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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) Cumulative distributions of link numbers in a log-log plot. Nationwide (black solid line), Osaka prefecture (magenta dotted line), and Miyagi prefecture (purple dot-dash line). The guideline (grey thin line) shows the slope for a power law with the cumulative exponent, 1.5. (b) Initial number of links M and the number of nodes N for seven networks of different sizes in a log-log plot. The slope of the guideline shows the power exponent ϕ = 1.3. Nationwide (black), Kanto region (green), Tokyo prefecture (orange), Osaka prefecture (magenta), Fukuoka prefecture (brown), Miyagi prefecture (purple), and Kagoshima prefecture (cyan).
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pone.0119979.g003: (a) Cumulative distributions of link numbers in a log-log plot. Nationwide (black solid line), Osaka prefecture (magenta dotted line), and Miyagi prefecture (purple dot-dash line). The guideline (grey thin line) shows the slope for a power law with the cumulative exponent, 1.5. (b) Initial number of links M and the number of nodes N for seven networks of different sizes in a log-log plot. The slope of the guideline shows the power exponent ϕ = 1.3. Nationwide (black), Kanto region (green), Tokyo prefecture (orange), Osaka prefecture (magenta), Fukuoka prefecture (brown), Miyagi prefecture (purple), and Kagoshima prefecture (cyan).

Mentions: In Fig 3(a), cumulative link number distributions are plotted for two regional parts of the whole network extracted by using the information regarding the location of the firms, Osaka prefecture (30,766 firms and 138,425 links), and Miyagi prefecture (4,686 firms and 15,443 links). Comparing with the distribution for the whole country, we find that the distributions shift depending on the system size, following the same power law implying that the business relation network has a scale-free property for link numbers larger than 10.


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) Cumulative distributions of link numbers in a log-log plot. Nationwide (black solid line), Osaka prefecture (magenta dotted line), and Miyagi prefecture (purple dot-dash line). The guideline (grey thin line) shows the slope for a power law with the cumulative exponent, 1.5. (b) Initial number of links M and the number of nodes N for seven networks of different sizes in a log-log plot. The slope of the guideline shows the power exponent ϕ = 1.3. Nationwide (black), Kanto region (green), Tokyo prefecture (orange), Osaka prefecture (magenta), Fukuoka prefecture (brown), Miyagi prefecture (purple), and Kagoshima prefecture (cyan).
© Copyright Policy
Related In: Results  -  Collection

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

pone.0119979.g003: (a) Cumulative distributions of link numbers in a log-log plot. Nationwide (black solid line), Osaka prefecture (magenta dotted line), and Miyagi prefecture (purple dot-dash line). The guideline (grey thin line) shows the slope for a power law with the cumulative exponent, 1.5. (b) Initial number of links M and the number of nodes N for seven networks of different sizes in a log-log plot. The slope of the guideline shows the power exponent ϕ = 1.3. Nationwide (black), Kanto region (green), Tokyo prefecture (orange), Osaka prefecture (magenta), Fukuoka prefecture (brown), Miyagi prefecture (purple), and Kagoshima prefecture (cyan).
Mentions: In Fig 3(a), cumulative link number distributions are plotted for two regional parts of the whole network extracted by using the information regarding the location of the firms, Osaka prefecture (30,766 firms and 138,425 links), and Miyagi prefecture (4,686 firms and 15,443 links). Comparing with the distribution for the whole country, we find that the distributions shift depending on the system size, following the same power law implying that the business relation network has a scale-free property for link numbers larger than 10.

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