predicting percolation thresholds in...

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Supplemental Material “Predicting percolation thresholds in networks” Filippo Radicchi Center for Complex Networks and Systems Research, School of Informatics and Computing, Indiana University, Bloomington, USA * 10 -3 10 -2 10 -1 10 0 ˜ p c 10 -3 10 -2 10 -1 10 0 ˆ p c a 10 -3 10 -2 10 -1 10 0 ˜ p c 10 -3 10 -2 10 -1 10 0 ¯ p c b 10 -3 10 -2 10 -1 10 0 ˆ p c 10 -3 10 -2 10 -1 10 0 ¯ p c c 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 connectance 10 -4 10 -3 10 -2 10 -1 10 0 absolute error f 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 connectance 0.0 0.5 1.0 relative error d ˜ pc ¯ pc ˆ pc 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 absolute error 0.0 0.5 1.0 cumulative distribution e ˜ pc ¯ pc ˆ pc FIG. 1: a) Comparison between the prediction values ˜ pc and ˆ pc, b) ˜ pc and ¯ pc, and c) ˆ pc and ¯ pc for all 109 real networks considered in our analysis. d) Relative errors of the predictions with respect to the best estimates pc as functions of the connectance of the network. e) Cumulative distribution of the absolute error committed in using a prediction value instead of the best estimate of the percolation threshold. f) Abosolute errors as functions of the graph connectance. * Electronic address: fi[email protected].

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Page 1: Predicting percolation thresholds in networkshomes.sice.indiana.edu/filiradi/Mypapers/percolation...network N E p c q c p~ c p c p^ c Fig. Refs. Url Social 3 32 80 0:2500 0:3125 0:2025

Supplemental Material “Predicting percolation thresholds in networks”

Filippo RadicchiCenter for Complex Networks and Systems Research,

School of Informatics and Computing, Indiana University, Bloomington, USA∗

10−3 10−2 10−1 100

pc

10−3

10−2

10−1

100

p c

a

10−3 10−2 10−1 100

pc

10−3

10−2

10−1

100

p c

b

10−3 10−2 10−1 100

pc

10−3

10−2

10−1

100

p c

c

10−610−510−410−310−210−1 100

connectance

10−4

10−3

10−2

10−1

100

abso

lute

erro

r

f10−610−510−410−310−210−1 100

connectance

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tive

erro

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d

pc

pc

pc

10−5 10−4 10−3 10−2 10−1 100

absolute error

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ulat

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ion e

pc

pc

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FIG. 1: a) Comparison between the prediction values pc and pc, b) pc and pc, and c) pc and pc for all 109 realnetworks considered in our analysis. d) Relative errors of the predictions with respect to the best estimates pc asfunctions of the connectance of the network. e) Cumulative distribution of the absolute error committed in usinga prediction value instead of the best estimate of the percolation threshold. f) Abosolute errors as functions ofthe graph connectance.

∗Electronic address: [email protected].

Page 2: Predicting percolation thresholds in networkshomes.sice.indiana.edu/filiradi/Mypapers/percolation...network N E p c q c p~ c p c p^ c Fig. Refs. Url Social 3 32 80 0:2500 0:3125 0:2025

10−1 100

relative error

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FIG. 2: Inverse participation ratio (IPR) multiplied by the network size (N) of the principal eigenvectors of theadjacency (red squares) and the non-backtracking matrices (black circles) as functions of the value of the bestestimate of the percolation threshold pc (panel a), the relative error (panel b), and the absolute error (panel c).Each point refers to numerical results obtained on one of the 109 real networks considered in our analysis.

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Page 3: Predicting percolation thresholds in networkshomes.sice.indiana.edu/filiradi/Mypapers/percolation...network N E p c q c p~ c p c p^ c Fig. Refs. Url Social 3 32 80 0:2500 0:3125 0:2025

