KONECT
KONECT > Statistics > LCC

## Largest connected component

Connectivity statistics measure to what extent a network is connected. Two nodes are said to be connected when they are either directly connected through an edge, or indirectly through a path of several edges. A connected component is a set of vertices all of which are connected, and unconnected to the other nodes in the network. The largest connected component in a network is usually very large and called the giant connected component. When it contains all nodes, the network is connected.

The size of the largest connected component is denoted $$N$$. \begin{align} N &= \max_{F \subseteq \mathcal C} |F| \\ \mathcal C &= \{ C \subseteq V \mid \forall u, v \in C: \exists w_1, w_2, \ldots \in V: u \sim w_1 \sim w_2 \sim \cdots \sim v \} \nonumber \end{align}

In bipartite networks, the number of left and right nodes in the largest connected components are denoted $$N_1$$ and $$N_2$$, with $$N_1 + N_2 = N$$.

The relative size of the largest connected component equals the size of the largest connected component divided by the size of the network \begin{align} N_{\mathrm{rel}} &= \frac N n. \end{align}

In directed networks, we additionally define the size of the largest strongly connected component $$N_{\mathrm s}$$. A strongly connected component is a set of vertices in a directed graph such that any node is reachable from any other node using a path following only directed edges in the forward direction. We always have $$N_{\mathrm s} \leq N$$.

 Code Name Category F. W. M. $$n$$ $$m$$ $$N$$ $$N_{rel}$$ $$N_s$$ RA Manufacturing emails ⬤ Communication 167 82,927 167 126 DN DNC emails co-recipients ⬤ OnlineContact 2,029 136,602 849 TC Air traffic control ⬤ Infrastructure 1,226 2,615 1,226 792 Shf Hamsterster friendships ⬤ Social 1,858 12,534 1,788 DNc DNC emails ⬤ Communication 2,029 39,264 1,833 520 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 DV Digg votes ⬤ Rating 282,371 3,018,197 Sh Hamsterster full ⬤ Social 2,426 16,631 2,000 MF Human protein (Figeys) ⬤ Metabolic 2,239 6,452 2,217 8 EF Facebook (NIPS) ⬤ Social 2,888 2,981 2,888 RC Reactome ⬤ Metabolic 6,327 147,547 5,973 EL Wikipedia elections ⬤ OnlineContact 7,118 103,675 7,066 1,300 AP arXiv astro-ph ⬤ Coauthorship 18,771 198,050 17,903 TL Twitter lists ⬤ Social 23,370 33,101 22,322 38 GP Google+ ⬤ Social 23,628 39,242 23,613 50 Lk Linux kernel mailing list replies ⬤ Communication 63,399 1,096,440 24,567 18,531 BK Brightkite ⬤ Social 58,228 214,078 56,739 GN Gnutella ⬤ Computer 62,586 147,892 62,561 14,149 Ol Facebook (WOSN) ⬤ Social 63,731 817,035 63,392 WD Wikipedia Threads (de) ⬤ Communication 91,340 2,435,731 89,146 52,570 PL Prosper loans ⬤ Interaction 89,269 3,394,979 89,171 3,513 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 W2 WikiSigned ⬤ OnlineContact 138,592 740,397 137,740 