TABLE I
Best Fitting Distributions and Their Avg. KSS, L. KSS, and H. KSS Values
No. | Distribution | Avg. KSS | L. KSS | H. KSS |
---|
1 | GEV | 0.1061 | 0.0360 | 0.1785 |
2 | Log-logistic (3P) | 0.1096 | 0.0476 | 0.2154 |
3 | Pearson 5 (3P) | 0.1098 | 0.0411 | 0.1877 |
4 | Log-normal (3P) | 0.1123 | 0.0411 | 0.1997 |
5 | Log Pearson 3 | 0.1133 | 0.0469 | 0.2101 |
6 | Frechet (3P) | 0.1145 | 0.0413 | 0.2034 |
7 | Inverse Gaussian (3P) | 0.1159 | 0.0403 | 0.2276 |
8 | General Pareto | 0.1164 | 0.0610 | 0.2107 |
9 | Burr | 0.1181 | 0.0337 | 0.3539 |
10 | Pearson 6 | 0.1226 | 0.0419 | 0.2246 |
11 | Weibull (3P) | 0.1251 | 0.0531 | 0.2581 |
12 | Weibull | 0.1318 | 0.0429 | 0.2567 |
13 | Gamma | 0.1324 | 0.0389 | 0.4169 |
14 | Log-gamma | 0.1359 | 0.0731 | 0.2461 |
15 | Log-logistic | 0.1392 | 0.0709 | 0.2815 |
(3P) indicates the three-parameter version of the distribution |
TABLE II
Real Network Topologies and Their Attributes ( and Are in km)
No. | Network | | | | | | |
---|
1 | VIA network [3] | 9 | 12 | 2.67 | 571 | 338 | 0.1744 |
2 | BREN [3] | 10 | 11 | 2.20 | 94 | 33 | 0.1785 |
3 | RNP [3] | 10 | 12 | 2.40 | 748 | 517 | 0.1709 |
4 | Abilene core [14] | 10 | 13 | 2.60 | 1067 | 516 | 0.1266 |
5 | LEARN [12] | 10 | 12 | 2.40 | 189 | 111 | 0.1476 |
6 | CompuServe [13] | 11 | 14 | 2.55 | 1161 | 859 | 0.1543 |
7 | vBNS [3] | 12 | 17 | 2.83 | 965 | 563 | 0.1005 |
8 | CESNET [3] | 12 | 19 | 3.17 | 91 | 31 | 0.1425 |
9 | NSFNET [3] | 14 | 21 | 3.00 | 1086 | 707 | 0.1014 |
10 | ITALY [3] | 14 | 29 | 4.14 | 280 | 146 | 0.0783 |
11 | ACONET [3] | 15 | 22 | 2.93 | 119 | 111 | 0.1397 |
12 | MZIMA [3] | 15 | 19 | 2.53 | 852 | 430 | 0.0896 |
13 | GARR-B [3] | 16 | 27 | 3.37 | 224 | 140 | 0.1354 |
14 | ARNES [3] | 17 | 20 | 3.25 | 38 | 19 | 0.1388 |
15 | GERMANY [3] | 17 | 26 | 3.06 | 143 | 78 | 0.1057 |
16 | REDIRIS [3] | 17 | 28 | 3.29 | 319 | 131 | 0.0820 |
17 | LambdaRail [3] | 19 | 23 | 2.42 | 671 | 378 | 0.0901 |
18 | MEMOREX [3] | 19 | 24 | 2.53 | 137 | 68 | 0.0865 |
19 | CANARIE [3] | 19 | 26 | 2.74 | 668 | 632 | 0.0685 |
20 | EON [3] | 19 | 37 | 3.89 | 754 | 356 | 0.1043 |
21 | ARPANET[3] | 20 | 32 | 3.20 | 839 | 491 | 0.0739 |
22 | OPTOSunet [16] | 20 | 24 | 2.40 | 100 | 34 | 0.1219 |
23 | Hibernia USA [17] | 20 | 27 | 2.70 | 279 | 207 | 0.0823 |
24 | PIONIER [3] | 21 | 25 | 2.38 | 131 | 55 | 0.1418 |
25 | COX [3] | 24 | 40 | 3.33 | 662 | 428 | 0.0673 |
26 | SANET [3] | 25 | 28 | 2.24 | 36 | 14 | 0.1553 |