network N E pc qc pc pc pc Fig. Refs. Url

Social 3 32 80 0.2500 0.3125 0.2025 0.1675 0.2109 3 [37] url

Karate club 34 78 0.2436 0.3824 0.1477 0.1487 0.1889 4 [56] url

Protein 2 53 123 0.3902 0.6038 0.2278 0.1722 0.2139 5 [37] url

Dolphins 62 159 0.2390 0.3871 0.1723 0.1390 0.1668 6 [32] url

Social 1 67 142 0.3099 0.4030 0.2351 0.1789 0.2294 7 [37] url

Les Miserables 77 254 0.1339 0.3377 0.0905 0.0833 0.0930 8 [23] url

Protein 1 95 213 0.6056 0.8000 0.2533 0.1865 0.2354 9 [37] url

E. Coli, transcription 97 212 0.5189 0.6495 0.2270 0.1531 0.1874 10 [33] url

Political books 105 441 0.1927 0.3429 0.0915 0.0838 0.0941 11 [1] url

David Copperfield 112 425 0.1035 0.1786 0.0783 0.0760 0.0867 12 [41] url

College football 115 613 0.1338 0.2087 0.1027 0.0928 0.1024 13 [14] url

S 208 122 189 0.4656 0.5492 0.3614 0.2435 0.3640 14 [37] url

High school, 2011 126 1, 709 0.0380 0.0873 0.0315 0.0294 0.0304 15 [13] url

Bay Dry 128 2, 106 0.0304 0.0469 0.0253 0.0249 0.0257 16 [24, 52] url

Bay Wet 128 2, 075 0.0308 0.0469 0.0256 0.0252 0.0260 17 [24] url

Radoslaw Email 167 3, 250 0.0203 0.0419 0.0158 0.0165 0.0168 18 [24, 36] url

High school, 2012 180 2, 220 0.0437 0.0944 0.0350 0.0332 0.0345 19 [13] url

Little Rock Lake 183 2, 434 0.0300 0.0874 0.0239 0.0242 0.0250 20 [24, 34] url

Jazz 198 2, 742 0.0314 0.0859 0.0266 0.0250 0.0258 21 [15] url

S 420 252 399 0.4511 0.5357 0.3437 0.2294 0.3462 22 [37] url

C. Elegans, neural 297 2, 148 0.0540 0.1044 0.0399 0.0410 0.0439 23 [54] url

Network Science 379 914 0.3982 0.6359 0.1424 0.0964 0.1148 24 [41] url

Dublin 410 2, 765 0.0778 0.1951 0.0564 0.0428 0.0450 25 [19, 24] url

US Air Trasportation 500 2, 980 0.0268 0.1080 0.0189 0.0208 0.0215 26 [11] url

S 838 512 819 0.4493 0.5449 0.3301 0.1996 0.3401 27 [37] url

Yeast, transcription 662 1, 062 0.2458 0.3414 0.0799 0.1002 0.1538 28 [38] url

URV email 1, 133 5, 451 0.0646 0.1068 0.0565 0.0482 0.0519 29 [17] url

Political blogs 1, 222 16, 714 0.0168 0.0442 0.0125 0.0135 0.0138 30 [1] url

Air traffic 1, 226 2, 408 0.1815 0.2871 0.1573 0.1086 0.1336 31 [24] url

Yeast, protein 1, 458 1, 948 0.2988 0.3752 0.1632 0.1327 0.1980 32 [20] url

Petster, hamster 1, 788 12, 476 0.0273 0.0587 0.0224 0.0217 0.0226 33 [24] url

UC Irvine 1, 893 13, 835 0.0248 0.0449 0.0183 0.0208 0.0216 34 [24, 44] url

Yeast, protein 2, 224 6, 609 0.0794 0.1245 0.0639 0.0525 0.0577 35 [9] url

Japanese 2, 698 7, 995 0.0318 0.1160 0.0093 0.0233 0.0262 36 [37] url

Open flights 2, 905 15, 645 0.0199 0.0792 0.0182 0.0159 0.0163 37 [24, 43] url

GR-QC, 1993-2003 4, 158 13, 422 0.1345 0.2244 0.0589 0.0219 0.0225 38 [29] url

Tennis 4, 338 81, 865 0.0073 0.0274 0.0063 0.0062 0.0062 39 [47] none

US Power grid 4, 941 6, 594 0.6583 0.7353 0.3483 0.1336 0.1606 40 [54] url

HT09 5, 352 18, 481 0.0283 0.1700 0.0050 0.0211 0.0244 41 [19] url

Hep-Th, 1995-1999 5, 835 13, 815 0.1081 0.1973 0.1231 0.0554 0.0588 42 [40] url

TABLE I: Analysis of real networks. From left to right, we report: name of the network, number of nodes in thegiant component, number of edges in the giant component, best estimate of the bond percolation threshold, bestestimate of the site percolation threshold, prediction values p, pc and pc, figure showing the numerical results ofthe simulations of the bond percolation process, references of the paper(s) where the network was analyzed, andurl where network data have been dowloaded.

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network N E pc qc pc pc pc Fig. Refs. Url

Reactome 5, 973 145, 778 0.0114 0.0591 0.0070 0.0048 0.0048 43 [21, 24] url

Jung 6, 120 50, 290 0.0093 0.1239 0.0010 0.0070 0.0078 44 [24, 50] url

Gnutella, Aug. 8, 2002 6, 299 20, 776 0.0455 0.1121 0.0600 0.0352 0.0377 45 [29, 49] url

JDK 6, 434 53, 658 0.0091 0.1173 0.0010 0.0070 0.0077 46 [24] url

AS Oregon 6, 474 12, 572 0.0356 0.1767 0.0061 0.0216 0.0285 47 [28] url

English 7, 377 44, 205 0.0114 0.1590 0.0031 0.0091 0.0096 48 [37] url

Gnutella, Aug. 9, 2002 8, 104 26, 008 0.0451 0.1099 0.0632 0.0351 0.0376 49 [29, 49] url

French 8, 308 23, 832 0.0236 0.1284 0.0046 0.0165 0.0191 50 [37] url

Hep-Th, 1993-2003 8, 638 24, 806 0.0755 0.1363 0.0834 0.0322 0.0333 51 [29] url

Gnutella, Aug. 6, 2002 8, 717 31, 525 0.0648 0.1005 0.0746 0.0447 0.0489 52 [29, 49] url

Gnutella, Aug. 5, 2002 8, 842 31, 837 0.0565 0.1017 0.0725 0.0425 0.0463 53 [29, 49] url

PGP 10, 680 24, 316 0.0642 0.1935 0.0559 0.0236 0.0244 54 [5] url

Gnutella, August 4 2002 10, 876 39, 994 0.0762 0.0942 0.0771 0.0585 0.0655 55 [29, 49] url

Hep-Ph, 1993-2003 11, 204 117, 619 0.0048 0.0565 0.0077 0.0041 0.0041 56 [29] url

Spanish 11, 558 43, 050 0.0127 0.2093 0.0022 0.0098 0.0107 57 [37] url

DBLP, citations 12, 495 49, 563 0.0325 0.0652 0.0234 0.0234 0.0263 58 [24, 31] url

Spanish 12, 643 55, 019 0.0119 0.2347 0.0012 0.0090 0.0100 59 [24] url

Cond-Mat, 1995-1999 13, 861 44, 619 0.0637 0.1649 0.0798 0.0400 0.0432 60 [40] url

Astrophysics 14, 845 119, 652 0.0182 0.0551 0.0225 0.0135 0.0138 61 [40] url

Google 15, 763 148, 585 0.0078 0.0636 0.0011 0.0057 0.0064 62 [45] url

AstroPhys, 1993-2003 17, 903 196, 972 0.0132 0.0323 0.0155 0.0106 0.0108 63 [29] url

Cond-Mat, 1993-2003 21, 363 91, 286 0.0367 0.0836 0.0466 0.0264 0.0279 64 [29] url

Gnutella, Aug. 25, 2002 22, 663 54, 693 0.1150 0.1265 0.1026 0.0916 0.1066 65 [29, 49] url

Internet 22, 963 48, 436 0.0193 0.1940 0.0038 0.0140 0.0155 66 none url

Thesaurus 23, 132 297, 094 0.0112 0.0174 0.0098 0.0100 0.0102 67 [22, 24] url

Cora 23, 166 89, 157 0.0454 0.1184 0.0441 0.0317 0.0342 68 [24, 51] url

Linux, mailing list 24, 567 158, 164 0.0052 0.2261 0.0029 0.0045 0.0045 69 [24] url

