20:
974:. The degree distribution does not define a graph uniquely. However under assumption that in all respects other than their degree distribution, the graphs are treated as entirely random, many results on finite/infinite-component sizes are known. In this model, the existence of the giant component depends only on the first two (mixed)
2518:
100:
More precisely, in graphs drawn randomly from a probability distribution over arbitrarily large graphs, a giant component is a connected component whose fraction of the overall number of vertices is bounded away from zero. In sufficiently dense graphs distributed according to the
1783:
1513:
1947:
1865:
1346:
2249:
648:
488:
1060:
2235:
2027:
2165:
263:
164:
1271:
970:
A similar sharp threshold between parameters that lead to graphs with all components small and parameters that lead to a giant component also occurs in random graphs with non-uniform
1619:
317:
1118:
1574:
440:
2072:
960:
1203:
190:
73:, showing a large component and many small ones. At this edge probability, the large component is not yet a giant component: it contains only a sublinear number of vertices.
1176:
1090:
1419:
714:
681:
569:
542:
344:
71:
1414:
811:
379:
918:
744:
515:
414:
1378:
773:
886:
850:
1614:
1594:
996:
214:
1870:
1788:
2513:{\displaystyle 2\mathbb {E} \mathbb {E} -\mathbb {E} \mathbb {E} -\mathbb {E} \mathbb {E} +\mathbb {E} \mathbb {E} -\mathbb {E} ^{2}>0}
1276:
852:
edges have been added that the graph contains a large component, and soon after that the component becomes giant. More precisely, when
1001:
574:
2770:
Kryven, Ivan (2016-07-27). "Emergence of the giant weak component in directed random graphs with arbitrary degree distributions".
2619:
19:
445:
1142:
weak component is a set of vertices that can be reached by recursively following all edges regardless of their direction.
2614:, Cambridge studies in advanced mathematics, vol. 73 (2nd ed.), Cambridge University Press, pp. 130–159,
86:
1968:
1949:. The criteria for giant component existence in directed and undirected random graphs are given in the table below:
2086:
227:
128:
1208:
1778:{\displaystyle {\mathcal {G}}(x,y)=\sum _{k_{in},k_{out}}\displaystyle P({k_{in},k_{out}})x^{k_{in}}y^{k_{out}}}
1516:
319:, intermediate between these two possibilities, the number of vertices in the largest component of the graph,
921:
276:
2170:
2845:
1146:
1095:
2661:
Molloy, Michael; Reed, Bruce (1995). "A critical point for random graphs with a given degree sequence".
2533:
1521:
419:
114:
102:
24:
1136:
out-component is a set of vertices that can be reached by recursively following all out-edges forward;
927:
1139:
in-component is a set of vertices that can be reached by recursively following all in-edges backward;
1508:{\displaystyle G_{1}(x)=\textstyle \sum _{k}\displaystyle {\frac {k}{\langle k\rangle }}P(k)x^{k-1}}
1181:
169:
1154:
1064:
2554:
2032:
686:
653:
2850:
547:
520:
322:
265:
there is with high probability a single giant component, with all other components having size
30:
1383:
778:
962:
edges are needed in order to have high probability that the whole random graph is connected.
349:
193:
94:
2789:
2726:
975:
891:
722:
493:
387:
1354:
749:
8:
1942:{\displaystyle f(y)={\frac {1}{c}}{\partial {\mathcal {G}} \over \partial y}\vert _{x=1}}
1860:{\displaystyle g(x)={\frac {1}{c}}{\partial {\mathcal {G}} \over \partial x}\vert _{y=1}}
1349:
1125:
971:
863:
827:
2793:
2730:
2821:
2779:
2716:
2584:
2560:
1599:
1579:
981:
199:
1147:
Criteria for giant component existence in directed and undirected configuration graphs
117:(ER) of random graphs, in which each possible edge connecting pairs of a given set of
2825:
2813:
2805:
2752:
2744:
2678:
2640:
2615:
1205:
out edges. By definition, the average number of in- and out-edges coincides so that
2797:
2734:
2670:
820:
Alternatively, if one adds randomly selected edges one at a time, starting with an
814:
384:
Giant component is also important in percolation theory. When a fraction of nodes,
2578:
2572:
2610:
Bollobás, Béla (2001), "6. The
Evolution of Random Graphs—The Giant Component",
2801:
2739:
2704:
2563: – Mathematical theory on behavior of connected clusters in a random graph
1129:
1121:
78:
2839:
2809:
2748:
2682:
2644:
2817:
2756:
2674:
2548:
90:
2705:"Random graphs with arbitrary degree distributions and their applications"
1120:
is the mean degree of the network. Similar expressions are also valid for
2721:
2566:
821:
121:
vertices is present, independently of the other edges, with probability
1128:
is two-dimensional. There are three types of connected components in
978:
of the degree distribution. Let a randomly chosen vertex have degree
1341:{\displaystyle G_{0}(x)=\textstyle \sum _{k}\displaystyle P(k)x^{k}}
2784:
2539:
216:
goes to infinity) all connected components of the graph have size
643:{\displaystyle P_{\inf }=p(1-\exp(-\langle k\rangle P_{\inf }))}
2536: – Two closely related models for generating random graphs
2703:
Newman, M. E. J.; Strogatz, S. H.; Watts, D. J. (2001-07-24).
