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KONECT > Statistics > Preferential attachment exponent

## Preferential attachment exponent

The term preferential attachment refers to the observation that in networks that grow over time, the probability that an edge is added to a node with $$d$$ neighbors is proportional to $$d$$. This linear relationship lies at the heart of BarabÃ¡si and Albert's scale-free network model [6], and has been used in a vast number of subsequent work to model networks, online and offline. The scale-free network model results in a distribution of degrees, i.e., number of neighbors of individual nodes, that follows a power law with negative exponent. In other words, the number of nodes with degree $$d$$ is proportional to $$d^{-\gamma}$$ in these networks, for a constant $$\gamma>1$$.
In basic preferential attachment, the probability that an edge attached to a vertex $$u$$ is propertional to its degree $$d(u)$$. An extension of this basic model uses a probability that is a power of the degree, i.e., $$d(u)^\beta$$. The exponent $$\beta$$ is a positive number, and can be measured empirically from a dataset [9]. The value of $$\beta$$ then determines the type of preferential attachment:

1. Constant case $$\beta=0$$. This case is equivalent to a constant probability of attachment, and thus this graph growth model results in networks in which each edge is equally likely and independent from other edges. This is the ErdÅs–RÃ©nyi model of random graphs [3].
2. Sublinear case $$0 < \beta < 1$$. In this case, the preferential attachment function is sublinear. This model gives rise to a stretched exponential degree distribution [4], whose exact expression is complex and given in [5].
3. Linear case $$\beta=1$$. This is the scale-free network model of BarabÃ¡si and Albert [6], in which attachment is proportional to the degree. This gives a power law degree distribution.
4. Superlinear case $$\beta > 1$$. In this case, a single node will acquire 100% of all edges asymptotically [7]. Networks with this behavior will however display power law degree distributions in the pre-asymptotic regime [8].
The following minimization problem gives an estimate for the exponent $$\beta$$ [9]. \begin{align} \min_{\alpha,\beta} \sum_{u\in V} \left( \alpha + \beta \ln[1 + d_1(u)] - \ln[\lambda + d_2(u)] \right)^2 \label{eq:min} \end{align} The resulting value of $$\beta$$ is the estimated preferential attachment exponent. To measure the error of the fit, the root-mean-square logarithmic error $$\epsilon$$ can be defined in the following way: \begin{align*} \epsilon &= \exp\left\{ \sqrt{ \frac 1 {|V|} \sum_{u \in V} \left(\alpha + \beta \ln[1 + d_1(u)] - \ln[\lambda + d_2(u)]\right)^2 } \right\} \end{align*} This gives the average factor by which the actual new number of edges differs from the predicted value, computed logarithmically. The value of $$\epsilon$$ is larger or equal to one by construction.

 [1] JÃ©rÃ´me Kunegis, Marcel Blattner, and Christine Moser. Preferential attachment in online networks: Measurement and explanations. In Proc. Web Science Conf., pages 205–214, 2013. [ .pdf ] [2] Albert-LÃ¡szlÃ³ BarabÃ¡si and RÃ©ka Albert. Emergence of scaling in random networks. Science, 286(5439):509–512, 1999. [3] Paul ErdÅs and AlfrÃ©d RÃ©nyi. On random graphs I. Publ. Math. Debrecen, 6:290–297, 1959. [4] Steffen Dereich and Peter MÃ¶rters. Random networks with sublinear preferential attachment: Degree evolutions. Electrical J. of Probability, 14:1222–1267, 2009. [5] Anna Rudas, BÃ¡lint TÃ³th, and Benedek ValkÃ³. Random trees and general branching processes. Random Struct. Algorithms, 31(2):186–202, 2007. [6] Paul L. Krapivsky and Dmitri Krioukov. Scale-free networks as preasymptotic regimes of superlinear preferential attachment. Phys. Rev. E, 78:026114, 2008. [7] S. N. Dorogovtsev and J. F. F. Mendes. Evolution of networks. Adv. Phys., 51:1079–1187, 2002.

