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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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Cumulative distributions of the survival rates.The red, green and blue lines represent values below (f = 0.950), at (f = 0.994), and above (f = 0.9999) the critical point, respectively. The values of survival rates are distributed most widely at the critical point.
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pone.0119979.g009: Cumulative distributions of the survival rates.The red, green and blue lines represent values below (f = 0.950), at (f = 0.994), and above (f = 0.9999) the critical point, respectively. The values of survival rates are distributed most widely at the critical point.

Mentions: In this section, we calculate the survival rate, Ps, of each node as a function of f by repeating multiple trials using different random numbers. We define the survival rate as the ratio of the number of trials, where the node belongs to the largest cluster, to the total number of trials. We perform 10,000 trials and calculate the survival rate function for each node. Fig 9 shows the distributions of survival rates for three values of f, i.e., f = 0.950 (below fc), f = 0.994 (at the critical point fc) and f = 0.9999 (above fc). Below the critical point (the broken red line), there are many nodes whose survival rates are very close to 1, and thus, we cannot characterize the differences among these nodes. Above the critical point (the broken blue line), many nodes (more than 90% of the total number of nodes) record survival rates less than the observation limit (10−4). At the critical point, the survival rates are most widely distributed, implying that the survival rate at the critical point can be a new measure of the robustness of nodes against random attacks.


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)

Cumulative distributions of the survival rates.The red, green and blue lines represent values below (f = 0.950), at (f = 0.994), and above (f = 0.9999) the critical point, respectively. The values of survival rates are distributed most widely at the critical point.
© Copyright Policy
Related In: Results  -  Collection

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

pone.0119979.g009: Cumulative distributions of the survival rates.The red, green and blue lines represent values below (f = 0.950), at (f = 0.994), and above (f = 0.9999) the critical point, respectively. The values of survival rates are distributed most widely at the critical point.
Mentions: In this section, we calculate the survival rate, Ps, of each node as a function of f by repeating multiple trials using different random numbers. We define the survival rate as the ratio of the number of trials, where the node belongs to the largest cluster, to the total number of trials. We perform 10,000 trials and calculate the survival rate function for each node. Fig 9 shows the distributions of survival rates for three values of f, i.e., f = 0.950 (below fc), f = 0.994 (at the critical point fc) and f = 0.9999 (above fc). Below the critical point (the broken red line), there are many nodes whose survival rates are very close to 1, and thus, we cannot characterize the differences among these nodes. Above the critical point (the broken blue line), many nodes (more than 90% of the total number of nodes) record survival rates less than the observation limit (10−4). At the critical point, the survival rates are most widely distributed, implying that the survival rate at the critical point can be a new measure of the robustness of nodes against random attacks.

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