Outline

Networks \(\times\) extreme value theory
Extending the power law for degrees
Mixture distribution
- even for data simulated from “true” random graph models
- finite sample behaviour
Compare results for real & simulated data
- make models more realistic?
Studying the evolution of real data

Slides: https://bit.ly/sunbelt2024
R package: https://cran.r-project.org/package=crandep
Paper: https://doi.org/10.1111/stan.12355

Why the power law?

Seemingly ubiquitous
Potentially generated from “nice” models
- Preferential attachment (Barabasi-Albert)
- Generalised random graphs

Left: http://konect.cc/networks/as-caida20071105/
Right: Valero, Pérez-Casany, and Duarte-López (2022)
Both available at https://github.com/adbroido/SFAnalysis

Really a straight line?

Survival function / complementary CDF

“Curved” downwards

Tackling various issues

Small degrees deviate from straight line
- subsets data above \(u\); chooses optimal \(u\) by K-S statistic (Clauset, Shalizi, and Newman 2009)
- basis of hypothesis testing framework (Broido and Clauset 2019)
- requires additional procedure
Large degrees deviate from straight line
- incorporates extreme value theory (Voitalov et al. 2019)
- only allows “heavy” tails; not always true for real data
“Curvature” shown by some data
- Double power law (Ayed, Lee, and Caron 2019)
- Zipf-polylog distribution (Valero, Pérez-Casany, and Duarte-López 2022)
- still quite restrictive
Related works
- de Solla Price model (Artico et al. 2020)
- mixture of Zipf distributions (Jung and Phoa 2021)

2. Extreme value mixture distribution

Primer: relationships

Schematic, on log-log scale

Slope: \(-\alpha(<-1)\)
Tail heaviness: \(1/(\alpha-1)\)

\(\theta\in(0,1]\)
Reduced to Zipf\((\alpha)\) when \(\theta=1\)

Blue’s tail heaviness: \(\xi\)
Brown’s tail heaviness: \(1/(\alpha-1)\)
- had the power law extended beyond \(u\)

Bayesian inference

Prior \(+\) data \(\rightarrow\) posterior
Uncertainty expressed naturally
Threshold \(u\) close to maximum - power law could be adequate
Can even test between Zipf (straight line) and polylog (curved)
- model selection within inference

CRAN dependencies

R packages available at https://cran.r-project.org/
imports between packages

Mixture distribution improves fit
Better than alternatives

Simulated data

library(igraph); set.seed(5772)
sample_pa(n = 500000, directed = FALSE,
          m = 1, zero.appeal = 1.0e-4)

Not clear-cut even when simulated from true model
Uncertainty & finite sample behaviour

Tail heaviness: actual vs implied

Indication whole of data could follow the power law
Huge uncertainty of \(\xi\) still

Away from \(y=x\) line
Difficult to have sustained growth according to \(\alpha\)

3. Evolution over time

Tail heaviness relatively stable

So is the power law exponent

Tail steadily lighter than implied by power law

Summary

Degrees seemingly follow the power law
- wholly or partially
Extreme value mixture distribution
- power law \((\alpha)\) in the body
- integer generalised Pareto \((\xi,\ldots)\) in the tail
Comparing \(\xi\) and \(1/(\alpha-1)\)
- indicates if power law applies to whole of data
Application: CRAN dependencies
- seeming stability over time
Next steps
- evolution of the raw data
- what model (modification) leads to such behaviour

Slides: https://bit.ly/sunbelt2024
R package: https://cran.r-project.org/package=crandep
Paper: https://doi.org/10.1111/stan.12355

Bibliography

Artico, I., I. Smolyarenko, V. Vinciotti, and E. C. Wit. 2020. “How Rare Are Power-Law Networks Really?” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476 (2241): 20190742. https://doi.org/10.1098/rspa.2019.0742.

Ayed, Fadhel, Juho Lee, and François Caron. 2019. “Beyond the Chinese Restaurant and Pitman-Yor Processes: Statistical Models with Double Power-Law Behavior.” ArXiv E-Print. http://arxiv.org/abs/1902.04714.

Broido, A. D., and A. Clauset. 2019. “Scale-Free Networks Are Rare.” Nature Communications 10 (1017). https://doi.org/10.1038/s41467-019-08746-5.

Clauset, A., C. R. Shalizi, and M. E. J. Newman. 2009. “Power-Law Distributions in Empirical Data.” SIAM Review 51 (4): 661–703. https://doi.org/10.1137/070710111.

Jung, Hohyun, and Frederick Kin Hing Phoa. 2021. “A Mixture Model of Truncated Zeta Distributions with Applications to Scientific Collaboration Networks.” Entropy 23 (5). https://doi.org/10.3390/e23050502.

Valero, Jordi, Marta Pérez-Casany, and Ariel Duarte-López. 2022. “The Zipf-Polylog Distribution: Modeling Human Interactions Through Social Networks.” Physica A 603. https://doi.org/10.1016/j.physa.2022.127680.

Voitalov, Ivan, Pim van der Hoorn, Remco van der Hofstad, and Dmitri Krioukov. 2019. “Scale-Free Networks Well Done.” Phys. Rev. Res. 1 (3): 033034. https://doi.org/10.1103/PhysRevResearch.1.033034.

From the power law to extreme value mixture distributions