Seeking Best-Balanced Patch-Injecting Strategies through Optimal Control Approach

<p>A synthetic small-world network <span class="nowrap"><svg height="11.927pt" id="M139" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 22.885 11.927" width="22.885pt" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.827,3.132)"><path d="M463 633C453 636 427 644 417 648C388 660 349 667 316 667C183 667 100 592 100 483C100 398 176 335 223 305L259 282C321 242 358 203 358 149C358 66 301 22 234 22C110 22 70 131 57 208L24 205L28 48C51 23 128 -17 215 -17C342 -17 446 60 446 174C446 262 394 310 317 361L284 383C229 420 183 455 183 515C183 569 222 628 298 628C384 628 418 563 428 481L461 487C459 527 458 593 463 633Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.259,3.132)"><path d="M1013 650H775L768 618L793 615C853 608 859 601 837 545C792 433 708 266 639 131H637L537 639H503L251 132H249L190 552C183 602 190 607 235 614L262 618L269 650H25L18 618C92 610 95 608 107 525L184 -9H221L475 494H479L576 -9H613L884 526C928 607 934 609 1005 618L1013 650Z"></path></g></svg>,</span> where <svg height="11.927pt" id="M140" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 22.6631 11.927" width="22.6631pt" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M715 650H476L470 622C550 619 562 610 562 570C562 536 553 481 535 384L523 321C485 121 408 37 283 37C178 37 99 109 134 297L178 532C193 612 199 616 278 622L284 650H29L23 622C105 616 109 609 94 532L48 291C23 160 51 83 98 40C143 -1 202 -16 260 -16C321 -16 387 3 439 49C509 111 542 204 563 312L577 384C595 479 608 533 619 569C629 603 637 620 709 622L715 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.606,3.132)"><path d="M463 633C453 636 427 644 417 648C388 660 349 667 316 667C183 667 100 592 100 483C100 398 176 335 223 305L259 282C321 242 358 203 358 149C358 66 301 22 234 22C110 22 70 131 57 208L24 205L28 48C51 23 128 -17 215 -17C342 -17 446 60 446 174C446 262 394 310 317 361L284 383C229 420 183 455 183 515C183 569 222 628 298 628C384 628 418 563 428 481L461 487C459 527 458 593 463 633Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.038,3.132)"><path d="M1013 650H775L768 618L793 615C853 608 859 601 837 545C792 433 708 266 639 131H637L537 639H503L251 132H249L190 552C183 602 190 607 235 614L262 618L269 650H25L18 618C92 610 95 608 107 525L184 -9H221L475 494H479L576 -9H613L884 526C928 607 934 609 1005 618L1013 650Z"></path></g></svg> consists of the red nodes.</p>

Security and Communication Networks

fig2

Figure 2

Figure 2: Seeking Best-Balanced Patch-Injecting Strategies through Optimal Control Approach 