AS Caida 26, 475 53, 381 0.0210 0.1640 0.0036 0.0144 0.0168 70 [28] url

Gnutella, Aug. 24, 2002 26, 498 65, 359 0.1060 0.1174 0.0906 0.0511 0.0927 71 [29, 49] url

Hep-Th, citations 27, 400 352, 021 0.0110 0.0245 0.0095 0.0090 0.0094 72 [24, 29] url

Cond-Mat, 1995-2003 27, 519 116, 181 0.0342 0.0814 0.0470 0.0248 0.0261 73 [40] url

Digg 29, 652 84, 781 0.0413 0.0577 0.0369 0.0324 0.0362 74 [12, 24] url

Linux, soft. 30, 817 213, 208 0.0076 0.0457 0.0012 0.0059 0.0065 75 [24] url

Enron 33, 696 180, 811 0.0100 0.4460 0.0071 0.0084 0.0087 76 [30] url

Hep-Ph, citations 34, 401 420, 784 0.0158 0.0332 0.0160 0.0131 0.0135 77 [24, 29] url

Cond-Mat, 1995-2005 36, 458 171, 735 0.0256 0.0585 0.0372 0.0195 0.0203 78 [40] url

Gnutella, Aug. 30, 2002 36, 646 88, 303 0.0968 0.1122 0.0956 0.0773 0.0878 79 [29, 49] url

Slashdot 51, 083 116, 573 0.0262 0.0435 0.0124 0.0170 0.0222 80 [16, 24] url

Gnutella, Aug. 31, 2002 62, 561 147, 878 0.0956 0.1082 0.0943 0.0759 0.0871 81 [29, 49] url

Facebook 63, 392 816, 886 0.0086 0.0192 0.0115 0.0075 0.0076 82 [53] url

TABLE II: Continuation of table I.

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network N E pc qc pc pc pc Fig. Refs. Url

Epinions 75, 877 405, 739 0.0062 0.0148 0.0055 0.0054 0.0055 83 [24, 48] url

Slashdot zoo 79, 116 467, 731 0.0088 0.0162 0.0069 0.0077 0.0078 84 [24, 26] url

Flickr 105, 722 2, 316, 668 0.0142 0.0662 0.0029 0.0016 0.0016 85 [24, 35] url

Wikipedia, edits 113, 123 2, 025, 910 0.0029 0.0085 0.0015 0.0025 0.0026 86 [7, 24] url

Petster, cats 148, 826 5, 447, 464 0.0010 0.0303 0.0001 0.0008 0.0009 87 [24] url

Gowalla 196, 591 950, 327 0.0073 0.0310 0.0033 0.0059 0.0063 88 [10, 24] url

Libimseti 220, 970 17, 233, 144 0.0011 0.0028 0.0006 0.0011 0.0011 89 [8, 24, 25] url

EU email 224, 832 339, 925 0.0119 0.5975 0.0018 0.0098 0.0103 90 [24, 29] url

Web Stanford 255, 265 1, 941, 926 0.0598 0.1932 0.0005 0.0022 0.0024 91 [30] url

Amazon, Mar. 2, 2003 262, 111 899, 792 0.0940 0.1539 0.0987 0.0425 0.0562 92 [27] url

DBLP, collaborations 317, 080 1, 049, 866 0.0337 0.0658 0.0482 0.0086 0.0087 93 [24, 31] url

Web Notre Dame 325, 729 1, 090, 108 0.0847 0.2037 0.0036 0.0054 0.0057 94 [2] url

MathSciNet 332, 689 820, 644 0.0478 0.0805 0.0648 0.0277 0.0298 95 [46] url

CiteSeer 365, 154 1, 721, 981 0.0250 0.0473 0.0211 0.0172 0.0190 96 [6, 24] url

Zhishi 372, 840 2, 318, 025 0.0301 0.0738 0.0000 0.0010 0.0011 97 [24, 42] url

Actor coll. net. 374, 511 15, 014, 839 0.0013 0.0047 0.0024 0.0012 0.0012 98 [4, 24] url

Amazon, Mar. 12, 2003 400, 727 2, 349, 869 0.0401 0.0761 0.0341 0.0178 0.0285 99 [27] url

Amazon, Jun. 6, 2003 403, 364 2, 443, 311 0.0364 0.0759 0.0338 0.0175 0.0248 100 [27] url

Amazon, May 5, 2003 410, 236 2, 439, 437 0.0360 0.0759 0.0334 0.0172 0.0248 101 [27] url

Petster, dogs 426, 485 8, 543, 321 0.0015 0.0380 0.0005 0.0013 0.0014 102 [24] url

Road network PA 1, 087, 562 1, 541, 514 0.6923 0.7571 0.4547 0.2263 0.3216 103 [30] url

YouTube friend. net. 1, 134, 890 2, 987, 624 0.0063 0.0171 0.0020 0.0048 0.0054 104 [24, 55] url

Road network TX 1, 351, 137 1, 879, 201 0.7362 0.7875 0.4661 0.2038 0.2809 105 [30] url

AS Skitter 1, 694, 616 11, 094, 209 0.0018 0.0130 0.0007 0.0015 0.0015 106 [28] url

Road network CA 1, 957, 027 2, 760, 388 0.6933 0.7582 0.4609 0.2156 0.3011 107 [30] url

Wikipedia, pages 2, 070, 367 42, 336, 614 0.0014 0.0046 0.0003 0.0012 0.0013 108 [46] url

US Patents 3, 764, 117 16, 511, 740 0.0290 0.0515 0.0492 0.0088 0.0091 109 [18, 24] url

DBpedia 3, 915, 921 12, 577, 253 0.0170 0.0380 0.0001 0.0014 0.0022 110 [3, 24] url

LiveJournal 5, 189, 809 48, 688, 097 0.0028 0.0116 0.0065 0.0019 0.0019 111 [24, 39] url

TABLE III: Continuation of tables I and II.

