Estimating risk propagation between interacting firms on inter-firm complex network - Fig 3

<p><b>(a)</b> Frequency of chain-reaction bankruptcies <i>A</i>(<i>k</i>_<i>f</i>, <i>k</i>_<i>c</i>)<i>p</i>(<i>k</i>_<i>f</i>)<i>p</i>(<i>k</i>_<i>c</i>) from 2006 to 2015, where <i>A</i>(<i>k</i>_<i>f</i>, <i>k</i>_<i>c</i>) shows the interaction kernel between the number of business transactions for the first bankrupted firm <i>k</i>_<i>f</i> and that of successive firm <i>k</i>_<i>c</i> and <i>p</i> shows the probability density. First bankruptcy firms tend to be larger than their chain-reaction bankruptcy firms. <b>(b)</b> Probability of chain-reaction bankruptcies <i>A</i>(<i>k</i>_<i>f</i>, <i>k</i>_<i>c</i>) from 2006 to 2015. The probability of chain-reaction bankruptcies between larger firms is higher than between smaller ones. <b>(c)</b> <i>A</i>(<i>k</i>_<i>f</i>, <i>k</i>_<i>c</i>)’s scaling laws with respect to <i>k</i>_<i>f</i> and <i>k</i>_<i>c</i>, respectively. Green triangles and red squares show the relation between the median of <i>A</i>(<i>k</i>_<i>f</i>, <i>k</i>_<i>c</i>) and <i>k</i>_<i>c</i> or <i>k</i>_<i>f</i> with fixed <i>k</i>_<i>f</i> or <i>k</i>_<i>c</i> from 2006 to 2015 in log-log scale, respectively. Each dashed-line shows a slope of <i>α</i> (red) and <i>β</i> (green). The probability of cascading failure is higher between larger first bankruptcy firms and smaller trading firms. We note that these exponents are observed by the least squares method with coefficient of determination <i>R</i>² ≥ 0.9 and the error-bars show first and third quartile, respectively.</p

Product categories and the coefficients of the variation (<i>σ</i>_<i>N</i>/<i>μ</i>_<i>N</i>) for each week’s totals for all stores.

<p>Product categories and the coefficients of the variation (<i>σ</i>_<i>N</i>/<i>μ</i>_<i>N</i>) for each week’s totals for all stores.</p

Scaling law in the magazines category.

<p>Solid and dashed lines show <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0157653#pone.0157653.e027" target="_blank">Eq (15)</a>, crossover value <i>A</i>_<i>m</i> respectively.</p

Scaling laws of median of bankrupt firms between pairs of (a) <i>S</i> and <i>E</i>, (b) <i>S</i> and <i>k</i>, (c) <i>E</i> and <i>k</i>, (d) <i>D</i> and <i>S</i>, (e) <i>D</i> and <i>E</i>, and (f) <i>D</i> and <i>k</i> in 2015 in log-log scale, respectively.

<p>There are scaling laws of continuing firms such that <i>S</i> ∝ <i>E</i>^1.3, <i>S</i> ∝ <i>k</i>^1.3, and <i>E</i> ∝ <i>k</i>^1.0, however, those of bankrupt firms are different than are observed as <i>S</i> ∝ <i>E</i>^0.9∼1.0, <i>S</i> ∝ <i>k</i>^1.0∼1.1, and <i>E</i> ∝ <i>k</i>^1.0∼1.1. We note that these exponents are observed by the least squares method with coefficient of determination <i>R</i>² ≥ 0.9 and the error-bars show first and third quartile, respectively.</p

Cumulative distribution of quantities by a purchase.

<p><b>(1)Magazines, (2)Eggs, (3)Salads and soup, (4)Gelled items, (5)Instant noodles, (6)Tissues, (7)Soft drinks, (8)Stamps</b>. Dashed line shows <i>x</i>⁻²</p

Cluster size distribution of chain-reaction bankruptcies in log-log plot.

<p>Black circles show the real distribution for each year. Red triangles, orange crosses, and pink x-marks show the Monte Carlo simulation results of the best parameters set, the second best parameters set, and the third best parameters set by judging from the median of <i>χ</i>² of each parameters set by 20 attempts, respectively. <b>(a)</b> <i>α</i> = 0.7, <i>β</i> = −0.3 in 2006. <b>(b)</b> <i>α</i> = 0.8, <i>β</i> = −0.3 in 2007. <b>(c)</b> <i>α</i> = 0.7, <i>β</i> = −0.2 in 2008. <b>(d)</b> <i>α</i> = 0.7, <i>β</i> = −0.4 in 2009. <b>(e)</b> <i>α</i> = 0.7, <i>β</i> = −0.3 in 2010. <b>(f)</b> <i>α</i> = 0.7, <i>β</i> = −0.2 in 2011. <b>(g)</b> <i>α</i> = 0.8, <i>β</i> = −0.3 in 2012. <b>(h)</b> <i>α</i> = 0.8, <i>β</i> = −0.2 in 2013. <b>(i)</b> <i>α</i> = 0.7, <i>β</i> = −0.2 in 2014. <b>(j)</b> <i>α</i> = 0.6, <i>β</i> = −0.2 in 2015. Each <i>α</i> and <i>β</i> of the interaction kernel is positive and negative, respectively, without time variant despite the fact that the number of bankruptcies in Japan has gradually decreased since 2008 in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0185712#pone.0185712.t001" target="_blank">Table 1</a>.</p

Frequency of bankruptcies and chain-reaction bankruptcies per year from 2006 to 2015 in Japan.

<p>There are about 1% bankruptcies of the total number of firms and about 4% chain-reaction bankruptcies of the total number of bankruptcies per year on average.</p

Property of Fluctuations of Sales Quantities by Product Category in Convenience Stores - Fig 3

<p><b>(a),(b) Sales-quantity time series and distribution of 1product, 1store, each day of the week. (c),(d) Sales-quantity time series and distribution of 1product, all stores within the city, each day of the week</b>. In (b),(d), green line shows a Poisson distribution whose mean is the same as the mean of real data. In (d), blue line is a Gaussian distribution whose mean is the same as the mean of real data and the standard deviation is 0.1 times of the mean of real data. In (a),(c), green and blue dashed lines represent the standard deviations of (b),(d), respectively, and black line is the mean of real data. <b>(e) Relationship between mean and standard deviation of sales quantities in the soft drinks category</b>. Regarding plot points, for each day of week: circle, weekly totals: cross, 1 product on 1 store: red, all products on 1 store: blue, 1 product within locality: green, all products within locality: magenta, 1 product within city: cyan, all products within city; black. Black solid line shows , and black dashed line shows <i>σ</i> = 0.15<i>μ</i>. <b>(f) Means of (e) with error bars given by the standard deviation</b>. Two red triangles indicate mean and standard deviation of (a) and (c).</p

Comparison of reproducibility for two item categories: cheese (red) and hair-styling (blue).

<p>Two solid line show <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0157653#pone.0157653.e027" target="_blank">Eq (15)</a> and dashed lines indicate crossover value <i>A</i>_<i>m</i> respectively. Plots are same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0157653#pone.0157653.g004" target="_blank">Fig 4</a>.</p