26,699 WO WordNet ⬤ Lexical 146,005 656,999 145,145 Sc Catster ⬤ Social 149,700 5,449,275 148,826 GW Gowalla ⬤ Social 196,591 950,327 196,591 Tnl Wikipedia talk, Dutch ⬤ Communication 225,749 1,554,699 224,185 18,598 BAr Baidu related ⬤ Hyperlink 415,641 3,284,387 372,840 208,222 Sd Dogster ⬤ Social 426,820 8,546,581 426,485 Tru Wikipedia talk, Russian ⬤ Communication 457,017 2,282,055 449,042 22,664 Tes Wikipedia talk, Spanish ⬤ Communication 497,446 2,702,879 476,465 42,144 Tde Wikipedia talk, German ⬤ Communication 519,403 6,729,794 505,468 69,121 Tpt Wikipedia talk, Portuguese ⬤ Communication 541,355 2,424,962 534,618 21,747 Scd Catster/Dogster ⬤ Social 623,766 15,699,276 601,213 Tit Wikipedia talk, Italian ⬤ Communication 863,846 3,067,680 862,214 36,356 Tar Wikipedia talk, Arabic ⬤ Communication 1,095,799 1,913,103 1,095,524 8,797 LY Youtube links ⬤ Social 1,138,499 4,942,297 1,134,885 509,245 Tzh Wikipedia talk, Chinese ⬤ Communication 1,219,241 2,284,546 1,217,365 10,831 Tfr Wikipedia talk, French ⬤ Communication 1,420,367 4,641,928 1,409,540 56,011 Wpl Wikipedia links, pl ⬤ Hyperlink 1,529,135 57,489,447 1,529,116 1,169,427 Wpt Wikipedia links, pt ⬤ Hyperlink 1,603,222 49,021,409 1,602,960 1,083,495 Wja Wikipedia links, ja ⬤ Hyperlink 1,610,638 71,055,717 1,610,592 1,242,374 YS Yahoo songs ⬤ Rating 2,626,941 256,804,235 1,625,951 100% Wit Wikipedia links, it ⬤ Hyperlink 1,865,965 91,555,008 1,865,905 1,448,715 WP Wikipedia, English ⬤ Hyperlink 1,870,709 39,953,145 1,870,521 1,629,321 HUi Hudong internal ⬤ Hyperlink 1,984,484 14,869,484 1,962,418 365,558 WK Wikipedia, English ⬤ Communication 2,394,385 5,021,410 2,388,953 111,881 HUr Hudong related ⬤ Hyperlink 2,452,715 18,854,882 2,415,542 185,668 YG YouTube ⬤ Affiliation 218,563 293,360 BAi Baidu internal ⬤ Hyperlink 2,141,300 17,794,839 2,107,689 609,905 Wru Wikipedia links, ru ⬤ Hyperlink 2,853,118 82,056,101 2,852,951 1,817,233 Ten Wikipedia talk, English ⬤ Communication 2,987,535 24,981,163 2,859,574 249,610 Wfr Wikipedia links, fr ⬤ Hyperlink 3,023,165 102,382,410 3,023,000 2,047,206 OR Orkut ⬤ Social 3,072,441 117,184,899 3,072,441 YT YouTube ⬤ Social 3,223,589 9,375,374 3,216,075 Wde Wikipedia links, de ⬤ Hyperlink 3,225,565 81,626,917 3,223,435 2,211,883 PC US patents ⬤ Citation 3,774,768 16,518,947 3,764,117 1 Wen Wikipedia links, en ⬤ Hyperlink 12,150,976 378,142,420 11,191,454 7,283,915 WC Wikipedia (en) ⬤ Feature 3,889,933 3,795,796 HT Highland tribes ⬤ HumanSocial 16 58 16 ZA Zachary karate club ⬤ HumanSocial 34 78 34 TF Twitter (MPI) ⬤ Social 52,579,682 1,963,263,821 TW Twitter (WWW) ⬤ Social 41,652,230 1,468,365,182 HY Hypertext 2009 ⬤ HumanContact 113 20,818 113 CR Chicago ⬤ Infrastructure 1,467 1,298 823 ET Euroroad ⬤ Infrastructure 1,174 1,417 1,039 Dj JUNG dependency ⬤ Software 6,120 138,706 6,120 77 DJ JDK dependency ⬤ Software 6,434 150,985 