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Precise calculation of a bond percolation transition and survival rates of nodes in a complex network.

Kawamoto H, Takayasu H, Jensen HJ, Takayasu M - PLoS ONE (2015)

Bottom Line: As an example of a real-world network, we apply our analysis to a business relation network consisting of approximately 3,000,000 links among 300,000 firms and observe the transition with critical exponents close to the mean-field values taking into account the finite size effect.We focus on the largest cluster at the critical point, and introduce survival probability as a new measure characterizing the robustness of each node.We also discuss the relation between survival probability and k-shell decomposition.

View Article: PubMed Central - PubMed

Affiliation: Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, Midori-ku, Yokohama, Japan.

ABSTRACT
Through precise numerical analysis, we reveal a new type of universal loopless percolation transition in randomly removed complex networks. As an example of a real-world network, we apply our analysis to a business relation network consisting of approximately 3,000,000 links among 300,000 firms and observe the transition with critical exponents close to the mean-field values taking into account the finite size effect. We focus on the largest cluster at the critical point, and introduce survival probability as a new measure characterizing the robustness of each node. We also discuss the relation between survival probability and k-shell decomposition.

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(a) Order parameter R for three networks: Nationwide (black solid line), Osaka pre-fecture (magenta dashed line), and Kagoshima prefecture (cyan chain line). The arrows indicate the corresponding critical points. (b) Cumulative cluster size distributions at the critical points in a log-log scale for the three cases shown in (a).
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pone.0119979.g004: (a) Order parameter R for three networks: Nationwide (black solid line), Osaka pre-fecture (magenta dashed line), and Kagoshima prefecture (cyan chain line). The arrows indicate the corresponding critical points. (b) Cumulative cluster size distributions at the critical points in a log-log scale for the three cases shown in (a).

Mentions: In Fig 4(a) and 4(b), we show the order parameters and the cumulative cluster size distributions at the critical point for two smaller size networks plotted with the case for the whole system. In both cases, the transition behaviors are less clear for smaller systems; however, we can confirm that the critical exponents are independent of the system size.


Precise calculation of a bond percolation transition and survival rates of nodes in a complex network.

Kawamoto H, Takayasu H, Jensen HJ, Takayasu M - PLoS ONE (2015)

(a) Order parameter R for three networks: Nationwide (black solid line), Osaka pre-fecture (magenta dashed line), and Kagoshima prefecture (cyan chain line). The arrows indicate the corresponding critical points. (b) Cumulative cluster size distributions at the critical points in a log-log scale for the three cases shown in (a).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4401659&req=5

pone.0119979.g004: (a) Order parameter R for three networks: Nationwide (black solid line), Osaka pre-fecture (magenta dashed line), and Kagoshima prefecture (cyan chain line). The arrows indicate the corresponding critical points. (b) Cumulative cluster size distributions at the critical points in a log-log scale for the three cases shown in (a).
Mentions: In Fig 4(a) and 4(b), we show the order parameters and the cumulative cluster size distributions at the critical point for two smaller size networks plotted with the case for the whole system. In both cases, the transition behaviors are less clear for smaller systems; however, we can confirm that the critical exponents are independent of the system size.

Bottom Line: As an example of a real-world network, we apply our analysis to a business relation network consisting of approximately 3,000,000 links among 300,000 firms and observe the transition with critical exponents close to the mean-field values taking into account the finite size effect.We focus on the largest cluster at the critical point, and introduce survival probability as a new measure characterizing the robustness of each node.We also discuss the relation between survival probability and k-shell decomposition.

View Article: PubMed Central - PubMed

Affiliation: Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, Midori-ku, Yokohama, Japan.

ABSTRACT
Through precise numerical analysis, we reveal a new type of universal loopless percolation transition in randomly removed complex networks. As an example of a real-world network, we apply our analysis to a business relation network consisting of approximately 3,000,000 links among 300,000 firms and observe the transition with critical exponents close to the mean-field values taking into account the finite size effect. We focus on the largest cluster at the critical point, and introduce survival probability as a new measure characterizing the robustness of each node. We also discuss the relation between survival probability and k-shell decomposition.

Show MeSH