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FIG. 3: Social 3, N = 32, E = 80, pc = 0.2500, qc = 0.3125, pc = 0.2025, pc = 0.1675, pc = 0.2109

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FIG. 4: Karate club, N = 34, E = 78, pc = 0.2436, qc = 0.3824, pc = 0.1477, pc = 0.1487, pc = 0.1889

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FIG. 5: Protein 2, N = 53, E = 123, pc = 0.3902, qc = 0.6038, pc = 0.2278, pc = 0.1722, pc = 0.2139

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FIG. 6: Dolphins, N = 62, E = 159, pc = 0.2390, qc = 0.3871, pc = 0.1723, pc = 0.1390, pc = 0.1668

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FIG. 7: Social 1, N = 67, E = 142, pc = 0.3099, qc = 0.4030, pc = 0.2351, pc = 0.1789, pc = 0.2294

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FIG. 8: Les Miserables, N = 77, E = 254, pc = 0.1339, qc = 0.3377, pc = 0.0905, pc = 0.0833, pc = 0.0930

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FIG. 9: Protein 1, N = 95, E = 213, pc = 0.6056, qc = 0.8000, pc = 0.2533, pc = 0.1865, pc = 0.2354

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FIG. 10: E. Coli, transcription, N = 97, E = 212, pc = 0.5189, qc = 0.6495, pc = 0.2270, pc = 0.1531,pc = 0.1874

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FIG. 11: Political books, N = 105, E = 441, pc = 0.1927, qc = 0.3429, pc = 0.0915, pc = 0.0838, pc = 0.0941

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FIG. 12: David Copperfield, N = 112, E = 425, pc = 0.1035, qc = 0.1786, pc = 0.0783, pc = 0.0760,pc = 0.0867

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0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.2 0.40.00

0.02

0.04

susc

eptib

ility

b

FIG. 13: College football, N = 115, E = 613, pc = 0.1338, qc = 0.2087, pc = 0.1027, pc = 0.0928, pc = 0.1024

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.00.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.0 0.5 1.00.00

0.01

0.02

0.03

susc

eptib

ility

b

FIG. 14: S 208, N = 122, E = 189, pc = 0.4656, qc = 0.5492, pc = 0.3614, pc = 0.2435, pc = 0.3640

11

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.10 0.150.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.05 0.10 0.150.00

0.01

0.02

susc

eptib

ility

b

FIG. 15: High school, 2011, N = 126, E = 1, 709, pc = 0.0380, qc = 0.0873, pc = 0.0315, pc = 0.0294,pc = 0.0304

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.05 0.100.00

0.01

0.02

susc

eptib

ility

b

FIG. 16: Bay Dry, N = 128, E = 2, 106, pc = 0.0304, qc = 0.0469, pc = 0.0253, pc = 0.0249, pc = 0.0257

12

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.05 0.100.00

0.01

0.02

susc

eptib

ility

b

FIG. 17: Bay Wet, N = 128, E = 2, 075, pc = 0.0308, qc = 0.0469, pc = 0.0256, pc = 0.0252, pc = 0.0260

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.050.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.050.00

0.01

0.02su

scep

tibili

tyb

FIG. 18: Radoslaw Email, N = 167, E = 3, 250, pc = 0.0203, qc = 0.0419, pc = 0.0158, pc = 0.0165, pc = 0.0168

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.10 0.150.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.05 0.10 0.150.00

0.01

0.02

0.03

susc

eptib

ility

b

FIG. 19: High school, 2012, N = 180, E = 2, 220, pc = 0.0437, qc = 0.0944, pc = 0.0350, pc = 0.0332,pc = 0.0345

13

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.05 0.100.00

0.01

0.02

susc

eptib

ility

b

FIG. 20: Little Rock Lake, N = 183, E = 2, 434, pc = 0.0300, qc = 0.0874, pc = 0.0239, pc = 0.0242,pc = 0.0250

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.05 0.100.00

0.01

0.02

susc

eptib

ility

b

FIG. 21: Jazz, N = 198, E = 2, 742, pc = 0.0314, qc = 0.0859, pc = 0.0266, pc = 0.0250, pc = 0.0258

14

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.00.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.5 1.00.00

0.01

0.02

0.03

susc

eptib

ility

b

FIG. 22: S 420, N = 252, E = 399, pc = 0.4511, qc = 0.5357, pc = 0.3437, pc = 0.2294, pc = 0.3462

15

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.20.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.1 0.20.00

0.01

0.02

susc

eptib

ility

b

FIG. 23: C. Elegans, neural, N = 297, E = 2, 148, pc = 0.0540, qc = 0.1044, pc = 0.0399, pc = 0.0410,pc = 0.0439

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.00.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.0 0.5 1.00.00

0.02

0.04

susc

eptib

ility

b

FIG. 24: Network Science, N = 379, E = 914, pc = 0.3982, qc = 0.6359, pc = 0.1424, pc = 0.0964, pc = 0.1148

16

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.30.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.1 0.2 0.30.00

0.01

0.02

susc

eptib

ility

b

FIG. 25: Dublin, N = 410, E = 2, 765, pc = 0.0778, qc = 0.1951, pc = 0.0564, pc = 0.0428, pc = 0.0450

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.05 0.100.000

0.005

susc

eptib

ility

b

FIG. 26: US Air Trasportation, N = 500, E = 2, 980, pc = 0.0268, qc = 0.1080, pc = 0.0189, pc = 0.0208,pc = 0.0215

17

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.00.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.5 1.00.00

0.01

0.02

0.03

susc

eptib

ility

b

FIG. 27: S 838, N = 512, E = 819, pc = 0.4493, qc = 0.5449, pc = 0.3301, pc = 0.1996, pc = 0.3401

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.50.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.0 0.50.00

0.01

0.02

susc

eptib

ility

b

FIG. 28: Yeast, transcription, N = 662, E = 1, 062, pc = 0.2458, qc = 0.3414, pc = 0.0799, pc = 0.1002,pc = 0.1538

18

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.20.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.1 0.20.000