2587: – Network whose degree distribution follows a power law
1515:. For directed networks, generating function assigned to the
965:
746:, the distribution of cluster sizes behaves as a power law,
93:
that contains a significant fraction of the entire graph's
517:
there exists a giant component (largest cluster) of size,
2698:
2696:
2694:
2692:
2544:
Pages displaying short descriptions of redirect targets
108:
2689:
1445:
1302:
2656:
2654:
2575: – Network with non-trivial topological features
2569: – Filtration of fluids through porous materials
2252:
2173:
2089:
2035:
1971:
1873:
1791:
1687:
1622:
1602:
1582:
1524:
1456:
1422:
1386:
1357:
1313:
1279:
1211:
1184:
1157:
1098:
1067:
1004:
984:
930:
894:
866:
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781:
752:
725:
689:
656:
577:
550:
523:
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448:
422:
390:
352:
325:
279:
230:
202:
172:
131:
33:
483:{\displaystyle p_{c}={\frac {1}{\langle k\rangle }}}
27:
with 1000 vertices at the critical edge probability
2702:
888:, the size of the giant component is approximately
416:, is removed randomly from an ER network of degree
2651:
2542: – Infinitely detailed mathematical structure
2512:
2229:
2159:
2066:
2021:
1941:
1859:
1777:
1608:
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1507:
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1197:
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642:
563:
536:
509:
482:
434:
408:
373:
338:
311:
257:
208:
184:
158:
105:, a giant component exists with high probability.
65:
2837:
1055:{\displaystyle \mathbb {E} -2\mathbb {E} >0.}
998:, then the giant component exists if and only if
695:
629:
583:
556:
529:
331:
224:, and there is no giant component. However, for
113:Giant components are a prominent feature of the
2022:{\displaystyle \mathbb {E} -2\mathbb {E} >0}
2160:{\displaystyle \mathbb {E} -\mathbb {E} >0}
258:{\displaystyle p\geq {\frac {1+\epsilon }{n}}}
159:{\displaystyle p\leq {\frac {1-\epsilon }{n}}}
1924:
1842:
1469:
1463:
1106:
1100:
621:
615:
474:
468:
429:
423:
1266:{\displaystyle c=\mathbb {E} =\mathbb {E} }
16:Large connected component of a random graph
2660:
966:Graphs with arbitrary degree distributions
2783:
2763:
2738:
2720:
2467:
2438:
2412:
2383:
2362:
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2312:
2278:
2257:
2131:
2091:
2000:
1973:
1243:
1219:
1069:
1033:
1006:
346:is with high probability proportional to
2609:
18:
2838:
2769:
2634:
312:{\displaystyle p=p_{c}={\frac {1}{n}}}
2639:. New York: Oxford University Press.
2230:{\displaystyle g'_{1}(1)=f'_{1}(1)=1}
1092:which is also written in the form of
856:edges have been added, for values of
824:, then it is not until approximately
716:, i.e., there is no giant component.
