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Percolation theory and fragmentation measures in social networks

Author

Listed:
  • Chen, Yiping
  • Paul, Gerald
  • Cohen, Reuven
  • Havlin, Shlomo
  • Borgatti, Stephen P.
  • Liljeros, Fredrik
  • Eugene Stanley, H.

Abstract

We study the statistical properties of a recently proposed social networks measure of fragmentation F after removal of a fraction q of nodes or links from the network. The measure F is defined as the ratio of the number of pairs of nodes that are not connected in the fragmented network to the total number of pairs in the original fully connected network. We compare this measure with the one traditionally used in percolation theory, P∞, the fraction of nodes in the largest cluster relative to the total number of nodes. Using both analytical and numerical methods, we study Erdős–Rényi (ER) and scale-free (SF) networks under various node removal strategies. We find that for a network obtained after removal of a fraction q of nodes above criticality, P∞≈(1-F)1/2. For fixed P∞ and close to criticality, we show that 1-F better reflects the actual fragmentation. For a given P∞, 1-F has a broad distribution and thus one can improve significantly the fragmentation of the network. We also study and compare the fragmentation measure F and the percolation measure P∞ for a real national social network of workplaces linked by the households of the employees and find similar results.

Suggested Citation

  • Chen, Yiping & Paul, Gerald & Cohen, Reuven & Havlin, Shlomo & Borgatti, Stephen P. & Liljeros, Fredrik & Eugene Stanley, H., 2007. "Percolation theory and fragmentation measures in social networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 378(1), pages 11-19.
    • Handle: RePEc:eee:phsmap:v:378:y:2007:i:1:p:11-19
    DOI: 10.1016/j.physa.2006.11.074
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    Citations

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    Cited by:

    1. Drago, Carlo & Ricciuti, Roberto, 2017. "Communities detection as a tool to assess a reform of the Italian interlocking directorship network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 466(C), pages 91-104.
    2. Zhukov, Dmitry & Khvatova, Tatiana & Lesko, Sergey & Zaltcman, Anastasia, 2018. "Managing social networks: Applying the percolation theory methodology to understand individuals' attitudes and moods," Technological Forecasting and Social Change, Elsevier, vol. 129(C), pages 297-307.
    3. Han, Jihui & Zhang, Ge & Dong, Gaogao & Zhao, Longfeng & Shi, Yuefeng & Zou, Yijiang, 2024. "Exact analysis of generalized degree-based percolation without memory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 642(C).
    4. Dong, Gaogao & Tian, Lixin & Du, Ruijin & Fu, Min & Stanley, H. Eugene, 2014. "Analysis of percolation behaviors of clustered networks with partial support–dependence relations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 394(C), pages 370-378.
    5. Ding, Li & Guan, Zhi-Hong, 2008. "Modeling wireless sensor networks using random graph theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(12), pages 3008-3016.
    6. Bu, Zhan & Xia, Zhengyou & Wang, Jiandong & Zhang, Chengcui, 2013. "A last updating evolution model for online social networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(9), pages 2240-2247.

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