0.005

0.010

susc

eptib

ility

b

FIG. 29: URV email, N = 1, 133, E = 5, 451, pc = 0.0646, qc = 0.1068, pc = 0.0565, pc = 0.0482, pc = 0.0519

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.04 0.060.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.02 0.04 0.060.000

0.005

0.010

susc

eptib

ility

b

FIG. 30: Political blogs, N = 1, 222, E = 16, 714, pc = 0.0168, qc = 0.0442, pc = 0.0125, pc = 0.0135,pc = 0.0138

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.60.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.0 0.2 0.4 0.60.000

0.005

0.010

susc

eptib

ility

b

FIG. 31: Air traffic, N = 1, 226, E = 2, 408, pc = 0.1815, qc = 0.2871, pc = 0.1573, pc = 0.1086, pc = 0.1336

19

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.00.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.5 1.00.000

0.005

0.010

0.015

susc

eptib

ility

b

FIG. 32: Yeast, protein, N = 1, 458, E = 1, 948, pc = 0.2988, qc = 0.3752, pc = 0.1632, pc = 0.1327,pc = 0.1980

20

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.05 0.100.000

0.005

susc

eptib

ility

b

FIG. 33: Petster, hamster, N = 1, 788, E = 12, 476, pc = 0.0273, qc = 0.0587, pc = 0.0224, pc = 0.0217,pc = 0.0226

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.050.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.050.000

0.005

susc

eptib

ility

b

FIG. 34: UC Irvine, N = 1, 893, E = 13, 835, pc = 0.0248, qc = 0.0449, pc = 0.0183, pc = 0.0208, pc = 0.0216

21

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.30.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.1 0.2 0.30.000

0.005

susc

eptib

ility

b

FIG. 35: Yeast, protein, N = 2, 224, E = 6, 609, pc = 0.0794, qc = 0.1245, pc = 0.0639, pc = 0.0525,pc = 0.0577

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.05 0.100.000

0.002

0.004

susc

eptib

ility

b

FIG. 36: Japanese, N = 2, 698, E = 7, 995, pc = 0.0318, qc = 0.1160, pc = 0.0093, pc = 0.0233, pc = 0.0262

22

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.04 0.060.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.02 0.04 0.060.000

0.002

0.004

susc

eptib

ility

b

FIG. 37: Open flights, N = 2, 905, E = 15, 645, pc = 0.0199, qc = 0.0792, pc = 0.0182, pc = 0.0159,pc = 0.0163

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.40.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.0 0.2 0.40.000

0.002

0.004

susc

eptib

ility

b

FIG. 38: GR-QC, 1993-2003, N = 4, 158, E = 13, 422, pc = 0.1345, qc = 0.2244, pc = 0.0589, pc = 0.0219,pc = 0.0225

23

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.01 0.020.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.01 0.020.000

0.002

0.004

0.006

susc

eptib

ility

b

FIG. 39: Tennis, N = 4, 338, E = 81, 865, pc = 0.0073, qc = 0.0274, pc = 0.0063, pc = 0.0062, pc = 0.0062

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.00.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.0 0.5 1.00.00

0.02

0.04

susc

eptib

ility

b

FIG. 40: US Power grid, N = 4, 941, E = 6, 594, pc = 0.6583, qc = 0.7353, pc = 0.3483, pc = 0.1336,pc = 0.1606

24

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.05 0.100.000

0.002

0.004

susc

eptib

ility

b

FIG. 41: HT09, N = 5, 352, E = 18, 481, pc = 0.0283, qc = 0.1700, pc = 0.0050, pc = 0.0211, pc = 0.0244

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.40.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.0 0.2 0.40.000

0.002

0.004

susc

eptib

ility

b

FIG. 42: Hep-Th, 1995-1999, N = 5, 835, E = 13, 815, pc = 0.1081, qc = 0.1973, pc = 0.1231, pc = 0.0554,pc = 0.0588

25

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.040.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.02 0.040.000

0.002

0.004

susc

eptib

ility

b

FIG. 43: Reactome, N = 5, 973, E = 145, 778, pc = 0.0114, qc = 0.0591, pc = 0.0070, pc = 0.0048, pc = 0.0048

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.01 0.02 0.030.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.01 0.02 0.030.000

0.001

0.002

0.003su

scep

tibili

tyb

FIG. 44: Jung, N = 6, 120, E = 50, 290, pc = 0.0093, qc = 0.1239, pc = 0.0010, pc = 0.0070, pc = 0.0078

26

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.10 0.150.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.05 0.10 0.150.000

0.001

0.002

0.003

susc

eptib

ility

b

FIG. 45: Gnutella, Aug. 8, 2002, N = 6, 299, E = 20, 776, pc = 0.0455, qc = 0.1121, pc = 0.0600,pc = 0.0352, pc = 0.0377

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.01 0.02 0.030.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.01 0.02 0.030.000

0.001

0.002

0.003

susc

eptib

ility

b

FIG. 46: JDK, N = 6, 434, E = 53, 658, pc = 0.0091, qc = 0.1173, pc = 0.0010, pc = 0.0070, pc = 0.0077

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.05 0.100.000

0.001

0.002

susc

eptib

ility

b

FIG. 47: AS Oregon, N = 6, 474, E = 12, 572, pc = 0.0356, qc = 0.1767, pc = 0.0061, pc = 0.0216, pc = 0.0285

27

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.040.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.02 0.040.000

0.001

0.002

0.003

susc

eptib

ility

b

FIG. 48: English, N = 7, 377, E = 44, 205, pc = 0.0114, qc = 0.1590, pc = 0.0031, pc = 0.0091, pc = 0.0096

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.10 0.150.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.05 0.10 0.150.000

0.001

0.002

susc

eptib

ility

b

FIG. 49: Gnutella, Aug. 9, 2002, N = 8, 104, E = 26, 008, pc = 0.0451, qc = 0.1099, pc = 0.0632,pc = 0.0351, pc = 0.0376

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.050.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.050.000

0.001

0.002

0.003

susc

eptib

ility

b

FIG. 50: French, N = 8, 308, E = 23, 832, pc = 0.0236, qc = 0.1284, pc = 0.0046, pc = 0.0165, pc = 0.0191

