Evidencing preferential attachment in dependency network evolution

Clement Lee (Newcastle University, UK)

2025-06-26

Introduction

  • Interest in degree distribution

  • Power law \(\rightarrow\) Scale-freeness

  • Associated with preferential attachment

    1. New node joins & brings new edges
    2. Node \(i\) selected with weight \(g(d_i)\)
    3. Barabási and Albert (1999): \(g(d_i) = d_i\)

 

A few issues

  1. General case \(\nrightarrow\) power law

    • \(g(d_i) = d_i^{\alpha}\) (Krapivsky and Redner 2001)

  2. Other models \(\rightarrow\) scale-freeness

    • Generalised random graphs (Hofstad 2016)

  3. Debate on ubiquity

    • Broido and Clauset (2019): rare in real networks
    • Voitalov et al. (2019): rebuttal
    • Lee, Eastoe, and Farrell (2024): partial scale-freeness

  4. Main cause: only snapshots available

 

Take a step back

Goals

If we have the evolution data:

  1. Can we evidence the preferential attachment?

  2. How do we model the data in a probabilistic / statistical way?

  3. What is the precise form of the weight function?

    • \(g(d_i)=d_i^{\alpha}\) or a more general form?

Daily increments of R packages on CRAN

##              from      to   add
## 1          glmmsr methods FALSE
## 2            GpGp     FNN  TRUE
## 3        pmxTools   stats FALSE
## 4  ceterisParibus   knitr  TRUE
## 5        pmxTools   xpose FALSE
## 6     mRchmadness   shiny FALSE
## 7          glmmsr    lme4 FALSE
## 8     mRchmadness   rvest FALSE
## 9           xpose   dplyr FALSE
## 10          xpose   utils FALSE

\[\vdots\]

 

  • From 2019-01-29 to 2024-12-31

  • Obtained by crandep::get_dep_all_packages()

  • … which uses tools::CRAN_package_db()

  • Change in Imports for one day

  • Some new packages, some existing ones

Aggregating

##    previous date indegree increment count
## 1     2019-01-29        0         0  9315
## 2     2019-01-29        0         1     1
## 3     2019-01-29        1         0  1306
## 4     2019-01-29        2         0   459
## 5     2019-01-29        3         0   226
## 6     2019-01-29        4         0   161
## 7     2019-01-29        5         0   103
## 8     2019-01-29        6         0    75
## 9     2019-01-29        7         0    49
## 10    2019-01-29        8         0    40

\[\vdots\qquad\qquad\qquad\qquad\]

Scatter plot

Regress increment on in-degree

Revisiting weight function

 

A more general one

 

Modelling recipe

  1. Regress increment on in-degree

    • \(Y_i\sim\text{Poisson}\left(\displaystyle\frac{\mu\,g(d_i)}{\sum_j g(d_j)}\right)\)

  2. Choice of \(g(d)\)

    • Piecewise function (previous slide)
    • Power function for benchmark: \(g(d) = d^{\alpha}\)
    • Include parameter for zero-appeal \(\delta\)

  3. Bayesian inference for \((\alpha,\beta,\gamma,\delta,\mu)\)

    • A set of parameters for each month of increments
    • Results of \(\alpha\) most important

Piecewise function, Imports, new packages

\(\alpha\) (super-/sub-linear?)

\(\beta\)

\(\gamma\)

\(\delta\)

Summary

Thank you

Bibliography

Barabási, Albert-László, and Réka Albert. 1999. “Emergence of Scaling in Random Networks.” Science 286 (5439): 509–12. https://doi.org/10.1126/science.286.5439.509.

Boughen, Thomas, Clement Lee, and Vianey Palacios Ramirez. 2025. “Tail Flexibility in the Degrees of Preferential Attachment Networks.” ArXiv E-Prints. https://doi.org/10.48550/arXiv.2506.18726.

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.

Hofstad, Remco van der. 2016. “Generalized Random Graphs.” In Random Graphs and Complex Networks, 183–215. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press. https://doi.org/10.1017/9781316779422.009.

Krapivsky, P. L., and S. Redner. 2001. “Organization of Growing Random Networks.” Physical Review E 63 (6): 066123.

Lee, Clement, Emma F Eastoe, and Aiden Farrell. 2024. “Degree Distributions in Networks: Beyond the Power Law.” Statistica Neerlandica, 1–17. https://doi.org/10.1111/stan.12355.

Oliveira, R, and J Spencer. 2005. “Connectivity Transitions in Networks with Super-Linear Preferential Attachment.” Internet Mathematics 2 (2): 121–63. https://doi.org/10.1080/15427951.2005.10129101.

Rudas, A, B Tóth, and B Valkó. 2007. “Random Trees and General Branching Processes.” Random Structures and Algorithms 31 (2): 186–202. https://doi.org/10.1002/rsa.20137.

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.