27 | NEWNET [3] | 26 | 31 | 2.38 | 528 | 268 | 0.1183 |
28 | PORTUGAL [3] | 26 | 36 | 2.77 | 203 | 325 | 0.1593 |
29 | RENATER [3] | 27 | 35 | 2.59 | 155 | 66 | 0.0790 |
30 | IBN31 [18] | 31 | 51 | 3.29 | 131 | 59 | 0.0914 |
31 | BULGARIA [19] | 32 | 33 | 2.06 | 51 | 21 | 0.1039 |
32 | GEANT2 [3] | 32 | 52 | 3.25 | 661 | 455 | 0.0716 |
33 | LONI [3] | 33 | 37 | 2.24 | 62 | 15 | 0.0886 |
34 | METRONA [3] | 33 | 41 | 2.48 | 73 | 40 | 0.0798 |
35 | COST37 [20] | 37 | 57 | 3.08 | 439 | 248 | 0.0840 |
36 | CERNET [21] | 37 | 53 | 2.86 | 636 | 503 | 0.0698 |
37 | OMNICOM [3] | 38 | 54 | 2.84 | 298 | 153 | 0.0796 |
38 | INTERNET2 [3] | 56 | 61 | 2.18 | 334 | 186 | 0.0694 |
39 | CORONET [22] | 75 | 99 | 2.64 | 326 | 265 | 0.0546 |
40 | USA100 [1] | 100 | 171 | 3.42 | 310 | 174 | 0.0360 |
Average: | 410.78 | | 0.1061 |
Standard deviation: | 332.83 | | – |
TABLE III
Different Areas of the 40 Real Networks (All Areas Are in Square km)
No. | Network | | | |
---|
1 | VIA network [3] | 1,267,876 | 824,464 | 1,915,456 |
2 | BREN [3] | 42,196 | 33,780 | 110,883 |
3 | RNP [3] | 1,814,676 | 1,077,222 | 8,514,965 |
4 | Abilene core [14] | 5,698,507 | 3,728,429 | 8,080,464 |
5 | LEARN [12] | 183,296 | 148,137 | 696,241 |
6 | CompuServe [13] | 5,442,148 | 4,661,688 | 8,080,464 |
7 | vBNS [3] | 6,279,993 | 4,182,861 | 8,080,464 |
8 | CESNET [3] | 45,957 | 32,357 | 78,909 |
9 | NSFNET [3] | 6,007,605 | 5,005,030 | 8,080,464 |
10 | ITALY [3] | 436,577 | 291,715 | 301,230 |
11 | ACONET [3] | 52,826 | 39,297 | 83,858 |
12 | MZIMA [3] | 6,805,153 | 4,278,891 | 8,080,464 |
13 | GARR-B [3] | 476,100 | 242,064 | 301,230 |
14 | ARNES [3] | 12,645 | 7112 | 20,273 |
15 | GERMANY [3] | 196,675 | 157,319 | 357,021 |
16 | REDIRIS [3] | 463,539 | 367,094 | 504,882 |
17 | LambdaRail [3] | 6,899,337 | 4,393,019 | 8,080,464 |
18 | MEMOREX [3] | 253,850 | 141,524 | 981,834 |
19 | CANARIE [3] | 3,462,688 | 2,095,793 | 9,985,140 |
20 | EON [3] | 5,800,056 | 2,524,156 | 6,550,943 |
21 | ARPANET [3] | 5,975,012 | 5,640,345 | 8,080,464 |
22 | OPTOSunet [16] | 122,980 | 81,063 | 449,964 |
23 | Hibernia USA [17] | 1,257,762 | 531,658 | 2,697,216 |
24 | PIONIER [3] | 194,178 | 163,888 | 312,685 |
25 | COX [3] | 4,870,065 | 3,531,799 | 8,080,464 |
26 | SANET [3] | 24,336 | 18,496 | 48,845 |
27 | NEWNET [3] | 7,033,452 | 4,200,490 | 8,080,464 |
28 | PORTUGAL [3] | 888,824 | 612,952 | 92,391 |
29 | RENATER [3] | 462,096 | 320,900 | 642,346 |
30 | IBN31 [18] | 426,658 | 200,382 | 300,448 |