28

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.30.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.1 0.2 0.30.000

0.002

0.004

susc

eptib

ility

b

FIG. 51: Hep-Th, 1993-2003, N = 8, 638, E = 24, 806, pc = 0.0755, qc = 0.1363, pc = 0.0834, pc = 0.0322,pc = 0.0333

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.20.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.0 0.1 0.20.000

0.002

0.004

susc

eptib

ility

b

FIG. 52: Gnutella, Aug. 6, 2002, N = 8, 717, E = 31, 525, pc = 0.0648, qc = 0.1005, pc = 0.0746,pc = 0.0447, pc = 0.0489

29

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.20.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.1 0.20.000

0.001

0.002

0.003

susc

eptib

ility

b

FIG. 53: Gnutella, Aug. 5, 2002, N = 8, 842, E = 31, 837, pc = 0.0565, qc = 0.1017, pc = 0.0725,pc = 0.0425, pc = 0.0463

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.20.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.0 0.1 0.20.0000

0.0005

0.0010

susc

eptib

ility

b

FIG. 54: PGP, N = 10, 680, E = 24, 316, pc = 0.0642, qc = 0.1935, pc = 0.0559, pc = 0.0236, pc = 0.0244

30

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.30.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.1 0.2 0.30.000

0.002

0.004

0.006

susc

eptib

ility

b

FIG. 55: Gnutella, August 4 2002, N = 10, 876, E = 39, 994, pc = 0.0762, qc = 0.0942, pc = 0.0771,pc = 0.0585, pc = 0.0655

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.005 0.010 0.0150.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.000 0.005 0.010 0.0150.0000

0.0005

0.0010

0.0015

susc

eptib

ility

b

FIG. 56: Hep-Ph, 1993-2003, N = 11, 204, E = 117, 619, pc = 0.0048, qc = 0.0565, pc = 0.0077, pc = 0.0041,pc = 0.0041

31

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.040.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.02 0.040.000

0.001

0.002

susc

eptib

ility

b

FIG. 57: Spanish, N = 11, 558, E = 43, 050, pc = 0.0127, qc = 0.2093, pc = 0.0022, pc = 0.0098, pc = 0.0107

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.05 0.100.000

0.001

0.002

0.003su

scep

tibili

tyb

FIG. 58: DBLP, citations, N = 12, 495, E = 49, 563, pc = 0.0325, qc = 0.0652, pc = 0.0234, pc = 0.0234,pc = 0.0263

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.040.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.02 0.040.000

0.001

0.002

susc

eptib

ility

b

FIG. 59: Spanish, N = 12, 643, E = 55, 019, pc = 0.0119, qc = 0.2347, pc = 0.0012, pc = 0.0090, pc = 0.0100

32

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.20.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.1 0.20.000

0.001

0.002

susc

eptib

ility

b

FIG. 60: Cond-Mat, 1995-1999, N = 13, 861, E = 44, 619, pc = 0.0637, qc = 0.1649, pc = 0.0798, pc = 0.0400,pc = 0.0432

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.04 0.060.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.02 0.04 0.060.000

0.001

0.002

susc

eptib

ility

b

FIG. 61: Astrophysics, N = 14, 845, E = 119, 652, pc = 0.0182, qc = 0.0551, pc = 0.0225, pc = 0.0135,pc = 0.0138

33

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.01 0.02 0.030.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.01 0.02 0.030.000

0.001

0.002

susc

eptib

ility

b

FIG. 62: Google, N = 15, 763, E = 148, 585, pc = 0.0078, qc = 0.0636, pc = 0.0011, pc = 0.0057, pc = 0.0064

34

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.040.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.02 0.040.000

0.001

0.002

0.003

susc

eptib

ility

b

FIG. 63: AstroPhys, 1993-2003, N = 17, 903, E = 196, 972, pc = 0.0132, qc = 0.0323, pc = 0.0155, pc = 0.0106,pc = 0.0108

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.05 0.100.000

0.001

0.002

susc

eptib

ility

b

FIG. 64: Cond-Mat, 1993-2003, N = 21, 363, E = 91, 286, pc = 0.0367, qc = 0.0836, pc = 0.0466, pc = 0.0264,pc = 0.0279

35

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.40.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.2 0.40.000

0.005

susc

eptib

ility

b

FIG. 65: Gnutella, Aug. 25, 2002, N = 22, 663, E = 54, 693, pc = 0.1150, qc = 0.1265, pc = 0.1026,pc = 0.0916, pc = 0.1066

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.04 0.060.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.02 0.04 0.060.0000

0.0005

0.0010

0.0015

susc

eptib

ility

b

FIG. 66: Internet, N = 22, 963, E = 48, 436, pc = 0.0193, qc = 0.1940, pc = 0.0038, pc = 0.0140, pc = 0.0155

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.040.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.02 0.040.000

0.002

0.004

susc

eptib

ility

b

FIG. 67: Thesaurus, N = 23, 132, E = 297, 094, pc = 0.0112, qc = 0.0174, pc = 0.0098, pc = 0.0100, pc = 0.0102

36

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.10 0.150.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.05 0.10 0.150.000

0.001

0.002

susc

eptib

ility

b

FIG. 68: Cora, N = 23, 166, E = 89, 157, pc = 0.0454, qc = 0.1184, pc = 0.0441, pc = 0.0317, pc = 0.0342

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.01 0.020.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.01 0.020.0000

0.0005

0.0010

0.0015

susc

eptib

ility

b

FIG. 69: Linux, mailing list, N = 24, 567, E = 158, 164, pc = 0.0052, qc = 0.2261, pc = 0.0029, pc = 0.0045,pc = 0.0045

37

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.050.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.050.0000

0.0005

0.0010

0.0015

susc

eptib

ility

b

FIG. 70: AS Caida, N = 26, 475, E = 53, 381, pc = 0.0210, qc = 0.1640, pc = 0.0036, pc = 0.0144, pc = 0.0168

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.40.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.0 0.2 0.40.000

