KONECT
KONECT > Statistics > Volume

## Volume

The volume of a network equals the number of edges and is defined as \begin{align} m &= |E|. \end{align} The volume can be expressed in terms of the adjacency or biadjacency matrix of the underlying unweighted graph as \begin{align} m &= \left\{ \begin{array}{ll} \frac 1 2 \| \mathbf A[\bar G] \|_{\mathrm F} ^2 & \text{when $$G$$ is undirected} \\ \| \mathbf A[\bar G] \| _{\mathrm F} ^2 & \text{when $$G$$ is directed} \\ \| \mathbf B[\bar G] \| _{\mathrm F} ^2 & \text{when $$G$$ is bipartite} \end{array} \right. \end{align} The number of edges in network is often considered a better measure of the size of a network than the number vertices, since a vertex unconnected to any other vertices may often be ignored. On the practical side, the volume is also a much better indicator of the amount of memory needed to represent a network. We will also make use of the number of edges without couting multiple edges. We will call this the unique volume of the graph. \begin{align} \bar{\bar m} &= m[\bar{\bar{G}}] \end{align}

 Code Name Category F. W. M. $$n$$ $$m$$ $$\bar{\bar m}$$ SC South African Companies ⬤ Affiliation 17 13 Sw Southern women 2 ⬤ Interaction 15 14 HT Highland tribes ⬤ HumanSocial 16 58 MT Taro exchange ⬤ HumanSocial 22 78 ZA Zachary karate club ⬤ HumanSocial 34 78 SW Southern women 1 ⬤ Interaction 50 89 MK Kangaroo ⬤ Animal 17 91 BM Club membership ⬤ Affiliation 65 95 BC Corporate Leadership ⬤ Affiliation 64 99 CU Contiguous USA ⬤ Infrastructure 49 107 MR Rhesus ⬤ Animal 16 111 MZ Zebra ⬤ Animal 27 111 DO Dolphins ⬤ Animal 62 159 Ar American Revolution ⬤ Affiliation 277 160 Ms Sampson ⬤ HumanSocial 18 189 MA Cattle ⬤ Animal 28 217 Mt Train bombing ⬤ HumanContact 64 243 MP PDZBase ⬤ Metabolic 212 244 Mc Sheep ⬤ Animal 28 250 Ml Les Misérables ⬤ Misc 77 254 MB Bison ⬤ Animal 26 314 MW Windsurfers ⬤ HumanContact 43 336 MH Highschool ⬤ HumanSocial 70 366 MX Seventh graders ⬤ HumanSocial 29 376 AN David Copperfield ⬤ Lexical 112 425 Mh Hens ⬤ Animal 32 496 MI Physicians ⬤ HumanSocial 241 1,098 MQ Macaques ⬤ Animal 62 1,187 UL Unicode languages ⬤ Feature 1,122 1,255 CR Chicago ⬤ Infrastructure 1,467 1,298 ET Euroroad ⬤ Infrastructure 1,174 1,417 MC Crime ⬤ Interaction 2,209 1,476 FW Florida ecosystem wet ⬤ Trophic 128 2,106 FD Florida ecosystem dry ⬤ Trophic 128 2,137 Mp Protein ⬤ Metabolic 1,870 2,277 ML Little Rock Lake ⬤ Trophic 183 2,494 TC Air traffic control ⬤ Infrastructure 1,226 2,615 MO Residence hall ⬤ HumanSocial 217 2,672 JZ Jazz musicians ⬤ HumanSocial 198 2,742 EF Facebook (NIPS) ⬤ Social 2,888 2,981 Mv Dutch college ⬤ HumanSocial 32 3,062 354 PM Caenorhabditis elegans ⬤ Metabolic 453 4,596 2,040 A@ U. Rovira i Virgili ⬤ Communication 1,133 5,451 MS Human protein (Stelzl) ⬤ Metabolic 