Dynamics of network growth and evolution: integrating non-constant edge growth, mixed attachment and reciprocity

Article Content

Abstract

Advancements in network science research have enriched our understanding of the mechanisms shaping the topological properties of networks over the past two decades. However, the existing models still grapple with limitations in fully capturing the diverse structures and behaviours observed in real-world networks. This paper addresses these limitations by examining network growth in networks characterised by a distinct in-degree distribution, exhibiting expected new edges in the head and an extended exponential or power-law behaviour in the tail. To overcome these complexities, we propose a comprehensive model encompassing non-constant edge establishment driven by mixed attachment and reciprocal mechanisms. This extension offers a more accurate representation of real-world networks. Utilising various discrete probability distributions, including Poisson, binomial, zeta and log-series, our model accommodates variations in the number of new edges, providing a realistic depiction of network evolution. Analytical expressions for the limit in- and out-degree distributions and evolving dynamics of cumulative complementary in- and out-degree distributions are derived. These findings enable a detailed assessment of each mechanism’s contribution to the head and tail of the in- and out-degree distributions. Furthermore, we validate the practical relevance of our model by fitting it to real-world networks, emphasising the impact of the number of new edges and reciprocity.

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

Complex Networks
Dynamic Networks
Network topology
Network Models
Network Research
Stochastic Networks

References

P Erdös and A Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960)

MATH Google Scholar
B Bollobás, Modern graph theory (Springer, New York, 1998)

MATH Google Scholar
D J Watts and S H Strogatz, Nature 393, 440 (1998)

ADS Google Scholar
A-L Barabási and R Albert, Science 286, 509 (1999)

ADS MathSciNet Google Scholar
G Bianconi and A-L Barabási, Europhys. Lett. 54, 436 (2001)

ADS Google Scholar
MO Jackson and BW Rogers, Am. Econ. Rev. 97, 890 (2007)

Google Scholar
A Clauset, C R Shalizi and M E Newman, SIAM Rev. 51, 661 (2009)

ADS MathSciNet Google Scholar
I Voitalov, P Van Der Hoorn, R Van Der Hofstad and D Krioukov, Phys. Rev. Res. 1, 033034 (2019)

Google Scholar
F Battiston, G Cencetti, I Iacopini, V Latora, M Lucas, A Patania, J-G Young and G Petri, Phys. Rep. 874, 1 (2020)

ADS MathSciNet Google Scholar
Z Zeng, M Feng, P Liu and J Kurths, IEEE Trans. Syst. Man Cybern. Syst. (2025)

Google Scholar
M E Newman and DJ Watts, Phys. Lett. A 263, 341 (1999)

ADS MathSciNet Google Scholar
A-L Barabási, R Albert and H Jeong, Physica A 272, 173 (1999)

ADS Google Scholar
PL Krapivsky, S Redner and F Leyvraz, Phys. Rev. Lett. 85, 4629 (2000)

ADS Google Scholar
S N Dorogovtsev, J F F Mendes and A N Samukhin, Phys. Rev. Lett. 85, 4633 (2000)

ADS Google Scholar
P Holme and J Saramäki, Phys. Rep. 519, 97 (2012)

ADS Google Scholar
M Barthélemy, Phys. Rep. 499, 1 (2011)

ADS MathSciNet Google Scholar
M Medo, G Cimini and S Gualdi, Phys. Rev. Lett. 107, 238701 (2011)

ADS Google Scholar
S N Dorogovtsev and J F F Mendes, Phys. Rev. E 63, 025101 (2001)

ADS Google Scholar
Y-B Xie, T Zhou and B-H Wang, Physica A 387, 1683 (2008)

ADS Google Scholar
S Fortunato, A Flammini and F Menczer, Phys. Rev. Lett. 96, 218701 (2006)

ADS Google Scholar
T Zhou, G Yan and B-H Wang, Phys. Rev. E 71, 046141 (2005)

ADS Google Scholar
P Sheridan, Y Yagahara and H Shimodaira, Ann. Inst. Stat. Math. 60, 747 (2008)

Google Scholar
V N Zadorozhnyi and E B Yudin, Physica A 428, 111 (2015)

ADS MathSciNet Google Scholar
I Fernandez, KM Passino and J Finke, J. Complex. Netw. 7, 421 (2018)