0.002

0.004

susc

eptib

ility

b

FIG. 71: Gnutella, Aug. 24, 2002, N = 26, 498, E = 65, 359, pc = 0.1060, qc = 0.1174, pc = 0.0906,pc = 0.0511, pc = 0.0927

38

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.040.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.02 0.040.000

0.001

0.002

0.003

susc

eptib

ility

b

FIG. 72: Hep-Th, citations, N = 27, 400, E = 352, 021, pc = 0.0110, qc = 0.0245, pc = 0.0095, pc = 0.0090,pc = 0.0094

39

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.05 0.100.0000

0.0005

0.0010

0.0015

susc

eptib

ility

b

FIG. 73: Cond-Mat, 1995-2003, N = 27, 519, E = 116, 181, pc = 0.0342, qc = 0.0814, pc = 0.0470, pc = 0.0248,pc = 0.0261

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.10 0.150.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.05 0.10 0.150.000

0.001

0.002

susc

eptib

ility

b

FIG. 74: Digg, N = 29, 652, E = 84, 781, pc = 0.0413, qc = 0.0577, pc = 0.0369, pc = 0.0324, pc = 0.0362

40

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.01 0.02 0.030.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.01 0.02 0.030.0000

0.0005

0.0010

0.0015

susc

eptib

ility

b

FIG. 75: Linux, soft., N = 30, 817, E = 213, 208, pc = 0.0076, qc = 0.0457, pc = 0.0012, pc = 0.0059,pc = 0.0065

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.01 0.02 0.03 0.040.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.01 0.02 0.03 0.040.0000

0.0005

0.0010

0.0015

susc

eptib

ility

b

FIG. 76: Enron, N = 33, 696, E = 180, 811, pc = 0.0100, qc = 0.4460, pc = 0.0071, pc = 0.0084, pc = 0.0087

41

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.04 0.060.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.02 0.04 0.060.000

0.001

0.002

0.003

susc

eptib

ility

b

FIG. 77: Hep-Ph, citations, N = 34, 401, E = 420, 784, pc = 0.0158, qc = 0.0332, pc = 0.0160, pc = 0.0131,pc = 0.0135

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.05 0.100.0000

0.0005

0.0010

0.0015

susc

eptib

ility

b

FIG. 78: Cond-Mat, 1995-2005, N = 36, 458, E = 171, 735, pc = 0.0256, qc = 0.0585, pc = 0.0372, pc = 0.0195,pc = 0.0203

42

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.30.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.1 0.2 0.30.000

0.002

0.004

susc

eptib

ility

b

FIG. 79: Gnutella, Aug. 30, 2002, N = 36, 646, E = 88, 303, pc = 0.0968, qc = 0.1122, pc = 0.0956,pc = 0.0773, pc = 0.0878

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.100.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.05 0.100.0000

0.0005

0.0010

susc

eptib

ility

b

FIG. 80: Slashdot, N = 51, 083, E = 116, 573, pc = 0.0262, qc = 0.0435, pc = 0.0124, pc = 0.0170, pc = 0.0222

43

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.30.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.0 0.1 0.2 0.30.000

0.002

0.004

susc

eptib

ility

b

FIG. 81: Gnutella, Aug. 31, 2002, N = 62, 561, E = 147, 878, pc = 0.0956, qc = 0.1082, pc = 0.0943,pc = 0.0759, pc = 0.0871

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.01 0.02 0.030.0

0.5

1.0

perc

olat

ion

stre

ngth a

0.00 0.01 0.02 0.030.0000

0.0005

0.0010

0.0015

susc

eptib

ility

b

FIG. 82: Facebook, N = 63, 392, E = 816, 886, pc = 0.0086, qc = 0.0192, pc = 0.0115, pc = 0.0075, pc = 0.0076

44

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.01 0.020.0

0.5

1.0pe

rcol

atio

nst

reng

th a

0.00 0.01 0.020.0000

0.0005

susc

eptib

ility

b

FIG. 83: Epinions, N = 75, 877, E = 405, 739, pc = 0.0062, qc = 0.0148, pc = 0.0055, pc = 0.0054, pc = 0.0055

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.01 0.02 0.030.0

0.5

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ion

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ngth a

0.00 0.01 0.02 0.030.0000

0.0005

0.0010

susc

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ility

b

FIG. 84: Slashdot zoo, N = 79, 116, E = 467, 731, pc = 0.0088, qc = 0.0162, pc = 0.0069, pc = 0.0077,pc = 0.0078

45

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

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0.00 0.02 0.040.0

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0.002

0.004

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FIG. 85: Flickr, N = 105, 722, E = 2, 316, 668, pc = 0.0142, qc = 0.0662, pc = 0.0029, pc = 0.0016, pc = 0.0016

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

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1.0

0.000 0.005 0.0100.0

0.5

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olat

ion

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ngth a

0.000 0.005 0.0100.0000

0.0005

0.0010

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ility

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FIG. 86: Wikipedia, edits, N = 113, 123, E = 2, 025, 910, pc = 0.0029, qc = 0.0085, pc = 0.0015, pc = 0.0025,pc = 0.0026

46

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

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0.000 0.001 0.002 0.0030.0

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atio

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0.000 0.001 0.002 0.0030.0000

0.0005

0.0010

susc

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ility

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FIG. 87: Petster, cats, N = 148, 826, E = 5, 447, 464, pc = 0.0010, qc = 0.0303, pc = 0.0001, pc = 0.0008,pc = 0.0009

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

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1.0

0.00 0.01 0.020.0

0.5

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perc

olat

ion

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0.00 0.01 0.020.0000

0.0002

0.0004

susc

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ility

b

FIG. 88: Gowalla, N = 196, 591, E = 950, 327, pc = 0.0073, qc = 0.0310, pc = 0.0033, pc = 0.0059, pc = 0.0063

47

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

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0.000 0.002 0.0040.0

0.5

1.0pe

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atio

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0.000 0.002 0.0040.0000

0.0005

0.0010

0.0015

susc

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ility

b

FIG. 89: Libimseti, N = 220, 970, E = 17, 233, 144, pc = 0.0011, qc = 0.0028, pc = 0.0006, pc = 0.0011,pc = 0.0011

