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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 distribution F(≥ k) of link numbers in log-log plot. The guideline (solid line) shows the slope of a power law with the cumulative exponent, 1.5. This distribution follows a power law on a large scale, F(≥ k) ∝ k−1.5. (b) Japanese business relation network for firms with more than 1,000 links. Hokkaido region (orange), Tohoku region (grey), Kanto region (including Tokyo) (green), Chubu region (including Nagoya) (red), Kansai region (including Osaka) (purple), Chugoku region (pink), Shikoku region (skyblue), Kyushu-Okinawa region (yellow).
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pone.0119979.g001: (a) Cumulative distribution F(≥ k) of link numbers in log-log plot. The guideline (solid line) shows the slope of a power law with the cumulative exponent, 1.5. This distribution follows a power law on a large scale, F(≥ k) ∝ k−1.5. (b) Japanese business relation network for firms with more than 1,000 links. Hokkaido region (orange), Tohoku region (grey), Kanto region (including Tokyo) (green), Chubu region (including Nagoya) (red), Kansai region (including Osaka) (purple), Chugoku region (pink), Shikoku region (skyblue), Kyushu-Okinawa region (yellow).

Mentions: Network data for our project is provided by TEIKOKU DATABANK, Ltd., a Japanese credit research company in Tokyo. This data contains information about the direction of money flow among businesses in Japan in 2011, defining the business relation network. From this network, we select the largest strongly connected component (LSCC), where LSCC is the largest part of the network in which every node is connected to every other node by at least one unidirectional path [26]. Our LSCC network consists of 327,721 nodes, and we connect trading pairs directly through undirected links for percolation analysis. The LSCC has 2,960,370 links, and the link number distribution of this network is approximated by a power law for large link numbers with a cumulative exponent of 1.5, as shown in Fig 1(a) and in Eq (1).F(≥k)∝k-1.5(1)Here, F(≥ k) denotes the cumulative distribution function of link numbers k. Similar power laws have been reported for business relation networks based on different data sources [27–30]. Not only is this network scale-free, it also has a small-world characteristic that the most likely distance between a pair of firms is 5 links; whereas, the longest pair distance is 21 links. In Fig 1(b), we plot the network structure for firms with more than 1,000 links.


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 distribution F(≥ k) of link numbers in log-log plot. The guideline (solid line) shows the slope of a power law with the cumulative exponent, 1.5. This distribution follows a power law on a large scale, F(≥ k) ∝ k−1.5. (b) Japanese business relation network for firms with more than 1,000 links. Hokkaido region (orange), Tohoku region (grey), Kanto region (including Tokyo) (green), Chubu region (including Nagoya) (red), Kansai region (including Osaka) (purple), Chugoku region (pink), Shikoku region (skyblue), Kyushu-Okinawa region (yellow).
© Copyright Policy
Related In: Results  -  Collection

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

pone.0119979.g001: (a) Cumulative distribution F(≥ k) of link numbers in log-log plot. The guideline (solid line) shows the slope of a power law with the cumulative exponent, 1.5. This distribution follows a power law on a large scale, F(≥ k) ∝ k−1.5. (b) Japanese business relation network for firms with more than 1,000 links. Hokkaido region (orange), Tohoku region (grey), Kanto region (including Tokyo) (green), Chubu region (including Nagoya) (red), Kansai region (including Osaka) (purple), Chugoku region (pink), Shikoku region (skyblue), Kyushu-Okinawa region (yellow).
Mentions: Network data for our project is provided by TEIKOKU DATABANK, Ltd., a Japanese credit research company in Tokyo. This data contains information about the direction of money flow among businesses in Japan in 2011, defining the business relation network. From this network, we select the largest strongly connected component (LSCC), where LSCC is the largest part of the network in which every node is connected to every other node by at least one unidirectional path [26]. Our LSCC network consists of 327,721 nodes, and we connect trading pairs directly through undirected links for percolation analysis. The LSCC has 2,960,370 links, and the link number distribution of this network is approximated by a power law for large link numbers with a cumulative exponent of 1.5, as shown in Fig 1(a) and in Eq (1).F(≥k)∝k-1.5(1)Here, F(≥ k) denotes the cumulative distribution function of link numbers k. Similar power laws have been reported for business relation networks based on different data sources [27–30]. Not only is this network scale-free, it also has a small-world characteristic that the most likely distance between a pair of firms is 5 links; whereas, the longest pair distance is 21 links. In Fig 1(b), we plot the network structure for firms with more than 1,000 links.

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