6,434 77 CC Cora citation ⬤ Citation 23,166 91,500 23,166 3,991 SO Stack Overflow ⬤ Rating 1,187,072 1,301,942 If Infectious ⬤ HumanContact 410 17,298 410 TO Internet topology ⬤ Computer 34,761 171,403 34,761 SD Slashdot ⬤ Communication 51,083 140,778 51,083 16,377 ES Epinions ⬤ Social 75,879 508,837 75,877 32,223 SZ Slashdot Zoo ⬤ Social 79,120 515,397 79,116 26,997 CO Wikipedia conflict ⬤ OnlineContact 118,100 2,917,785 113,123 WU WebUni Magdeburg ⬤ Text 212,552 3,869,707 SF Stanford ⬤ Hyperlink 281,903 2,312,497 255,265 150,532 ND Notre Dame ⬤ Hyperlink 325,729 1,497,134 325,729 53,968 BS Berkeley/Stanford ⬤ Hyperlink 685,230 7,600,595 654,782 334,857 GO Google ⬤ Hyperlink 875,713 5,105,039 855,802 434,818 UL Unicode languages ⬤ Feature 1,122 1,255 RD Pennsylvania ⬤ Infrastructure 1,088,092 1,541,898 1,087,562 R1 Texas ⬤ Infrastructure 1,379,917 1,921,660 1,351,137 WT TREC WT10g ⬤ Hyperlink 1,601,787 8,063,026 1,458,316 470,441 WT Web trackers ⬤ Hyperlink 68,087,704 140,613,762 PL Pokec ⬤ Social 1,632,803 30,622,564 1,632,803 1,304,537 RO California ⬤ Infrastructure 1,965,206 2,766,607 1,957,027 RE Reuters ⬤ Text 1,846,441 96,903,520 LJ LiveJournal ⬤ Social 4,847,571 68,475,391 4,843,953 3,828,682 OG Orkut ⬤ Affiliation 14,297,249 327,037,487 AF US airports ⬤ Infrastructure 1,574 28,236 1,572 1,402 UC UC Irvine messages ⬤ Communication 1,899 59,835 1,893 1,294 UF UC Irvine forum ⬤ Interaction 2,320 33,720 SW Southern women 1 ⬤ Interaction 50 89 OF OpenFlights ⬤ Infrastructure 2,939 30,501 2,905 2,868 OF OpenFlights ⬤ Infrastructure 3,425 67,663 3,397 3,354 UG US power grid ⬤ Infrastructure 4,941 6,594 4,941 AC arXiv cond-mat ⬤ Authorship 55,467 58,595 NX Netflix ⬤ Rating 978,148 100,480,507 Ws Twitter (ICWSM) ⬤ Social 465,017 834,797 465,017 1,726 Wut Twitter u-t ⬤ Folksonomy 880,846 4,664,605 Wui Twitter u-i ⬤ Folksonomy 9,618,743 12,656,613 Wti Twitter t-i ⬤ Folksonomy 1,773,193 2,635,885 DG Digg ⬤ Communication 30,398 87,627 29,652 6,746 Wa Twitter ⬤ OnlineContact 2,919,613 12,887,063 2,893,623 98,784 M2 MovieLens 1M ⬤ Rating 15,786 1,000,209 Mut MovieLens u-t ⬤ Folksonomy 24,546 95,580 Mui MovieLens u-i ⬤ Folksonomy 15,619 95,580 Mti MovieLens t-i ⬤ Folksonomy 40,657 95,580 M3 MovieLens 10M ⬤ Rating 150,433 10,000,054 M1 MovieLens 100k ⬤ Rating 3,568 100,000 MR Rhesus ⬤ Animal 16 111 16 16 MK Kangaroo ⬤ Animal 17 91 17 Ms Sampson ⬤ HumanSocial 18 189 18 18 MT Taro exchange ⬤ HumanSocial 22 78 22 22 MZ Zebra ⬤ Animal 27 111 23 Mc Sheep ⬤ Animal 28 250 28 22 MX Seventh graders ⬤ HumanSocial 29 376 29 29 Mh Hens ⬤ Animal 32 496 32 31 Mv Dutch college ⬤ HumanSocial 32 3,062 32 31 MQ Macaques ⬤ Animal 62 1,187 62 38 Mt Train bombing ⬤ HumanContact 64 243 64 MH Highschool ⬤ HumanSocial 70 366 70 67 Ml Les Misérables ⬤ Misc 77 254 77 MI Physicians ⬤ HumanSocial 241 1,098 117 95 MO Residence hall ⬤ HumanSocial 217 2,672 217 