Free-Energy-Based Design Policy for Robust Network Control against Environmental Fluctuation

<table class="table-group" id="tab4"><tr><td><table class="table"><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td align="left">Symbol</td><td align="center"> Value </td><td align="center"> Description </td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td align="left"><span style="width: 9.08855ptpx;"><svg height="11.927pt" id="M333" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 9.08855 11.927" width="9.08855pt" 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="M405 650H141L135 622C222 616 230 610 215 535L133 116C118 41 113 33 29 28L23 0H289L295 28C209 33 205 40 219 116L298 535C312 609 317 616 399 622L405 650Z" id="g113-74"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="428" vert-adv-y="428"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,4.719,3.132)"><path d="M387 400C387 425 348 451 303 451C247 451 176 414 132 376C69 322 24 228 24 148C24 43 74 -12 147 -12C211 -12 301 33 363 103L346 128C319 99 249 51 193 51C148 51 112 84 112 165C112 230 130 287 154 330C170 359 199 400 243 400C277 400 304 383 326 354C333 345 343 343 354 348C378 360 387 382 387 400Z" id="g50-100"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="411" vert-adv-y="411"></glyph.data></g></svg></span></td><td align="center"> 1 s </td><td align="center"> Interval for path selection </td></tr><tr><td align="left"><span style="width: 8.81454ptpx;"><svg height="11.927pt" id="M334" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 8.81454 11.927" width="8.81454pt" 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="M405 650H141L135 622C222 616 230 610 215 535L133 116C118 41 113 33 29 28L23 0H289L295 28C209 33 205 40 219 116L298 535C312 609 317 616 399 622L405 650Z" id="g113-74"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="428" vert-adv-y="428"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,4.719,3.132)"><path d="M357 391C357 416 325 451 270 451C241 451 179 431 146 400C109 365 97 334 97 307C97 248 143 213 196 180C250 146 261 123 261 101C261 75 237 40 190 40C153 40 110 72 86 109C81 116 68 119 55 112C37 102 24 87 24 66C24 30 85 -12 137 -12C229 -12 331 62 331 140C331 189 304 218 236 260C197 284 164 309 164 346C164 381 194 400 220 400C259 400 282 380 304 350C310 342 321 339 330 344C347 353 357 372 357 391Z" id="g50-116"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="381" vert-adv-y="381"></glyph.data></g></svg></span></td><td align="center"> 1 s </td><td align="center"> Interval for observing connectivity</td></tr><tr><td align="left"><span style="width: 10.0294ptpx;"><svg height="11.927pt" id="M335" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 10.0294 11.927" width="10.0294pt" 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="M405 650H141L135 622C222 616 230 610 215 535L133 116C118 41 113 33 29 28L23 0H289L295 28C209 33 205 40 219 116L298 535C312 609 317 616 399 622L405 650Z" id="g113-74"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="428" vert-adv-y="428"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,4.719,3.132)"><path d="M490 97L476 124C442 96 405 70 398 70C392 70 390 78 396 114C419 243 448 379 463 432L457 436C446 436 431 439 418 442C393 447 368 451 343 451C281 451 204 418 155 381C74 320 24 206 24 107C24 23 59 -12 88 -12C118 -12 155 5 191 34C236 70 290 122 328 177H330L312 84C296 0 311 -12 331 -12C355 -12 425 24 490 97ZM374 387C371 367 360 299 347 264C323 202 187 53 142 53C128 53 113 73 113 120C113 224 157 332 221 380C241 395 274 403 303 403C330 403 360 395 374 387Z" id="g50-98"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="514" vert-adv-y="514"></glyph.data></g></svg></span></td><td align="center"> 1 s </td><td align="center"> Interval for updating activity </td></tr><tr><td align="left"><span style="width: 10.4861ptpx;"><svg height="11.927pt" id="M336" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 10.4861 11.927" width="10.4861pt" 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="M405 650H141L135 622C222 616 230 610 215 535L133 116C118 41 113 33 29 28L23 0H289L295 28C209 33 205 40 219 116L298 535C312 609 317 616 399 622L405 650Z" id="g113-74"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="428" vert-adv-y="428"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,4.719,3.132)"><path d="M538 684C542 703 539 710 529 710C512 710 455 682 363 674L361 644H396C444 644 448 642 438 594L405 438C373 450 347 451 336 451C288 451 200 416 147 374C69 312 24 207 24 113C24 