31 | BULGARIA [19] | 68,008 | 52,069 | 111,910 |
32 | GEANT2 [3] | 7,703,318 | 3,383,604 | 770,326 |
33 | LONI [3] | 68,008 | 52,069 | 129,578 |
34 | METRONA [3] | 158,837 | 75,727 | 244,820 |
35 | COST37 [20] | 5,371,917 | 3,329,573 | 6,228,201 |
36 | CERNET [21] | 6,719,547 | 2,628,942 | 9,644,552 |
37 | OMNICOM [3] | 2,596,279 | 2,091,020 | 2,646,447 |
38 | INTERNET2 [3] | 7,169,590 | 5,974,161 | 8,080,464 |
39 | CORONET [22] | 7,903,482 | 6,046,394 | 8,080,464 |
40 | USA100 [1] | 8,191,044 | 5,992,704 | 8,080,464 |
TABLE IV
Estimation of Average Link Lengths From Different Areas and Their Comparisons (, , , and Are in km)
No. | Network | | | | | | | | Diff.1 | Diff.2 | Best Area |
---|
1 | VIA network [3] | 563 | 454 | 692 | 571 | 1.40 | 20.49 | | 19.09 | 19.79 | |
2 | BREN [3] | 95 | 85 | 154 | 94 | | 9.57 | | 7.86 | 62.77 | |
3 | RNP [3] | 623 | 480 | 1350 | 748 | 16.71 | 35.83 | | 19.12 | 63.77 | |
4 | Abilene core [14] | 1104 | 893 | 1315 | 1067 | | 16.31 | | 12.84 | 19.77 | |
5 | LEARN [12] | 198 | 178 | 386 | 189 | | 5.82 | | 1.06 | 99.47 | |
6 | CompuServe [13] | 1007 | 932 | 1227 | 1161 | 13.26 | 19.72 | | 6.46 | | |
7 | vBNS [3] | 1017 | 830 | 1154 | 965 | | 13.99 | | 8.60 | 14.20 | |
8 | CESNET [3] | 87 | 73 | 114 | 91 | 4.40 | 19.78 | | 15.38 | 20.87 | |
9 | NSFNET [3] | 894 | 816 | 1037 | 1086 | 17.68 | 24.86 | 4.51 | 7.18 | | |
10 | ITALY [3] | 241 | 197 | 200 | 280 | 13.93 | 29.64 | 28.57 | 15.71 | 14.64 | |
11 | ACONET [3] | 80 | 69 | 101 | 119 | 32.77 | 42.02 | 15.13 | 9.25 | | |
12 | MZIMA [3] | 908 | 720 | 989 | 852 | | 15.49 | | 8.92 | 9.51 | |
13 | GARR-B [3] | 230 | 164 | 183 | 224 | | 26.79 | 18.30 | 24.11 | 15.62 | |
14 | ARNES [3] | 36 | 27 | 46 | 38 | 5.26 | 28.95 | | 23.69 | 15.79 | |
15 | GERMANY [3] | 142 | 127 | 191 | 143 | 0.70 | 11.19 | | 10.49 | 32.87 | |
16 | REDIRIS [3] | 218 | 194 | 228 | 319 | 31.66 | 39.18 | 28.53 | 7.52 | | |
17 | LambdaRail [3] | 782 | 624 | 846 | 671 | | 7.00 | | | 9.54 | |
18 | MEMOREX [3] | 150 | 112 | 295 | 137 | | 18.25 | | 8.76 | 105.84 | |
19 | CANARIE [3] | 554 | 431 | 941 | 668 | 17.07 | 35.48 | | 18.41 | 23.80 | |
20 | EON [3] | 717 | 473 | 762 | 754 | 4.91 | 37.27 | | 32.36 | | |
21 | ARPANET [3] | 704 | 684 | 819 | 839 | 16.09 | 18.47 | 2.38 | 2.38 | | |
22 | OPTOSunet [16] | 101 | 82 | 193 | 100 | | 18.00 | | 17.00 | 92.00 | |
23 | Hibernia USA [17] | 323 | 210 | 473 | 279 | | 24.73 | | 8.96 | 53.76 | |
