fields.

Key Findings

1. Empirical Rarity of Scale-Free Networks:
   • Direct Evidence: Only 4% of the networks show the strongest possible evidence of scale-free structure, fitting the definition with $\alpha \in [2, 3]$. Direct statistical evidence of power-law degree distributions is present in only 33% of data sets in the weakest form.
   • Indirect Evidence: Approximately 52% of the networks fall into the category where power-law distributions are at least marginally plausible compared to alternative distributions. However, this does not constitute direct evidence of scale-free structure.
2. Domain-Specific Variation:
   • Biological Networks: These are less likely to exhibit scale-free structure, with 61% showing no evidence and only 6% demonstrating the strongest evidence.
   • Social Networks: The majority (71%) show only indirect evidence with no networks showing strong or strongest levels of direct evidence. This suggests social networks are at best weakly scale-free.
   • Technological Networks: These exhibit the highest likelihood of scale-free structure with 43% showing some direct evidence. However, only 1% reach the strongest evidence threshold.
3. Alternative Models:
   • Comparative analysis indicates that alternative models, such as the exponential, log-normal, and Weibull distributions often fit the degree distributions better than the power-law models. For instance, the log-normal distribution was preferred three times more often than the power law.

Implications and Future Directions

The findings challenge the longstanding belief that scale-free networks are ubiquitous across real-world systems. This has substantial implications for network theory and the various applications reliant on the scale-free hypothesis. The paper suggests that:

• Reevaluation of Mechanisms: There is a need to reassess theoretical mechanisms for network formation, given the empirical rarity of scale-free networks. This includes preferential attachment and other generative models which may not universally apply across different domains.
• Development of New Models: Future research should focus on developing and validating alternative models that better capture the structural diversity observed in real-world networks.
• Impact on Dynamical Processes: The assumed scale-free structure significantly influences the understanding of dynamics over networks, such as the spread of epidemics, resilience to attacks, and information diffusion. Given the rarity of scale-free networks, these studies might need revisiting to consider alternative structures.

In conclusion, this comprehensive analysis provides a rigorous empirical evaluation of the scale-free hypothesis across a diverse set of network data. The results indicate that genuinely scale-free networks are rare, prompting a need for new theoretical frameworks and models to explain the rich structural diversity of real-world networks. This paper paves the way for future investigations that will undoubtedly enhance our understanding of network structures and their underlying mechanisms.