# Giant component in a configuration-model power-law graph with a variable number of links

**Authors:** Heung Kyung Kim, Mi Jin Lee, Matthieu Barbier, Sung-Gook Choi, Min Seok Kim, Hyung-Ha Yoo, Deok-Sun Lee  
**arXiv:** [1911.10489](https://arxiv.org/abs/1911.10489)  
**Date:** 2019-11-24  
**Summary page:** https://www.summarizepaper.com/en/paper/1911.10489v1/

## Key points

- Generalization of algorithm in configuration model for generating power-law degree sequences with independently controllable degree exponent λ and number of links per node K
- Different form of degree distribution compared to static model or random removal of links, while still sharing same λ and K values
- Critical point Kc for appearance of giant component remains zero not only for λ ≤ 3 but also for 3 < λ < λl ≈ 3.81
- Critical exponents and cluster-size distribution different from known results for λ < λl
- Asymptotic behavior, degree exponent, and functional form of degree distribution relevant to understanding critical phenomena
- Implications for designing and assessing networked systems' structure and robustness
- Ability to generate power-law graphs with independently tunable parameters provides flexibility in modeling real-world networks
- Understanding how different properties of degree distribution affect critical phenomena can enhance system resilience.

## Summary

The paper presents a generalization of an algorithm used in the configuration model to generate power-law degree sequences with independently controllable degree exponent λ and number of links per node K. The authors show that this generalized power-law degree distribution yields a different form of the degree distribution compared to the static model or random removal of links, while still sharing the same λ and K values. One important finding is that the critical point Kc for the appearance of the giant component remains zero not only for λ ≤ 3 but also for 3 < λ < λl ≈ 3.81. This is in contrast to the static model and random link removal, where Kc=0 only for λ ≤ 3. The authors also observe that the critical exponents and cluster-size distribution for λ < λl are different from known results. By analyzing moments and generating functions of the degree distribution, as well as comparing them with other models, the authors argue that not only the asymptotic behavior and degree exponent but also the whole functional form of the degree distribution can be relevant to understanding critical phenomena. The results presented in this paper have implications for designing and assessing networked systems' structure and robustness. The ability to generate power-law graphs with independently tunable parameters provides more flexibility in modeling real-world networks. Additionally, understanding how different properties of the degree distribution affect critical phenomena can help improve our understanding of network dynamics and inform strategies for enhancing system resilience.