1,706 6,207 MF Human protein (Figeys) ⬤ Metabolic 2,239 6,452 UG US power grid ⬤ Infrastructure 4,941 6,594 MV Human protein (Vidal) ⬤ Metabolic 3,133 6,726 Shf Hamsterster friendships ⬤ Social 1,858 12,534 ME Adolescent health ⬤ HumanSocial 2,539 12,969 AS Route views ⬤ Computer 6,474 13,895 MN Bible ⬤ Lexical 1,773 16,401 9,131 Sh Hamsterster full ⬤ Social 2,426 16,631 If Infectious ⬤ HumanContact 410 17,298 2,765 Mg Blogs ⬤ Hyperlink 1,224 19,025 HY Hypertext 2009 ⬤ HumanContact 113 20,818 2,196 PS Prosper.com ⬤ Affiliation 14,177 21,017 PG Pretty Good Privacy ⬤ OnlineContact 10,680 24,316 AF US airports ⬤ Infrastructure 1,574 28,236 HA Haggle ⬤ HumanContact 274 28,244 2,899 OF OpenFlights ⬤ Infrastructure 2,939 30,501 TL Twitter lists ⬤ Social 23,370 33,101 UF UC Irvine forum ⬤ Interaction 2,320 33,720 7,089 PW Prosper.com ⬤ Interaction 25,697 35,377 GP Google+ ⬤ Social 23,628 39,242 DNc DNC emails ⬤ Communication 2,029 39,264 5,598 Pi DBLP ⬤ Citation 12,591 49,743 SX Sexual escorts ⬤ Rating 26,836 50,632 35,051 AD Advogato ⬤ Social 6,541 51,127 IN CAIDA ⬤ Computer 26,475 53,381 AC arXiv cond-mat ⬤ Authorship 55,467 58,595 UC UC Irvine messages ⬤ Communication 1,899 59,835 20,296 CH Chess ⬤ Interaction 7,301 65,053 34,564 OF OpenFlights ⬤ Infrastructure 3,425 67,663 37,595 RA Manufacturing emails ⬤ Communication 167 82,927 5,784 Ff Filmtipset ⬤ Social 39,199 87,415 DG Digg ⬤ Communication 30,398 87,627 86,404 CC Cora citation ⬤ Citation 23,166 91,500 Mti MovieLens t-i ⬤ Folksonomy 40,657 95,580 71,154 Mui MovieLens u-i ⬤ Folksonomy 15,619 95,580 55,484 Mut MovieLens u-t ⬤ Folksonomy 24,546 95,580 43,760 M1 MovieLens 100k ⬤ Rating 3,568 100,000 EL Wikipedia elections ⬤ OnlineContact 7,118 103,675 FO FOLDOC ⬤ Hyperlink 13,356 125,207 120,238 DN DNC emails co-recipients ⬤ OnlineContact 2,029 136,602 12,085 Dj JUNG dependency ⬤ Software 6,120 138,706 50,535 SD Slashdot ⬤ Communication 51,083 140,778 131,175 WR Writers ⬤ Authorship 224,925 144,340 RC Reactome ⬤ Metabolic 6,327 147,547 GN Gnutella ⬤ Computer 62,586 147,892 DJ JDK dependency ⬤ Software 6,434 150,985 53,892 GC Google.com internal ⬤ Hyperlink 15,763 171,206 TO Internet topology ⬤ Computer 34,761 171,403 114,496 nfr Wikinews (fr) ⬤ Authorship 27,954 193,618 68,703 AP arXiv astro-ph ⬤ Coauthorship 18,771 198,050 bfr Wikibooks (fr) ⬤ Authorship 33,881 201,727 67,613 PR Producers ⬤ Authorship 236,510 207,268 LX Linux ⬤ Software 30,837 213,954 BK Brightkite ⬤ Social 58,228 214,078 RL Record labels ⬤ Affiliation 355,095 233,286 OC Occupation ⬤ Affiliation 356,884 250,945 ST Movies ⬤ Feature 233,283 281,396 YG YouTube ⬤ Affiliation 218,563 293,360 LO Location ⬤ Feature 397,589 293,697 DB Douban ⬤ Social 154,908 327,162 THc arXiv hep-th ⬤ Citation 27,770 352,807 EU EU institution ⬤ Communication 