 Code Name Category F. W. M. $$n$$ $$m$$ $$\beta$$ HUr Hudong related ⬤ Hyperlink 2,452,715 18,854,882 HUi Hudong internal ⬤ Hyperlink 1,984,484 14,869,484 BAr Baidu related ⬤ Hyperlink 415,641 3,284,387 BAi Baidu internal ⬤ Hyperlink 2,141,300 17,794,839 LK Linux kernel mailing list threads ⬤ Interaction 421,599 1,565,683 1.58109 × 10–2 (Îµ = 1.16245 × 100) Bui BibSonomy u-i ⬤ Folksonomy 779,035 2,555,080 4.40401 × 10–2 (Îµ = 1.26467 × 100) Bti BibSonomy t-i ⬤ Folksonomy 1,176,793 2,555,080 4.90021 × 10–2 (Îµ = 1.25201 × 100) PL Prosper loans ⬤ Interaction 89,269 3,394,979 1.14084 × 10–1 (Îµ = 2.77921 × 100) Cti CiteULike t-i ⬤ Folksonomy 1,038,323 2,411,819 1.35116 × 10–1 (Îµ = 1.69413 × 100) CS CiteSeer ⬤ Citation 384,413 1,751,463 GC Google.com internal ⬤ Hyperlink 15,763 171,206 nen Wikinews (en) ⬤ Authorship 184,536 901,416 1.80022 × 10–1 (Îµ = 1.74212 × 100) nfr Wikinews (fr) ⬤ Authorship 27,954 193,618 3.69423 × 10–1 (Îµ = 3.42547 × 100) bfr Wikibooks (fr) ⬤ Authorship 33,881 201,727 5.38978 × 10–1 (Îµ = 2.73999 × 100) mde Wiktionary (de) ⬤ Authorship 157,806 1,229,501 5.50364 × 10–1 (Îµ = 3.66671 × 100) ben Wikibooks (en) ⬤ Authorship 200,108 1,164,576 5.61451 × 10–1 (Îµ = 2.98704 × 100) YT YouTube ⬤ Social 3,223,589 9,375,374 5.65382 × 10–1 (Îµ = 2.61153 × 100) LY Youtube links ⬤ Social 1,138,499 4,942,297 YG YouTube ⬤ Affiliation 218,563 293,360 Tfr Wikipedia talk, French ⬤ Communication 1,420,367 4,641,928 4.10606 × 10–1 (Îµ = 1.66958 × 100) Tit Wikipedia talk, Italian ⬤ Communication 863,846 3,067,680 4.68929 × 10–1 (Îµ = 1.57189 × 100) Ten Wikipedia talk, English ⬤ Communication 2,987,535 24,981,163 5.70522 × 10–1 (Îµ = 2.07098 × 100) Tde Wikipedia talk, German ⬤ Communication 519,403 6,729,794 6.04064 × 10–1 (Îµ = 2.39662 × 100) But BibSonomy u-t ⬤ Folksonomy 216,261 2,555,080 6.09356 × 10–1 (Îµ = 2.52773 × 100) YS Yahoo songs ⬤ Rating 2,626,941 256,804,235 7.38143 × 10–1 (Îµ = 3.77882 × 100) WO WordNet ⬤ Lexical 146,005 656,999 PW Prosper.com ⬤ Interaction 25,697 35,377 PS Prosper.com ⬤ Affiliation 14,177 21,017 Vut vi.sualize.us u-t ⬤ Folksonomy 116,279 2,298,816 Vui vi.sualize.us u-i ⬤ Folksonomy 529,646 2,298,816 Vti vi.sualize.us t-i ⬤ Folksonomy 659,472 2,298,816 Sh Hamsterster full ⬤ Social 2,426 16,631 Shf Hamsterster friendships ⬤ Social 1,858 12,534 Sd Dogster ⬤ Social 426,820 8,546,581 Sc Catster ⬤ Social 149,700 5,449,275 Scd Catster/Dogster ⬤ Social 623,766 15,699,276 PC US patents ⬤ Citation 3,774,768 16,518,947 GN Gnutella ⬤ Computer 