29 60 -12 94 -12C126 -12 156 4 196 30C231 53 296 100 346 166H348L328 81C311 10 325 -12 348 -12C380 -12 450 24 512 98L495 125C466 97 431 70 417 70C408 70 408 80 411 98L538 684ZM390 373L361 240C336 196 210 56 145 56C132 56 113 73 113 131C113 216 156 338 220 379C243 394 266 402 303 402C331 402 375 388 390 373Z" id="g50-101"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="564" vert-adv-y="564"></glyph.data></g></svg></span></td><td align="center"> 1 ms </td><td align="center"> Interval for sending of data packets </td></tr><tr><td align="left"><span style="width: 10.1481ptpx;"><svg height="11.927pt" id="M337" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 10.1481 11.927" width="10.1481pt" 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="M405 650H141L135 622C222 616 230 610 215 535L133 116C118 41 113 33 29 28L23 0H289L295 28C209 33 205 40 219 116L298 535C312 609 317 616 399 622L405 650Z" id="g113-74"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="428" vert-adv-y="428"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,4.719,3.132)"><path d="M509 90L493 118C465 91 427 70 418 70S409 77 415 103L469 327C494 429 466 451 436 451C412 451 387 444 356 427C307 400 222 342 160 264H158L246 667C253 701 252 710 243 710C228 710 169 680 87 673L83 644H117C155 644 160 640 149 588L24 -4L32 -12C57 -4 86 4 108 9L143 187C198 275 324 386 375 386C394 386 399 367 385 311L331 87C315 22 322 -12 351 -12C387 -12 452 24 509 90Z" id="g50-105"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="527" vert-adv-y="527"></glyph.data></g></svg></span></td><td align="center"><svg height="8.63352pt" id="M338" style="vertical-align:-0.2455397pt" version="1.1" viewbox="-0.0498162 -8.38798 12.6264 8.63352" width="12.6264pt" 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="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g121-47"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g121-46"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg> packets </td><td align="center"> Length of history </td></tr><tr><td align="left"><span style="width: 6.63704ptpx;"><svg height="9.39034pt" id="M339" 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><glyph.data ascent="3473" descent="-2876" horiz-adv-x="501" vert-adv-y="501"></glyph.data></g></svg></span></td><td align="center"><span style="width: 15.6015ptpx;"><svg height="8.63352pt" id="M340" style="vertical-align:-0.2455397pt" version="1.1" viewbox="-0.0498162 -8.38798 15.6015 8.63352" width="15.6015pt" 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="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g121-47"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z" id="g113-47"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,9.204,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g121-46"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg></span></td><td align="center"> Magnitude of attraction </td></tr><tr><td align="left"><span style="width: 6.84582ptpx;"><svg height="9.49473pt" id="M341" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 6.84582 9.49473" width="6.84582pt" 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="M494 514C482 587 419 712 303 712C238 712 174 667 174 603C174 561 205 514 249 449C219 438 187 422 162 407C93 366 23 283 23 177C23 69 87 -12 190 -12C244 -12 288 5 328 33C406 87 444 170 444 249C444 329 404 391 331 475C265 550 222 605 222 627C222 647 238 657 267 657C355 657 421 585 484 499L494 514ZM359 234C359 143 319 30 219 30C172 30 114 75 114 178C114 275 163 343 195 378C212 397 241 415 269 425C305 382 359 313 359 234Z" id="g113-226"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="517" vert-adv-y="517"></glyph.data></g></svg></span></td><td align="center"><span style="width: 15.6015ptpx;"><svg height="8.63352pt" id="M342" style="vertical-align:-0.2455397pt" version="1.1" viewbox="-0.0498162 -8.38798 15.6015 8.63352" width="15.6015pt" 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="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g121-46"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z" id="g113-47"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,9.204,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g121-47"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg></span></td><td align="center"> Smoothing coefficient </td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

The Scientific World Journal

tab4

Table 4

Table 4: Free-Energy-Based Design Policy for Robust Network Control against Environmental Fluctuation 