24 | PIONIER [3] | 123 | 113 | 156 | 131 | 6.11 | 13.74 | | 7.63 | 12.97 | |
25 | COX [3] | 566 | 482 | 729 | 662 | 14.50 | 27.19 | | 12.69 | | |
26 | SANET [3] | 39 | 34 | 55 | 36 | | 5.56 | | | 44.45 | |
27 | NEWNET [3] | 647 | 500 | 693 | 528 | | 5.30 | | | 8.71 | |
28 | PORTUGAL [3] | 230 | 191 | 74 | 203 | | 5.91 | 63.55 | | 50.25 | |
29 | RENATER [3] | 162 | 135 | 191 | 155 | | 12.90 | | 8.38 | 18.71 | |
30 | IBN31 [18] | 143 | 98 | 120 | 131 | | 25.19 | 8.40 | 16.03 | | |
31 | BULGARIA [19] | 65 | 57 | 71 | 51 | | | | | 11.77 | |
32 | GEANT2 [3] | 596 | 395 | 596 | 661 | 9.83 | 40.24 | 9.83 | 30.41 | 0.00 | |
33 | LONI [3] | 56 | 49 | 77 | 62 | 9.68 | 20.97 | | 11.29 | 14.51 | |
34 | METRONA [3] | 84 | 58 | 104 | 73 | | 20.55 | | 5.48 | 27.40 | |
35 | COST37 [20] | 456 | 359 | 491 | 439 | | 18.22 | | 14.35 | 7.98 | |
36 | CERNET [21] | 510 | 319 | 611 | 636 | 19.81 | 49.84 | 3.93 | 30.03 | | |
37 | OMNICOM [3] | 312 | 280 | 315 | 298 | | 6.04 | | 1.34 | 1.00 | |
38 | INTERNET2 [3] | 413 | 377 | 438 | 334 | | | | | 7.49 | |
39 | CORONET [22] | 367 | 321 | 371 | 326 | | 1.53 | | | 1.22 | |
40 | USA100 [1] | 318 | 272 | 316 | 310 | | 12.26 | | 9.68 | | |
Average: | 396.53 | 322.38 | 477.60 | 410.78 | 0.53 | 18.99 | | 8.97 | 19.99 | |
Absolute average: | 396.53 | 322.38 | 477.60 | 410.78 | 11.26 | 20.22 | 31.25 | 12.69 | 24.03 | |
TABLE V
Comparison of Errors With Optimized Multiplying Factors in Eqs. (15) and (16) (Column 3, 4, 5, and 6). Estimation of the Parameters of the Proposed Model From the Average Link Length (Column , , , ), the KSS Values of the Networks (Column ), and Their Evaluation (Column Acceptable?)
No. | Network | () | () | Diff. | Best Area | | | | | | Acceptable? () |
---|
1 | VIA network [3] | 4.38 | 2.98 | | | 546 | 368 | 228 | 0.167 | 0.2692 | Yes |
2 | BREN [3] | 2.13 | | 8.51 | | 92 | 69 | 28 | 0.167 | 0.3084 | Yes |
3 | RNP [3] | 19.25 | | 2.41 | | 604 | 406 | 254 | 0.167 | 0.2231 | Yes |
4 | Abilene core [14] | | | 1.69 | | 1071 | 713 | 460 | 0.167 | 0.1351 | Yes |
5 | LEARN [12] | | | 13.22 | | 192 | 135 | 72 | 0.167 | 0.1243 | Yes |
6 | CompuServe [13] | 15.85 | 2.07 | | | 977 | 651 | 418 | 0.167 | 0.1872 | Yes |
7 | vBNS [3] | | | 2.79 | | 986 | 657 | 422 | 0.167 | 0.1438 | Yes |
8 | CESNET [3] | 7.69 | 2.20 | | | 84 | 64 | 25 | 0.167 | 0.3254 | No |
9 | NSFNET [3] | 20.17 | 8.29 | | | 867 | 579 | 370 | 0.167 | 0.1849 | Yes |
10 | ITALY [3] | 16.43 | | | | 234 | 163 | 91 | 0.167 | 0.1951 | Yes |