62,586 147,892 OR Orkut ⬤ Social 3,072,441 117,184,899 OG Orkut ⬤ Affiliation 14,297,249 327,037,487 AF US airports ⬤ Infrastructure 1,574 28,236 UC UC Irvine messages ⬤ Communication 1,899 59,835 8.72694 × 10–1 (Îµ = 5.86284 × 100) UF UC Irvine forum ⬤ Interaction 2,320 33,720 1.30513 × 100 (Îµ = 6.05656 × 100) SW Southern women 1 ⬤ Interaction 50 89 UG US power grid ⬤ Infrastructure 4,941 6,594 OF OpenFlights ⬤ Infrastructure 2,939 30,501 AC arXiv cond-mat ⬤ Authorship 55,467 58,595 OF OpenFlights ⬤ Infrastructure 3,425 67,663 Tar Wikipedia talk, Arabic ⬤ Communication 1,095,799 1,913,103 1.83031 × 10–1 (Îµ = 1.86856 × 100) Tnl Wikipedia talk, Dutch ⬤ Communication 225,749 1,554,699 5.30214 × 10–1 (Îµ = 1.97699 × 100) WK Wikipedia, English ⬤ Communication 2,394,385 5,021,410 W2 WikiSigned ⬤ OnlineContact 138,592 740,397 Wru Wikipedia links, ru ⬤ Hyperlink 2,853,118 82,056,101 Wpt Wikipedia links, pt ⬤ Hyperlink 1,603,222 49,021,409 Wpl Wikipedia links, pl ⬤ Hyperlink 1,529,135 57,489,447 Wja Wikipedia links, ja ⬤ Hyperlink 1,610,638 71,055,717 Wit Wikipedia links, it ⬤ Hyperlink 1,865,965 91,555,008 Wfr Wikipedia links, fr ⬤ Hyperlink 3,023,165 102,382,410 Wen Wikipedia links, en ⬤ Hyperlink 12,150,976 378,142,420 Wde Wikipedia links, de ⬤ Hyperlink 3,225,565 81,626,917 qen Wikiquote (en) ⬤ Authorship 137,970 549,210 8.18022 × 10–1 (Îµ = 2.61096 × 100) men Wiktionary (en) ⬤ Authorship 2,163,240 8,998,641 8.83404 × 10–1 (Îµ = 3.45231 × 100) CH Chess ⬤ Interaction 7,301 65,053 9.32937 × 10–1 (Îµ = 4.53583 × 100) WP Wikipedia, English ⬤ Hyperlink 1,870,709 39,953,145 1.01057 × 100 (Îµ = 4.08479 × 100) WD Wikipedia Threads (de) ⬤ Communication 91,340 2,435,731 WC Wikipedia (en) ⬤ Feature 3,889,933 3,795,796 CO Wikipedia conflict ⬤ OnlineContact 118,100 2,917,785 4.65020 × 10–1 (Îµ = 4.18165 × 100) WU WebUni Magdeburg ⬤ Text 212,552 3,869,707 6.42062 × 10–1 (Îµ = 2.05982 × 100) SF Stanford ⬤ Hyperlink 281,903 2,312,497 ND Notre Dame ⬤ Hyperlink 325,729 1,497,134 GO Google ⬤ Hyperlink 875,713 5,105,039 BS Berkeley/Stanford ⬤ Hyperlink 685,230 7,600,595 UL Unicode languages ⬤ Feature 1,122 1,255 ZA Zachary karate club ⬤ HumanSocial 34 78 HT Highland tribes ⬤ HumanSocial 16 58 TF Twitter (MPI) ⬤ Social 52,579,682 1,963,263,821 TW Twitter (WWW) ⬤ Social 41,652,230 1,468,365,182 WT TREC WT10g ⬤ Hyperlink 1,601,787 8,063,026 WT Web trackers ⬤ Hyperlink 68,087,704 140,613,762 TO Internet topology ⬤ Computer 34,761 171,403 1.27898 × 100 (Îµ = 3.37529 × 100) CR Chicago ⬤ Infrastructure 1,467 1,298 Dj JUNG dependency ⬤ Software 6,120 138,706 DJ JDK dependency ⬤ Software 6,434 150,985 ET Euroroad ⬤ Infrastructure 1,174 1,417 CC Cora citation ⬤ Citation 23,166 91,500 SO Stack Overflow ⬤ Rating 1,187,072 1,301,942 7.10220 × 10–1 (Îµ = 4.25938 × 100) PL Pokec ⬤ Social 1,632,803 30,622,564 LJ LiveJournal ⬤ Social 4,847,571 68,475,391 If Infectious ⬤ HumanContact 410 17,298 9.20776 × 10–2 (Îµ = 7.21049 × 100) HY Hypertext 2009 ⬤ HumanContact 113 20,818 1.28539 × 100 (Îµ = 9.54577 × 100) ES Epinions ⬤ Social 75,879 508,837 SZ Slashdot Zoo ⬤ Social 79,120 515,397 SD Slashdot ⬤ Communication 51,083 140,778 4.45834 × 10–1 (Îµ = 2.35086 × 100) R1 Texas ⬤ Infrastructure 1,379,917 1,921,660 RD Pennsylvania ⬤ Infrastructure 1,088,092 1,541,898 RO California ⬤ Infrastructure 1,965,206 2,766,607 RE Reuters ⬤ Text 1,846,441 96,903,520 RC Reactome ⬤ Metabolic 6,327 147,547 Tzh Wikipedia talk, Chinese ⬤ Communication 1,219,241 2,284,546 4.50239 × 10–1 (Îµ = 1.87417 × 100) Tes Wikipedia talk, Spanish ⬤ Communication 497,446 2,702,879 4.51376 × 10–1 (Îµ = 1.95522 × 100) Tpt Wikipedia talk, Portuguese ⬤ Communication 541,355 2,424,962 Wti Twitter t-i ⬤ Folksonomy 1,773,193 2,635,885 1.41236 × 10–1 (Îµ = 1.34047 × 100) Wui Twitter u-i ⬤ Folksonomy 9,618,743 12,656,613 2.80744 × 10–1 (Îµ = 1.29996 × 100) DG Digg ⬤ Communication 30,398 87,627 5.02360 × 10–1 (Îµ = 3.08696 × 100) Tru Wikipedia talk, Russian ⬤ Communication 457,017 2,282,055 5.77545 × 10–1 (Îµ = 1.80742 × 100) M3 MovieLens 10M ⬤ Rating 150,433 10,000,054 7.45097 × 10–1 (Îµ = 3.25646 × 100) Mut MovieLens u-t ⬤ Folksonomy 24,546 95,580 7.58827 × 10–1 (Îµ = 2.82191 × 100) Lk Linux kernel mailing list replies ⬤ Communication 63,399 1,096,440 7.66088 × 10–1 (Îµ = 3.50603 × 100) Mui MovieLens u-i ⬤ Folksonomy 15,619 95,580 8.39461 × 10–1 (Îµ = 4.11304 × 100) NX Netflix ⬤ Rating 978,148 100,480,507 8.73160 × 10–1 (Îµ = 2.85036 × 100) M2 MovieLens 1M ⬤ Rating 15,786 1,000,209 9.63746 × 10–1 (Îµ = 2.83098 × 100) Wa Twitter ⬤ OnlineContact 2,919,613 12,887,063 1.03792 × 100 (Îµ = 2.99086 × 100) Mti MovieLens t-i ⬤ Folksonomy 40,657 95,580 1.09157 × 100 (Îµ = 4.09860 × 100) RA Manufacturing emails ⬤ Communication 167 82,927 1.22839 × 100 (Îµ = 2.49547 × 100) Ws Twitter (ICWSM) ⬤ Social 465,017 834,797 Wut Twitter u-t ⬤ Folksonomy 880,846 4,664,605 8.43049 × 10–1 (Îµ = 2.47827 × 100) LM Livemocha ⬤ Social 104,103 2,193,083 Lj LiveJournal links ⬤ Social 5,204,176 49,174,464 M1 MovieLens 100k ⬤ Rating 3,568 