265,214 420,045 PHc arXiv hep-ph ⬤ Citation 34,546 421,578 Bx BookCrossing (ratings) ⬤ Rating 341,559 433,652 GH Github ⬤ Authorship 233,905 440,237 GE DBpedia genre ⬤ Feature 525,651 463,497 ES Epinions ⬤ Social 75,879 508,837 EA Edinburgh Associative Thesaurus ⬤ Lexical 23,132 511,764 312,310 SZ Slashdot Zoo ⬤ Social 79,120 515,397 qen Wikiquote (en) ⬤ Authorship 137,970 549,210 238,714 CN Countries ⬤ Affiliation 1,182,526 637,134 624,402 WO WordNet ⬤ Lexical 146,005 656,999 W2 WikiSigned ⬤ OnlineContact 138,592 740,397 Ol Facebook (WOSN) ⬤ Social 63,731 817,035 Ws Twitter (ICWSM) ⬤ Social 465,017 834,797 EP Epinions ⬤ Social 131,828 841,372 Ow Facebook ⬤ Communication 46,952 876,993 274,086 nen Wikinews (en) ⬤ Authorship 184,536 901,416 354,157 CA Amazon (MDS) ⬤ Misc 334,863 925,872 GW Gowalla ⬤ Social 196,591 950,327 M2 MovieLens 1M ⬤ Rating 15,786 1,000,209 CD DBLP co-authorship ⬤ Coauthorship 317,080 1,049,866 RM Reality Mining ⬤ HumanContact 96 1,086,404 2,539 Lk Linux kernel mailing list replies ⬤ Communication 63,399 1,096,440 242,976 EN Enron ⬤ Communication 87,273 1,148,072 321,918 BX BookCrossing (implicit) ⬤ Rating 551,079 1,149,739 ben Wikibooks (en) ⬤ Authorship 200,108 1,164,576 387,283 mde Wiktionary (de) ⬤ Authorship 157,806 1,229,501 687,214 Fc Filmtipset ⬤ Interaction 104,890 1,266,753 1,204,831 SO Stack Overflow ⬤ Rating 1,187,072 1,301,942 TM Teams ⬤ Affiliation 1,836,793 1,366,466 R2 Reuters-21578 ⬤ Text 81,791 1,464,182 978,446 AM Actor movies ⬤ Affiliation 639,286 1,470,404 ND Notre Dame ⬤ Hyperlink 325,729 1,497,134 RD Pennsylvania ⬤ Infrastructure 1,088,092 1,541,898 Tnl Wikipedia talk, Dutch ⬤ Communication 225,749 1,554,699 565,477 LK Linux kernel mailing list threads ⬤ Interaction 421,599 1,565,683 599,858 Us Wikipedia, simple en (dynamic) ⬤ Hyperlink 100,312 1,627,472 746,086 J2 Jester 150 ⬤ Rating 101,524 1,728,847 DF Digg friends ⬤ Social 279,630 1,731,653 CS CiteSeer ⬤ Citation 384,413 1,751,463 Tar Wikipedia talk, Arabic ⬤ Communication 1,095,799 1,913,103 1,564,598 R1 Texas ⬤ Infrastructure 1,379,917 1,921,660 LM Livemocha ⬤ Social 104,103 2,193,083 Tru Wikipedia talk, Russian ⬤ Communication 457,017 2,282,055 919,790 Tzh Wikipedia talk, Chinese ⬤ Communication 1,219,241 2,284,546 1,735,118 Vti vi.sualize.us t-i ⬤ Folksonomy 659,472 2,298,816 1,800,330 Vui vi.sualize.us u-i ⬤ Folksonomy 529,646 2,298,816 997,840 Vut vi.sualize.us u-t ⬤ Folksonomy 116,279 2,298,816 449,503 SF Stanford ⬤ Hyperlink 281,903 2,312,497 FI Flickr ⬤ Misc 105,938 2,316,948 Cti CiteULike t-i ⬤ Folksonomy 1,038,323 2,411,819 2,338,554 Cui CiteULike u-i ⬤ Folksonomy 777,199 2,411,819 842,426 Cut CiteULike u-t ⬤ Folksonomy 198,707 2,411,819 538,761 Tpt Wikipedia talk, Portuguese ⬤ Communication 541,355 2,424,962 1,463,308 WD Wikipedia