4 analytically derived cascade windows for

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Unformatted text preview: ectivity regime— corresponding to the lower and upper phase ted, with nsitions respectively. The nature of the phase transitions at the he distrioat small l boundaries is different, and this has important consequences the apparent stability of the systems involved. As Fig. 3 (open ponding cascade ares) demonstrates, the cumulative distribution of cascades at lower boundary of the cascade window follows a power law, alogous to the distribution of avalanches in models of self- Fig. 4. Analytically derived cascade windows for heterogeneous networks. The solid lines are the same as Fig. 1. (a) The dashed lines represent cascade windows for uniform random graphs, but where the threshold distributions ( ) are normally distributed with mean and SD 0.05 and 0.1. (b) The dashed line represents the cascade window for a random graph with a degree distribution that is a power law with exponent and exponential cut-off 0, where has been fixed at 2.5 and 0 has been adjusted to generate graphs with variable z. • H...
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