100,000 9.31708 × 10–1 (Îµ = 3.12379 × 100) MZ Zebra ⬤ Animal 27 111 Mv Dutch college ⬤ HumanSocial 32 3,062 6.12189 × 10–1 (Îµ = 1.11198 × 100) Mt Train bombing ⬤ HumanContact 64 243 MT Taro exchange ⬤ HumanSocial 22 78 Mc Sheep ⬤ Animal 28 250 MX Seventh graders ⬤ HumanSocial 29 376 Ms Sampson ⬤ HumanSocial 18 189 MR Rhesus ⬤ Animal 16 111 Mp Protein ⬤ Metabolic 1,870 2,277 MO Residence hall ⬤ HumanSocial 217 2,672 MN Bible ⬤ Lexical 1,773 16,401 MQ Macaques ⬤ Animal 62 1,187 Ml Les Misérables ⬤ Misc 77 254 MK Kangaroo ⬤ Animal 17 91 MI Physicians ⬤ HumanSocial 241 1,098 MH Highschool ⬤ HumanSocial 70 366 Mh Hens ⬤ Animal 32 496 ME Adolescent health ⬤ HumanSocial 2,539 12,969 MC Crime ⬤ Interaction 2,209 1,476 MA Cattle ⬤ Animal 28 217 Mg Blogs ⬤ Hyperlink 1,224 19,025 MB Bison ⬤ Animal 26 314 MW Windsurfers ⬤ HumanContact 43 336 RM Reality Mining ⬤ HumanContact 96 1,086,404 1.53150 × 100 (Îµ = 3.23015 × 101) MV Human protein (Vidal) ⬤ Metabolic 3,133 6,726 MS Human protein (Stelzl) ⬤ Metabolic 1,706 6,207 MP PDZBase ⬤ Metabolic 212 244 ML Little Rock Lake ⬤ Trophic 183 2,494 MF Human protein (Figeys) ⬤ Metabolic 2,239 6,452 TC Air traffic control ⬤ Infrastructure 1,226 2,615 GW Gowalla ⬤ Social 196,591 950,327 BK Brightkite ⬤ Social 58,228 214,078 LG LiveJournal ⬤ Affiliation 13,891,479 112,307,385 LX Linux ⬤ Software 30,837 213,954 Ud Wikipedia, nl (dynamic) ⬤ Hyperlink 1,039,252 20,070,561 8.71927 × 10–1 (Îµ = 4.70301 × 100) Up Wikipedia, pl (dynamic) ⬤ Hyperlink 1,033,050 25,026,208 9.63288 × 10–1 (Îµ = 4.38802 × 100) Ui Wikipedia, it (dynamic) ⬤ Hyperlink 1,204,009 34,826,283 9.98447 × 10–1 (Îµ = 4.78728 × 100) Ug Wikipedia, de (dynamic) ⬤ Hyperlink 2,166,669 86,337,879 1.00066 × 100 (Îµ = 4.78850 × 100) Uf Wikipedia, fr (dynamic) ⬤ Hyperlink 2,212,682 59,008,831 1.01639 × 100 (Îµ = 4.06262 × 100) Us Wikipedia, simple en (dynamic) ⬤ Hyperlink 100,312 1,627,472 1.06373 × 100 (Îµ = 3.96137 × 100) LI Libimseti.cz ⬤ Social 220,970 17,359,346 EL Wikipedia elections ⬤ OnlineContact 7,118 103,675 1.01217 × 10–2 (Îµ = 2.74238 × 100) Ol Facebook (WOSN) ⬤ Social 63,731 817,035 5.16567 × 10–1 (Îµ = 6.09425 × 100) SX Sexual escorts ⬤ Rating 26,836 50,632 7.94200 × 10–1 (Îµ = 2.83835 × 100) Ow Facebook ⬤ Communication 46,952 876,993 8.72097 × 10–1 (Îµ = 5.00818 × 100) Fc Filmtipset ⬤ Interaction 104,890 1,266,753 1.00448 × 100 (Îµ = 3.78276 × 100) Lb Last.fm band ⬤ Interaction 176,061 19,150,868 1.10444 × 100 (Îµ = 5.17573 × 100) Ls Last.fm song ⬤ Interaction 1,086,604 