Google Scholar
D Ruiz, J Campos and J Finke, J. Stat. Phys. 181, 673 (2020)

ADS MathSciNet Google Scholar
I J Gomez Portillo and P M Gleiser, PloS ONE 4, e6863 (2009)
J Ohkubo, M Yasuda and K Tanaka, Phys. Rev. E 72, 065104 (2005)

ADS Google Scholar
K Judd, M Small and T Stemler, Europhys. Lett. 103, 58004 (2013)

ADS Google Scholar
M Feng, H Qu, Z Yi, X Xie and J Kurths, IEEE Trans. Cybern. 46, 1144 (2015)

Google Scholar
A Murcia, N Pérez and D Ruiz, Pramana – J. Phys. 97, 142 (2023)

Google Scholar
F Safaei, H Yeganloo and M Moudi, Int. J. Model. Simul. 40, 456 (2020)

Google Scholar
A M ElBahrawy, L Alessandretti and A Baronchelli, Front. Phys. 9, 720708 (2021)

Google Scholar
T Wang and S Resnick, Methodol. Comput. Appl. Probab. 25, 8 (2023)

Google Scholar
J Medina, J Finke and C Rocha, J. Phys. A: Math. Theor. 52, 095001 (2019)

ADS Google Scholar
V Batagelj, A Mrvar and M Zaversnik, Proc. Language Technologies (Ljubljana, Slovenia, 2002)

ADS MathSciNet Google Scholar
J Kunegis, Proceedings of 22nd International Conference WWW (ACM, New York, 2013)
D Ruiz and J Finke, J. Stat. Phys. 172, 1127 (2018)

Download references

Acknowledgements

This research was supported in part by the Colombian ‘Fondo de Ciencia, Tecnología e Innovación del Sistema General de Regalías FCTeI-SGR’ of the Cauca Department throughout the project ‘Fortalecimiento de las Capacidades de las EBT-TIC del Cauca para Competir en un Mercado Global-Cluster CreaTIC’ under Grant No. TH 2017-01.

Author information

Authors and Affiliations

Department of Electronics and Computer Sciences, Pontificia Universidad Javeriana Cali, Cali, Colombia

Jan Medina-López
Department of Mathematics, Universidad del Cauca, Popayán, Colombia

Diego Ruiz

Corresponding author

Correspondence to Jan Medina-López.

Appendix A. Additional simulations

In this appendix, we present additional simulations to complement the main analysis discussed in the preceding sections. The supplementary figures included here showcase simulations based on different probability distributions, including Poisson (figure 5), geometric (figure 6) and logarithmic series (figure 7). These simulations provide further validation for the theoretical analysis conducted in the main text. Each figure offers insights into how the simulated networks align with theoretical predictions across various probability distributions. By examining these additional simulations, we gain a comprehensive understanding of the robustness and applicability of our theoretical framework.

Figures 5a, 5c, 6a, 6c and 7a illustrate how the simulated network converges with the theoretical predictions as the number of iterations increases. In figures 5b, 5d, 6b, 6d and 7b, you can observe the dynamics of $F_{t} (k)$ and ${\hat{F}}_{t} (k)$ for nodes with in-degree and out-degree values of $k = {0, 1, 2, 3, 4, 5}$ . The results show that the simulated values of F(k), $\hat{F} (k)$ , $F_{t} (k)$ and ${\hat{F}}_{t} (k)$ gradually approach the corresponding theoretical values as the number of iterations increases.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Cite this article

Medina-López, J., Ruiz, D. Dynamics of network growth and evolution: integrating non-constant edge growth, mixed attachment and reciprocity. Pramana – J Phys 99, 102 (2025). https://doi.org/10.1007/s12043-025-02935-2

Received12 December 2024
Revised 18 February 2025
Accepted 19 February 2025
Published 15 July 2025
DOI https://doi.org/10.1007/s12043-025-02935-2

Keywords

Network modelling
formation models
power-law
preferential attachment
uniform attachment
reciprocity
non-constant edge growth
structural analysis
degree distribution
stability

PACS

02.10.Ox; 02.50.−r; 89.75.Fb
05.45.−a

Related Articles

Contact us