214 Mp Protein ⬤ Metabolic 1,870 2,277 1,458 MN Bible ⬤ Lexical 1,773 16,401 1,707 ME Adolescent health ⬤ HumanSocial 2,539 12,969 2,539 2,155 MC Crime ⬤ Interaction 2,209 1,476 MB Bison ⬤ Animal 26 314 26 26 MA Cattle ⬤ Animal 28 217 28 20 MW Windsurfers ⬤ HumanContact 43 336 43 RM Reality Mining ⬤ HumanContact 96 1,086,404 96 MP PDZBase ⬤ Metabolic 212 244 161 ML Little Rock Lake ⬤ Trophic 183 2,494 183 22 Mg Blogs ⬤ Hyperlink 1,224 19,025 1,222 793 MS Human protein (Stelzl) ⬤ Metabolic 1,706 6,207 1,615 1,493 MV Human protein (Vidal) ⬤ Metabolic 3,133 6,726 2,783 LK Linux kernel mailing list threads ⬤ Interaction 421,599 1,565,683 LM Livemocha ⬤ Social 104,103 2,193,083 104,103 DF Digg friends ⬤ Social 279,630 1,731,653 261,489 34,826 Lj LiveJournal links ⬤ Social 5,204,176 49,174,464 5,189,808 LG LiveJournal ⬤ Affiliation 13,891,479 112,307,385 LX Linux ⬤ Software 30,837 213,954 30,817 913 Us Wikipedia, simple en (dynamic) ⬤ Hyperlink 100,312 1,627,472 99,636 54,524 LI Libimseti.cz ⬤ Social 220,970 17,359,346 220,970 81,145 Up Wikipedia, pl (dynamic) ⬤ Hyperlink 1,033,050 25,026,208 1,031,602 723,953 Ud Wikipedia, nl (dynamic) ⬤ Hyperlink 1,039,252 20,070,561 1,038,209 719,510 Ui Wikipedia, it (dynamic) ⬤ Hyperlink 1,204,009 34,826,283 1,202,469 920,293 Ug Wikipedia, de (dynamic) ⬤ Hyperlink 2,166,669 86,337,879 2,165,679 1,464,096 Uf Wikipedia, fr (dynamic) ⬤ Hyperlink 2,212,682 59,008,831 2,208,840 1,323,072 Ls Last.fm song ⬤ Interaction 1,086,604 19,150,868 Lb Last.fm band ⬤ Interaction 176,061 19,150,868 YD Yahoo advertisers ⬤ Lexical 653,260 2,931,708 653,260 2,754 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 1,402,673 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 FD Florida ecosystem dry ⬤ Trophic 128 2,137 128 103 FW Florida ecosystem wet ⬤ Trophic 128 2,106 128 103 FO FOLDOC ⬤ Hyperlink 13,356 125,207 13,356 13,274 FX Flixster ⬤ Social 2,523,386 7,918,801 2,523,386 SX Sexual escorts ⬤ Rating 26,836 50,632 ER Epinions ⬤ Rating 996,744 13,668,320 EN Enron ⬤ Communication 87,273 1,148,072 84,384 9,164 EP Epinions ⬤ Social 131,828 841,372 119,130 41,441 EU EU institution ⬤ Communication 265,214 420,045 224,832 34,203 LF Flickr links ⬤ Social 1,715,255 15,551,250 1,624,991 FL Flickr ⬤ Social 2,302,925 33,140,017 2,173,370 1,605,184 FG Flickr ⬤ Affiliation 895,589 8,545,307 FI Flickr ⬤ Misc 105,938 2,316,948 105,722 Fr Filmtipset ⬤ Rating 225,153 19,554,219 Ff Filmtipset ⬤ Social 39,199 87,415 36,475 Fc Filmtipset ⬤ Interaction 104,890 1,266,753 Ow Facebook ⬤ Communication 46,952 876,993 43,953 30,793 it Wikipedia (it) ⬤ Authorship 2,531,261 26,241,217 mfr Wiktionary (fr) ⬤ Authorship 1,917,281 7,399,298 nfr Wikinews (fr) ⬤ Authorship 27,954 193,618 fr Wikipedia (fr) ⬤ Authorship 4,598,826 46,168,355 bfr Wikibooks (fr) ⬤ Authorship 33,881 201,727 es Wikipedia (es) ⬤ Authorship 3,623,742 27,011,506 