19,150,868 1.15370 × 100 (Îµ = 3.95966 × 100) YD Yahoo advertisers ⬤ Lexical 653,260 2,931,708 J2 Jester 150 ⬤ Rating 101,524 1,728,847 J1 Jester 100 ⬤ Rating 146,942 4,136,360 HY Hyves ⬤ Social 1,402,673 2,777,419 TR TREC (disks 4–5) ⬤ Text 2,285,379 151,632,178 R2 Reuters-21578 ⬤ Text 81,791 1,464,182 EX Wikipedia (en) ⬤ Text 279,519 7,846,807 GH Github ⬤ Authorship 233,905 440,237 FR Friendster ⬤ Social 68,349,466 2,586,147,869 FW Florida ecosystem wet ⬤ Trophic 128 2,106 FD Florida ecosystem dry ⬤ Trophic 128 2,137 FO FOLDOC ⬤ Hyperlink 13,356 125,207 FX Flixster ⬤ Social 2,523,386 7,918,801 LF Flickr links ⬤ Social 1,715,255 15,551,250 FL Flickr ⬤ Social 2,302,925 33,140,017 9.60503 × 10–1 (Îµ = 2.78687 × 100) FG Flickr ⬤ Affiliation 895,589 8,545,307 FI Flickr ⬤ Misc 105,938 2,316,948 Fr Filmtipset ⬤ Rating 225,153 19,554,219 9.68650 × 10–1 (Îµ = 4.12458 × 100) Ff Filmtipset ⬤ Social 39,199 87,415 TL Twitter lists ⬤ Social 23,370 33,101 GP Google+ ⬤ Social 23,628 39,242 EF Facebook (NIPS) ⬤ Social 2,888 2,981 ER Epinions ⬤ Rating 996,744 13,668,320 3.42512 × 10–1 (Îµ = 2.81777 × 100) EP Epinions ⬤ Social 131,828 841,372 5.91685 × 10–1 (Îµ = 2.87147 × 100) EN Enron ⬤ Communication 87,273 1,148,072 7.85895 × 10–1 (Îµ = 3.81678 × 100) EU EU institution ⬤ Communication 265,214 420,045 it Wikipedia (it) ⬤ Authorship 2,531,261 26,241,217 1.08684 × 100 (Îµ = 3.51966 × 100) en Wikipedia (en) ⬤ Authorship 29,143,573 266,769,613 1.10598 × 100 (Îµ = 3.36988 × 100) fr Wikipedia (fr) ⬤ Authorship 4,598,826 46,168,355 1.11031 × 100 (Îµ = 3.38954 × 100) es Wikipedia (es) ⬤ Authorship 3,623,742 27,011,506 1.17118 × 100 (Îµ = 3.05932 × 100) de Wikipedia (de) ⬤ Authorship 4,046,832 57,323,775 1.17251 × 100 (Îµ = 3.63093 × 100) mfr Wiktionary (fr) ⬤ Authorship 1,917,281 7,399,298 1.20956 × 100 (Îµ = 3.54192 × 100) EA Edinburgh Associative Thesaurus ⬤ Lexical 23,132 511,764 DB Douban ⬤ Social 154,908 327,162 DO Dolphins ⬤ Animal 62 159 DNc DNC emails ⬤ Communication 2,029 39,264 1.18369 × 100 (Îµ = 3.12748 × 100) DN DNC emails co-recipients ⬤ OnlineContact 2,029 136,602 Dt Discogs artist–style ⬤ Feature 3,236,269 24,085,580 Ds Discogs label–style ⬤ Feature 487,911 5,255,950 Dr Discogs label–genre ⬤ Feature 541,557 4,147,665 Da Discogs artist–genre ⬤ Feature 3,509,661 19,033,891 Dl Discogs ⬤ Affiliation 3,780,417 14,414,659 Cui CiteULike u-i ⬤ Folksonomy 777,199 2,411,819 9.00989 × 10–2 (Îµ = 1.55255 × 100) DF Digg friends ⬤ Social 279,630 1,731,653 6.54108 × 10–1 (Îµ = 2.49387 × 