Threads (de) ⬤ Communication 91,340 2,435,731 987,092 Bti BibSonomy t-i ⬤ Folksonomy 1,176,793 2,555,080 2,499,057 Bui BibSonomy u-i ⬤ Folksonomy 779,035 2,555,080 801,784 But BibSonomy u-t ⬤ Folksonomy 216,261 2,555,080 453,987 Wti Twitter t-i ⬤ Folksonomy 1,773,193 2,635,885 1,996,158 TH arXiv hep-th ⬤ Coauthorship 22,908 2,673,133 2,444,798 Tes Wikipedia talk, Spanish ⬤ Communication 497,446 2,702,879 1,250,097 RO California ⬤ Infrastructure 1,965,206 2,766,607 HY Hyves ⬤ Social 1,402,673 2,777,419 CO Wikipedia conflict ⬤ OnlineContact 118,100 2,917,785 2,014,074 YD Yahoo advertisers ⬤ Lexical 653,260 2,931,708 CY Youtube friendship ⬤ Social 1,134,890 2,987,624 DV Digg votes ⬤ Rating 282,371 3,018,197 3,010,898 Tit Wikipedia talk, Italian ⬤ Communication 863,846 3,067,680 1,661,453 DBT TV Tropes ⬤ Feature 216,508 3,232,134 BAr Baidu related ⬤ Hyperlink 415,641 3,284,387 3,284,335 Am Amazon (TWEB) ⬤ Misc 403,394 3,387,388 PL Prosper loans ⬤ Interaction 89,269 3,394,979 3,330,225 WC Wikipedia (en) ⬤ Feature 3,889,933 3,795,796 WU WebUni Magdeburg ⬤ Text 212,552 3,869,707 1,948,004 J1 Jester 100 ⬤ Rating 146,942 4,136,360 Dr Discogs label–genre ⬤ Feature 541,557 4,147,665 481,661 PH arXiv hep-ph ⬤ Coauthorship 28,093 4,596,803 3,148,447 Tfr Wikipedia talk, French ⬤ Communication 1,420,367 4,641,928 2,471,501 Wut Twitter u-t ⬤ Folksonomy 880,846 4,664,605 1,890,661 LY Youtube links ⬤ Social 1,138,499 4,942,297 WK Wikipedia, English ⬤ Communication 2,394,385 5,021,410 GO Google ⬤ Hyperlink 875,713 5,105,039 Ds Discogs label–style ⬤ Feature 487,911 5,255,950 1,064,853 Sc Catster ⬤ Social 149,700 5,449,275 AR Amazon ratings ⬤ Rating 5,523,029 5,838,041 5,743,258 Tde Wikipedia talk, German ⬤ Communication 519,403 6,729,794 1,751,343 mfr Wiktionary (fr) ⬤ Authorship 1,917,281 7,399,298 4,794,894 BS Berkeley/Stanford ⬤ Hyperlink 685,230 7,600,595 EX Wikipedia (en) ⬤ Text 279,519 7,846,807 2,941,902 FX Flixster ⬤ Social 2,523,386 7,918,801 WT TREC WT10g ⬤ Hyperlink 1,601,787 8,063,026 FG Flickr ⬤ Affiliation 895,589 8,545,307 Sd Dogster ⬤ Social 426,820 8,546,581 Pa DBLP ⬤ Authorship 6,851,776 8,649,016 men Wiktionary (en) ⬤ Authorship 2,163,240 8,998,641 5,573,038 YT YouTube ⬤ Social 3,223,589 9,375,374 M3 MovieLens 10M ⬤ Rating 150,433 10,000,054 SK Skitter ⬤ Computer 1,696,415 11,095,298 Wui Twitter u-i ⬤ Folksonomy 9,618,743 12,656,613 10,214,177 Wa Twitter ⬤ OnlineContact 2,919,613 12,887,063 7,301,101 ER Epinions ⬤ Rating 996,744 13,668,320 DB DBpedia ⬤ Misc 3,966,924 13,820,853 12,860,157 Dl Discogs ⬤ Affiliation 3,780,417 14,414,659 5,302,276 HUi Hudong internal ⬤ Hyperlink 1,984,484 14,869,484 LF Flickr links ⬤ Social 1,715,255 15,551,250 Scd Catster/Dogster ⬤ Social 623,766 15,699,276 PC US patents ⬤ Citation 3,774,768 16,518,947 LI Libimseti.cz ⬤ Social 220,970 17,359,346 BAi Baidu internal ⬤ Hyperlink 2,141,300 17,794,839 17,643,697 HUr Hudong related ⬤ Hyperlink 2,452,715 18,854,882 Pc DBLP ⬤ Coauthorship 1,314,050 18,986,618 10,724,828 Da Discogs artist–genre ⬤ Feature 3,509,661 19,033,891 3,142,059 Lb Last.fm band ⬤ Interaction 176,061 19,150,868 898,062 Ls Last.fm song ⬤ Interaction 1,086,604 19,150,868 4,413,834 Fr Filmtipset ⬤ Rating 225,153 19,554,219 19,553,591 Ud Wikipedia, nl (dynamic) ⬤ Hyperlink 1,039,252 20,070,561 10,612,491 Dt Discogs artist–style ⬤ Feature 3,236,269 24,085,580 5,740,842 Ten Wikipedia talk, English ⬤ Communication 2,987,535 24,981,163 9,379,561 Up Wikipedia, pl (dynamic) ⬤ Hyperlink 1,033,050 25,026,208 12,587,086 it Wikipedia (it) ⬤ Authorship 2,531,261 26,241,217 12,644,802 es Wikipedia (es) ⬤ Authorship 3,623,742 27,011,506 13,011,547 PL Pokec ⬤ Social 1,632,803 30,622,564 CL Actor collaborations ⬤ Misc 382,219 33,115,812 30,076,166 FL Flickr ⬤ Social 2,302,925 33,140,017 Ui Wikipedia, it (dynamic) ⬤ Hyperlink 1,204,009 34,826,283 17,086,845 WP Wikipedia, English ⬤ Hyperlink 1,870,709 39,953,145 fr Wikipedia (fr) ⬤ Authorship 4,598,826 46,168,355 22,090,703 Wpt Wikipedia links, pt ⬤ Hyperlink 1,603,222 49,021,409 Lj LiveJournal links ⬤ Social 5,204,176 49,174,464 de Wikipedia (de) ⬤ Authorship 4,046,832 57,323,775 26,011,353 Wpl Wikipedia links, pl ⬤ Hyperlink 1,529,135 57,489,447 Uf Wikipedia, fr (dynamic) ⬤ Hyperlink 2,212,682 59,008,831 24,440,537 LJ LiveJournal ⬤ Social 4,847,571 68,475,391 Wja Wikipedia links, ja ⬤ Hyperlink 1,610,638 71,055,717 Wde Wikipedia links, de ⬤ Hyperlink 3,225,565 81,626,917 Wru Wikipedia links, ru ⬤ Hyperlink 2,853,118 82,056,101 Ug Wikipedia, de (dynamic) ⬤ Hyperlink 2,166,669 86,337,879 31,105,755 Wit Wikipedia links, it ⬤ Hyperlink 1,865,965 91,555,008 RE Reuters ⬤ Text 1,846,441 96,903,520 60,569,726 NX Netflix ⬤ Rating 978,148 100,480,507 Wfr Wikipedia links, fr ⬤ Hyperlink 3,023,165 102,382,410 LG LiveJournal ⬤ Affiliation 13,891,479 112,307,385 OR Orkut ⬤ Social 3,072,441 117,184,899 WT Web trackers ⬤ Hyperlink 68,087,704 140,613,762 TR TREC (disks 4–5) ⬤ Text 2,285,379 151,632,178 83,629,405 DL Wikipedia, English ⬤ Hyperlink 18,268,992 172,183,984 136,537,566 YS Yahoo songs ⬤ Rating 2,626,941 256,804,235 en Wikipedia (en) ⬤ Authorship 29,143,573 266,769,613 122,075,170 Dti Delicious ti ⬤ Folksonomy 42,801,712 301,183,605 137,240,382 Dui Delicious ui ⬤ Folksonomy 35,444,383 301,186,579 101,798,957 Dut Delicious ut ⬤ Folksonomy 6,178,261 301,186,579 81,989,133 OG Orkut ⬤ Affiliation 14,297,249 327,037,487 Wen Wikipedia links, en ⬤ Hyperlink 12,150,976 378,142,420 TW Twitter (WWW) ⬤ Social 41,652,230 1,468,365,182 TF Twitter (MPI) ⬤ Social 52,579,682 1,963,263,821 FR Friendster ⬤ Social 68,349,466 2,586,147,869