Fig 1.
(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).
Fig 2.
(a) Order parameter R in the whole range of f. (b) Order parameter R in the range of f between 0.97 and 1.00. Circles and squares specify values below and above the critical point, respectively. (Inset) Log-log plots of R vs. f′−f. f′ < fc (circles), f′ = fc (triangles), and f′ > fc (squares). The grey guideline shows the power law with critical exponent β = 1.0. Error bars estimated by the interquartile range (IQR) from 100 trials using different random number seeds are plotted (all error bars are within the size of plotted squares.). (c) Normalized second largest cluster size T below the critical point (circles), and the average cluster size S above the critical point (squares). (Inset) Average cluster size S and f−fc in log-log scale. The grey guideline shows the slope for the critical exponent γ = 1.0. (d) Cumulative cluster size distributions in log-log scale. The dot-dash, bold and dash lines show values below, at, and above the critical point, respectively. The guideline shows a slope of 1.5, corresponding to the critical exponent τ = 2.5. The results are a superposition of 10 trials.
Fig 3.
(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).
Fig 4.
(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).
Fig 5.
Examples of typical clusters; (a) f < fc, (b) f = fc and (c) f > fc. Bridge links and loop links are shown in blue and red, respectively. (d) Ratio of bridge links (blue line) and loop links (broken red line) in the largest cluster. (e) Probability that the largest cluster has a loop. The broken black line shows the critical point. Results are estimated for 100 trials.
Fig 6.
Size dependence of the probability that the largest cluster has a loop for the three cases shown in Fig 4.
The dotted lines indicate the corresponding critical points.
Fig 7.
(a) Normalized size of the largest cluster Rc at the critical point for each network shown in Fig 3(b). The line shows the slope for Rc ∝ M−δ, δ = 0.50. Colors are the same as in Fig 3(b). (b) Number of clusters at the critical point for each network shown in Fig 3(b). The line shows the slope for Ns ∝ Mρ, ρ = 0.77. (c) Critical link density as a function of M for each network shown in Fig 3(b). The line shows the slope for 1−fc ∝ M−ε, ε = 0.23. The number of trials ranges from 1,000 to 100,000, depending on convergence speed for each network. The error bars indicate the interquartile range (IQR). (d) Critical link density on Erdös-Rényi graph (ER-graph), ϕ = 1.5 (triangles), ϕ = 1.7 (squares), and ϕ = 2.0 (circles). The number of trials ranges from 1,000 to 100,000, depending on convergence speed for each network.
Fig 8.
Schematic figures of the percolation process of a complete graph.
In our simulation, an initial state (f = 0) is chosen as an ER-graph with a link density p between p = pc and 1, and we consider removal process of links toward p = 0(f = 1).
Fig 9.
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.
Fig 10.
(a) PDF of nodes belonging to the ks-th shell. The total number of shells is 25 and the most populated shell is ks = 7. (b) Ratio of the number of nodes for each shell that did not survive the 10,000 trials. (c) Median of the survival rate for each shell for ranging from ks ≥ 9. Error bars are plotted using quartile deviation. The guideline shows Ps ∝ exp(Bks) where B = 0.16. (d) Schematic figure of the degree of decomposition in k-shell decomposition analysis. Each plate shows the shell (ks = 1 (blue); 2 (green); 3 (pink)). Focusing on the white node, the red links are oriented towards a higher shell, and their number is denoted by ku. The green links are oriented in the same shell, and their number is km. The blue links are oriented to a lower shell, and their number is kd.
Fig 11.
(a) Cumulative distributions of the survival rate at the critical point (fc = 0.994) of nodes belonging to the largest shell, ks = 25, in the initial state. (b) Schematic figure of calculating the survival rate. Each link is supposed to be removed with the same probability and we compare the sizes of separated clusters. The gray nodes belong to the largest cluster. (c) Cumulative distribution of link numbers at the critical point in a log-log plot. The solid line is calculated only in the largest cluster, and a superposition of 100 trials. The dotted line is calculated for all clusters, and we take superposition of 10 trials. The guide line shows the slope of 1.5, the same slope as Fig 1(a).