100) Dui Delicious ui ⬤ Folksonomy 35,444,383 301,186,579 6.87183 × 10–1 (Îµ = 2.68027 × 100) Dti Delicious ti ⬤ Folksonomy 42,801,712 301,183,605 7.91114 × 10–1 (Îµ = 2.96128 × 100) DBT TV Tropes ⬤ Feature 216,508 3,232,134 WR Writers ⬤ Authorship 224,925 144,340 TM Teams ⬤ Affiliation 1,836,793 1,366,466 ST Movies ⬤ Feature 233,283 281,396 RL Record labels ⬤ Affiliation 355,095 233,286 PR Producers ⬤ Authorship 236,510 207,268 OC Occupation ⬤ Affiliation 356,884 250,945 LO Location ⬤ Feature 397,589 293,697 DL Wikipedia, English ⬤ Hyperlink 18,268,992 172,183,984 GE DBpedia genre ⬤ Feature 525,651 463,497 CN Countries ⬤ Affiliation 1,182,526 637,134 DB DBpedia ⬤ Misc 3,966,924 13,820,853 Pc DBLP ⬤ Coauthorship 1,314,050 18,986,618 1.07566 × 100 (Îµ = 6.26927 × 100) Pi DBLP ⬤ Citation 12,591 49,743 Pa DBLP ⬤ Authorship 6,851,776 8,649,016 CU Contiguous USA ⬤ Infrastructure 49 107 HA Haggle ⬤ HumanContact 274 28,244 1.19421 × 100 (Îµ = 7.83017 × 100) CY Youtube friendship ⬤ Social 1,134,890 2,987,624 CD DBLP co-authorship ⬤ Coauthorship 317,080 1,049,866 CA Amazon (MDS) ⬤ Misc 334,863 925,872 THc arXiv hep-th ⬤ Citation 27,770 352,807 PHc arXiv hep-ph ⬤ Citation 34,546 421,578 Cut CiteULike u-t ⬤ Folksonomy 198,707 2,411,819 8.13852 × 10–1 (Îµ = 2.85268 × 100) Dut Delicious ut ⬤ Folksonomy 6,178,261 301,186,579 1.08441 × 100 (Îµ = 3.27077 × 100) DV Digg votes ⬤ Rating 282,371 3,018,197 1.23849 × 100 (Îµ = 5.60307 × 100) TH arXiv hep-th ⬤ Coauthorship 22,908 2,673,133 PH arXiv hep-ph ⬤ Coauthorship 28,093 4,596,803 AP arXiv astro-ph ⬤ Coauthorship 18,771 198,050 SC South African Companies ⬤ Affiliation 17 13 Ar American Revolution ⬤ Affiliation 277 160 BC Corporate Leadership ⬤ Affiliation 64 99 Sw Southern women 2 ⬤ Interaction 15 14 BM Club membership ⬤ Affiliation 65 95 Bx BookCrossing (ratings) ⬤ Rating 341,559 433,652 BX BookCrossing (implicit) ⬤ Rating 551,079 1,149,739 SK Skitter ⬤ Computer 1,696,415 11,095,298 IN CAIDA ⬤ Computer 26,475 53,381 AS Route views ⬤ Computer 6,474 13,895 PG Pretty Good Privacy ⬤ OnlineContact 10,680 24,316 PM Caenorhabditis elegans ⬤ Metabolic 453 4,596 JZ Jazz musicians ⬤ HumanSocial 198 2,742 A@ U. Rovira i Virgili ⬤ Communication 1,133 5,451 AR Amazon ratings ⬤ Rating 5,523,029 5,838,041 6.04802 × 10–1 (Îµ = 2.52309 × 100) Am Amazon (TWEB) ⬤ Misc 403,394 3,387,388 AD Advogato ⬤ Social 6,541 51,127 AN David Copperfield ⬤ Lexical 112 425 AM Actor movies ⬤ Affiliation 639,286 1,470,404 CL Actor collaborations ⬤ Misc 382,219 33,115,812