A Cascading Failure Model for Command and Control Networks with Hierarchy Structure

<table class="fixed-width table-group" id="tab2"><tr><td><table class="table"><colgroup><col style="width:3.59em"/><col style="width:6.62em"/><col style="width:7.15em"/><col style="width:6.95em"/><col style="width:6.96em"/></colgroup><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr class="thead"><td class="align_left"> </td><td class="align_center">The 1st group</td><td class="align_center">The 2nd group</td><td class="align_center">The 3rd group</td><td class="align_center">The 4th group</td></tr><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr><td class="align_left"><span style="width: 16.3453ptpx;"><svg height="8.8423pt" id="M145" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -8.6359 16.3453 8.8423" width="16.3453pt" 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="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z" id="g113-68"></path></g><g transform="matrix(.013,0,0,-0.013,8.293,0)"><path d="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z" id="g113-71"></path></g></svg></span></td><td class="align_center">0.2332</td><td class="align_center">0.2320</td><td class="align_center">0.2315</td><td class="align_center">0.2218</td></tr><tr><td class="align_left"><span style="width: 29.0417ptpx;"><svg height="14.5316pt" id="M146" style="vertical-align:-3.9384pt" version="1.1" viewbox="-0.0498162 -10.5932 29.0417 14.5316" width="29.0417pt" xmlns="http://www.w3.org/2000/svg" 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<div>The value of <svg height="9.39034pt" id="M142" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -5.96091 6.63704 9.39034" width="6.63704pt" 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="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 478 372Z" id="g113-225"></path></g></svg> and <svg height="9.49473pt" id="M143" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 7.30254 9.49473" width="7.30254pt" 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="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z" id="g113-233"></path></g></svg> when <svg height="8.8423pt" id="M144" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -8.6359 16.3453 8.8423" width="16.3453pt" 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="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z" id="g113-68"></path></g><g transform="matrix(.013,0,0,-0.013,8.293,0)"><path d="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z" id="g113-71"></path></g></svg> is maximum.</div>

Security and Communication Networks

tab2

Table 2

Table 2: A Cascading Failure Model for Command and Control Networks with Hierarchy Structure 