Threshold for the Outbreak of Cascading Failures in Degree-Degree Uncorrelated Networks

<table class="figure-group"><tr class="fig-image" id="a"><td><object data="https://static.hindawi.com/articles/mpe/volume-2015/752893/figures/752893.fig.002a.svgz" name="752893.fig.002a" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(a) </b>The independent absorbing nodes</td></tr><tr class="fig-image" id="b"><td><object data="https://static.hindawi.com/articles/mpe/volume-2015/752893/figures/752893.fig.002b.svgz" name="752893.fig.002b" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(b) </b>The dependent absorbing nodes</td></tr></table>

<div>Two examples of the absorbing nodes. In the procedure of cascading failures, an unfailed node (<svg height="7.35473pt" id="M70" style="vertical-align:-0.3499303pt" version="1.1" viewbox="-0.0498162 -7.0048 4.94072 7.35473" width="4.94072pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M326 255C326 330 263 391 186 391S46 330 46 255S109 119 186 119S326 180 326 255ZM274 255C274 204 235 163 186 163S98 204 98 255S137 347 186 347S274 306 274 255Z" id="g117-73"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="371" vert-adv-y="371"></glyph.data></g></svg>) may turn to be a failed node (<svg height="7.35473pt" id="M71" style="vertical-align:-0.3499303pt" version="1.1" viewbox="-0.0498162 -7.0048 4.94072 7.35473" width="4.94072pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M320 255C320 327 260 385 186 385S52 327 52 255S112 125 186 125S320 183 320 255Z" id="g117-74"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="371" vert-adv-y="371"></glyph.data></g></svg>) when it overloads or does not fail as it is an absorbing node (<svg height="7.34169pt" id="M72" style="vertical-align:-0.3499303pt" version="1.1" viewbox="-0.0498162 -6.99176 7.75925 7.34169" width="7.75925pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M573 255C573 407 448 530 294 530S15 407 15 255S140 -20 294 -20S573 103 573 255ZM521 255C521 127 419 24 294 24S67 127 67 255S169 486 294 486S521 383 521 255ZM437 255C437 332 373 394 294 394S151 332 151 255S215 116 294 116S437 178 437 255ZM379 255C379 206 341 166 294 166S209 206 209 255S247 344 294 344S379 304 379 255Z" id="g114-4"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g></svg>).</div>

Mathematical Problems in Engineering

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Figure 2

Figure 2: Threshold for the Outbreak of Cascading Failures in Degree-Degree Uncorrelated Networks 