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.02 0.040.0

0.5

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perc

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ion

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ngth a

0.00 0.02 0.040.0000

0.0002

0.0004

susc

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ility

b

FIG. 90: EU email, N = 224, 832, E = 339, 925, pc = 0.0119, qc = 0.5975, pc = 0.0018, pc = 0.0098, pc = 0.0103

48

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0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

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0.0 0.1 0.20.0

0.5

1.0pe

rcol

atio

nst

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0.0 0.1 0.20.000

0.001

0.002

susc

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ility

b

FIG. 91: Web Stanford, N = 255, 265, E = 1, 941, 926, pc = 0.0598, qc = 0.1932, pc = 0.0005, pc = 0.0022,pc = 0.0024

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

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1.0

0.0 0.1 0.2 0.30.0

0.5

1.0

perc

olat

ion

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ngth a

0.0 0.1 0.2 0.30.0000

0.0005

0.0010

susc

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ility

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FIG. 92: Amazon, Mar. 2, 2003, N = 262, 111, E = 899, 792, pc = 0.0940, qc = 0.1539, pc = 0.0987,pc = 0.0425, pc = 0.0562

49

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occupation probability

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FIG. 93: DBLP, collaborations, N = 317, 080, E = 1, 049, 866, pc = 0.0337, qc = 0.0658, pc = 0.0482,pc = 0.0086, pc = 0.0087

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

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1.0

0.0 0.1 0.2 0.30.0

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olat

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ngth a

0.0 0.1 0.2 0.30.000

0.001

0.002

0.003

susc

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FIG. 94: Web Notre Dame, N = 325, 729, E = 1, 090, 108, pc = 0.0847, qc = 0.2037, pc = 0.0036, pc = 0.0054,pc = 0.0057

50

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occupation probability

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0.00 0.05 0.10 0.150.0

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atio

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FIG. 95: MathSciNet, N = 332, 689, E = 820, 644, pc = 0.0478, qc = 0.0805, pc = 0.0648, pc = 0.0277,pc = 0.0298

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

0.6

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1.0

0.00 0.05 0.100.0

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ion

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0.0002

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susc

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FIG. 96: CiteSeer, N = 365, 154, E = 1, 721, 981, pc = 0.0250, qc = 0.0473, pc = 0.0211, pc = 0.0172,pc = 0.0190

51

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occupation probability

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FIG. 97: Zhishi, N = 372, 840, E = 2, 318, 025, pc = 0.0301, qc = 0.0738, pc = 0.0000, pc = 0.0010, pc = 0.0011

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

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0.4

0.6

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1.0

0.000 0.002 0.0040.0

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0.000 0.002 0.0040.0000

0.0001

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susc

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ility

b

FIG. 98: Actor coll. net., N = 374, 511, E = 15, 014, 839, pc = 0.0013, qc = 0.0047, pc = 0.0024,pc = 0.0012, pc = 0.0012

52

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occupation probability

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FIG. 99: Amazon, Mar. 12, 2003, N = 400, 727, E = 2, 349, 869, pc = 0.0401, qc = 0.0761, pc = 0.0341,pc = 0.0178, pc = 0.0285

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

0.0

0.2

0.4

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0.00 0.05 0.100.0

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FIG. 100: Amazon, Jun. 6, 2003, N = 403, 364, E = 2, 443, 311, pc = 0.0364, qc = 0.0759, pc = 0.0338,pc = 0.0175, pc = 0.0248

53

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FIG. 101: Amazon, May 5, 2003, N = 410, 236, E = 2, 439, 437, pc = 0.0360, qc = 0.0759, pc = 0.0334,pc = 0.0172, pc = 0.0248

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

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0.000 0.002 0.0040.0

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susc

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FIG. 102: Petster, dogs, N = 426, 485, E = 8, 543, 321, pc = 0.0015, qc = 0.0380, pc = 0.0005, pc = 0.0013,pc = 0.0014

54

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FIG. 103: Road network PA, N = 1, 087, 562, E = 1, 541, 514, pc = 0.6923, qc = 0.7571, pc = 0.4547, pc =0.2263, pc = 0.3216

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

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0.00 0.01 0.020.0

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0.00005

0.00010

0.00015

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FIG. 104: YouTube friend. net., N = 1, 134, 890, E = 2, 987, 624, pc = 0.0063, qc = 0.0171, pc = 0.0020,pc = 0.0048, pc = 0.0054

55

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FIG. 105: Road network TX, N = 1, 351, 137, E = 1, 879, 201, pc = 0.7362, qc = 0.7875, pc = 0.4661, pc =0.2038, pc = 0.2809

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

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0.000 0.002 0.004 0.0060.0

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0.000 0.002 0.004 0.0060.00000

0.00005

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FIG. 106: AS Skitter, N = 1, 694, 616, E = 11, 094, 209, pc = 0.0018, qc = 0.0130, pc = 0.0007, pc = 0.0015,pc = 0.0015

56

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FIG. 107: Road network CA, N = 1, 957, 027, E = 2, 760, 388, pc = 0.6933, qc = 0.7582, pc = 0.4609, pc =0.2156, pc = 0.3011

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

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FIG. 108: Wikipedia, pages, N = 2, 070, 367, E = 42, 336, 614, pc = 0.0014, qc = 0.0046, pc = 0.0003,pc = 0.0012, pc = 0.0013

57

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FIG. 109: US Patents, N = 3, 764, 117, E = 16, 511, 740, pc = 0.0290, qc = 0.0515, pc = 0.0492, pc = 0.0088,pc = 0.0091

0.0 0.2 0.4 0.6 0.8 1.0

occupation probability

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0.00 0.02 0.04 0.060.0

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FIG. 110: DBpedia, N = 3, 915, 921, E = 12, 577, 253, pc = 0.0170, qc = 0.0380, pc = 0.0001, pc = 0.0014,pc = 0.0022

58

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0.00002

0.00004

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FIG. 111: LiveJournal, N = 5, 189, 809, E = 48, 688, 097, pc = 0.0028, qc = 0.0116, pc = 0.0065, pc = 0.0019,pc = 0.0019

59

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