11 | ACONET [3] | 34.45 | | | | 78 | 60 | 22 | 0.167 | 0.4421 | No |
12 | MZIMA [3] | | | | | 881 | 588 | 376 | 0.167 | 0.1468 | Yes |
13 | GARR-B [3] | 0.45 | 10.71 | 10.26 | | 223 | 155 | 86 | 0.167 | 0.1551 | Yes |
14 | ARNES [3] | 7.89 | | 5.27 | | 35 | 32 | 3 | 0.167 | 0.5283 | No |
15 | GERMANY [3] | 3.50 | | 4.89 | | 138 | 99 | 48 | 0.167 | 0.1731 | Yes |
16 | REDIRIS [3] | 33.86 | 25.71 | | | 211 | 147 | 81 | 0.167 | 0.3782 | No |
17 | LambdaRail [3] | | | 0.30 | | 759 | 508 | 322 | 0.167 | 0.1235 | Yes |
18 | MEMOREX [3] | | 0.00 | | | 146 | 105 | 52 | 0.167 | 0.1203 | Yes |
19 | CANARIE [3] | 19.61 | 21.26 | 1.65 | | 537 | 362 | 224 | 0.167 | 0.1170 | Yes |
20 | EON [3] | 7.82 | 23.47 | 15.65 | | 695 | 466 | 294 | 0.167 | 0.1223 | Yes |
21 | ARPANET [3] | 18.59 | 0.60 | | | 683 | 458 | 289 | 0.167 | 0.2062 | Yes |
22 | OPTOSunet [16] | 2.00 | 0.00 | | | 98 | 73 | 31 | 0.167 | 0.2619 | Yes |
23 | Hibernia USA [17] | | 8.24 | | | 313 | 215 | 126 | 0.167 | 0.1756 | Yes |
24 | PIONIER [3] | 9.16 | | | | 119 | 87 | 40 | 0.167 | 0.2047 | Yes |
25 | COX [3] | 17.07 | 11.18 | | | 549 | 370 | 230 | 0.167 | 0.0956 | Yes |
26 | SANET [3] | | | 8.33 | | 38 | 34 | 4 | 0.167 | 0.3885 | No |
27 | NEWNET [3] | | | | | 628 | 422 | 265 | 0.167 | 0.1178 | Yes |
28 | PORTUGAL [3] | | | 4.93 | | 223 | 155 | 86 | 0.167 | 0.6653 | No |
29 | RENATER [3] | | | 5.16 | | 157 | 112 | 57 | 0.167 | 0.1488 | Yes |
30 | IBN31 [18] | | 8.40 | 2.29 | | 139 | 100 | 49 | 0.167 | 0.1067 | Yes |
31 | BULGARIA [19] | | | 13.72 | | 62 | 49 | 15 | 0.167 | 0.2929 | No |
32 | GEANT2 [3] | 12.56 | 27.08 | 14.52 | | 578 | 389 | 243 | 0.167 | 0.1053 | Yes |
33 | LONI [3] | 12.90 | 3.23 | | | 54 | 44 | 11 | 0.167 | 0.0902 | Yes |
34 | METRONA [3] | | 2.74 | | | 81 | 62 | 23 | 0.167 | 0.2278 | Yes |
35 | COST37 [20] | | 0.23 | | | 442 | 299 | 183 | 0.167 | 0.1447 | Yes |
36 | CERNET [21] | 22.17 | 38.84 | 16.67 | | 495 | 334 | 206 | 0.167 | 0.1726 | Yes |
37 | OMNICOM [3] | | | 13.09 | | 303 | 208 | 122 | 0.167 | 0.1411 | Yes |
38 | INTERNET2 [3] | | | 17.66 | | 401 | 272 | 164 | 0.167 | 0.1615 | Yes |
39 | CORONET [22] | | | 11.05 | | 356 | 243 | 145 | 0.167 | 0.1221 | Yes |
40 | USA100 [1] | 0.65 | | 6.45 | | 308 | 212 | 123 | 0.167 | 0.0707 | Yes |
Average: | | 1.13 | 1.76 | | 384.70 | – | – | – | 0.2076 | Yes |
Absolute average: | 10.90 | 12.65 | 7.27 | | 384.70 | – | – | – | – | Yes |
Standard deviation: | 13.72 | 16.70 | 8.86 | – | 303.25 | – | – | – | – | – |