men Wiktionary (en) ⬤ Authorship 2,163,240 8,998,641 qen Wikiquote (en) ⬤ Authorship 137,970 549,210 nen Wikinews (en) ⬤ Authorship 184,536 901,416 en Wikipedia (en) ⬤ Authorship 29,143,573 266,769,613 24,749,669 97.7% ben Wikibooks (en) ⬤ Authorship 200,108 1,164,576 mde Wiktionary (de) ⬤ Authorship 157,806 1,229,501 de Wikipedia (de) ⬤ Authorship 4,046,832 57,323,775 DO Dolphins ⬤ Animal 62 159 62 EA Edinburgh Associative Thesaurus ⬤ Lexical 23,132 511,764 23,132 7,751 DB Douban ⬤ Social 154,908 327,162 154,908 Dut Delicious ut ⬤ Folksonomy 6,178,261 301,186,579 5,362,822 99.9% Dti Delicious ti ⬤ Folksonomy 42,801,712 301,183,605 33,792,611 100% Dui Delicious ui ⬤ Folksonomy 35,444,383 301,186,579 34,323,021 99.2% 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 HA Haggle ⬤ HumanContact 274 28,244 274 Pi DBLP ⬤ Citation 12,591 49,743 12,495 240 CD DBLP co-authorship ⬤ Coauthorship 317,080 1,049,866 317,080 CA Amazon (MDS) ⬤ Misc 334,863 925,872 334,863 CY Youtube friendship ⬤ Social 1,134,890 2,987,624 1,134,890 Pc DBLP ⬤ Coauthorship 1,314,050 18,986,618 1,167,956 Pa DBLP ⬤ Authorship 6,851,776 8,649,016 CU Contiguous USA ⬤ Infrastructure 49 107 49 DL Wikipedia, English ⬤ Hyperlink 18,268,992 172,183,984 18,265,512 3,796,073 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 3,915,921 178,593 SC South African Companies ⬤ Affiliation 17 13 TH arXiv hep-th ⬤ Coauthorship 22,908 2,673,133 22,721 THc arXiv hep-th ⬤ Citation 27,770 352,807 27,400 7,464 PH arXiv hep-ph ⬤ Coauthorship 28,093 4,596,803 28,045 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 PHc arXiv hep-ph ⬤ Citation 34,546 421,578 34,401 12,711 Cut CiteULike u-t ⬤ Folksonomy 198,707 2,411,819 Cui CiteULike u-i ⬤ Folksonomy 777,199 2,411,819 Cti CiteULike t-i ⬤ Folksonomy 1,038,323 2,411,819 CH Chess ⬤ Interaction 7,301 65,053 7,115 4,738 GC Google.com internal ⬤ Hyperlink 15,763 171,206 15,763 12,354 CS CiteSeer ⬤ Citation 384,413 1,751,463 365,154 16,208 Bx BookCrossing (ratings) ⬤ Rating 341,559 433,652 BX BookCrossing (implicit) ⬤ Rating 551,079 1,149,739 But BibSonomy u-t ⬤ Folksonomy 216,261 2,555,080 Bui BibSonomy u-i ⬤ Folksonomy 779,035 2,555,080 Bti BibSonomy t-i ⬤ Folksonomy 1,176,793 2,555,080 JZ Jazz musicians ⬤ HumanSocial 198 2,742 198 PM Caenorhabditis elegans ⬤ Metabolic 453 4,596 453 A@ U. Rovira i Virgili ⬤ Communication 1,133 5,451 1,133 AS Route views ⬤ Computer 6,474 13,895 6,474 PG Pretty Good Privacy ⬤ OnlineContact 10,680 24,316 10,680 AR Amazon ratings ⬤ Rating 5,523,029 5,838,041 AN David Copperfield ⬤ Lexical 112 425 112 AD Advogato ⬤ Social 6,541 51,127 5,042 3,140 AM Actor movies ⬤ Affiliation 639,286 1,470,404 IN CAIDA ⬤ Computer 26,475 53,381 26,475 CL Actor collaborations ⬤ Misc 382,219 33,115,812 374,511 Am Amazon (TWEB) ⬤ Misc 403,394 3,387,388 403,364 395,234 SK Skitter ⬤ Computer 1,696,415 11,095,298 1,694,616