442:, there exists a critical threshold,
2551: – Area of discrete mathematics
109:Giant component in Erdős–Rényi model
2605:
2603:
2601:
2557: – Subfield of network science
1113:{\displaystyle {\langle k\rangle }}
13:
2663:Random Structures & Algorithms
1914:
1907:
1902:
1832:
1825:
1820:
1625:
1576:can be written with two valuables
1348:is the generating function of the
931:
14:
2862:
2628:
1569:{\displaystyle P(k_{in},k_{out})}
1151:Let a randomly chosen vertex has
683:the solution of this equation is
435:{\displaystyle \langle k\rangle }
2598:
1380:for an undirected network, then
1132:. For a randomly chosen vertex:
955:{\displaystyle \Theta (n\log n)}
25:Erdős–Rényi–Gilbert random graph
2637:Networks : an introduction
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2010:
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1990:
1977:
1883:
1877:
1801:
1795:
1728:
1691:
1642:
1630:
1563:
1528:
1517:joint probability distribution
1484:
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1439:
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1403:
1397:
1367:
1361:
1323:
1317:
1296:
1290:
1260:
1247:
1236:
1223:
1198:{\displaystyle k_{\text{out}}}
1079:
1073:
1043:
1037:
1023:
1010:
949:
934:
762:
756:
637:
634:
609:
594:
185:{\displaystyle \epsilon >0}
60:
48:
1:
2591:
1171:{\displaystyle k_{\text{in}}}
2246:
2083:
1965:
1963:undirected: giant component
1957:
1954:
1085:{\displaystyle \mathbb {E} }
920:. However, according to the
7:
2527:
2067:{\displaystyle G'_{1}(1)=1}
709:{\displaystyle P_{\inf }=0}
10:
2867:
2802:10.1103/physreve.94.012315
2740:10.1103/physreve.64.026118
2240:
2077:
1962:
922:coupon collector's problem
676:{\displaystyle p<p_{c}}
2635:Newman, M. E. J. (2010).
860:close to but larger than
564:{\displaystyle P_{\inf }}
537:{\displaystyle P_{\inf }}
339:{\displaystyle P_{\inf }}
66:{\displaystyle p=1/(n-1)}
1409:{\displaystyle G_{1}(x)}
806:{\displaystyle s^{-5/2}}
2555:Interdependent networks
374:{\displaystyle n^{2/3}}
2675:10.1002/rsa.3240060204
2581: – Academic field
2514:
2231:
2161:
2080:giant in/out-component
2068:
2023:
1943:
1861:
1785:, then one can define
1779:
1610:
1590:
1570:
1509:
1410:
1374:
1342:
1267:
1199:
1172:
1114:
1086:
1056:
992:
956:
914:
882:
846:
813:which is a feature of
807:
769:
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710:
677:
644:
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538:
511:
484:
436:
410:
375:
340:
313:
259:
210:
186:
160:
74:
67:
2515:
2232:
2162:
2069:
2024:
1944:
1862:
1780:
1611:
1591:
1571:
1510:
1411:
1375:
1343:
1268:
1200:
1173:
1115:
1087:
1057:
993:
957:
915:
913:{\displaystyle 4t-2n}
883:
847:
808:
770:
741:
739:{\displaystyle p_{c}}
711:
678:
645:
566:
539:
512:
510:{\displaystyle p_{c}}
485:
437:
411:
409:{\displaystyle q=1-p}
376:
341:
314:
260:
211:
194:with high probability
187:
161:
68:
22:
2250:
2243:giant weak component
2171:
2087:
2033:
1969:
1871:
1789:
1620:
1600:
1580:
1522:
1420:
1384:
1373:{\displaystyle P(k)}
1355:
1277:
1209:
1182:
1155:
1124:, in which case the
1096:
1065:
1002:
982:
972:degree distributions
928:
892:
864:
828:
779:
768:{\displaystyle n(s)}
750:
723:
687:
654:
575:
548:
521:
494:
446:
420:
388:
350:
323:
277:
228:
200:
170:
129:
125:. In this model, if
31:
2794:2016PhRvE..94a2315K
2731:2001PhRvE..64b6118N
2524:
2459:
2433:
2404:
2354:
2211:
2186:
2048:
1350:degree distribution
1126:degree distribution
881:{\displaystyle n/2}
845:{\displaystyle n/2}
87:connected component
2846:Graph connectivity
2585:Scale-free network
2561:Percolation theory
2523:
2510:
2445:
2419:
2390:
2340:
2227:
2199:
2174:
2157:
2064:
2036:
2019:
1939:
1857:
1775:
1774:
1686:
1606:
1586:
1566:
1505:
1504:
1503:
1455:
1416:can be defined as
1406:
1370:
1338:
1337:
1336:
1312:
1263:
1195:
1168:
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206:
182:
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63:
2772:Physical Review E
2709:Physical Review E
2621:978-0-521-79722-1
2534:Erdős–Rényi model
2525:
2522:
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2481:
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2426:
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1815:
1648:
1609:{\displaystyle y}
1589:{\displaystyle x}
1473:
1446:
1303:
1257:
1233:
1192:
1165:
991:{\displaystyle k}
478:
307:
253:
209:{\displaystyle n}
196:(in the limit as
166:for any constant
154:
115:Erdős–Rényi model
103:Erdős–Rényi model
2858:
2830:
2829:
2787:
2767:
2761:
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2742:
2724:
2722:cond-mat/0007235
2700:
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2669:(2–3): 161–180.
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815:phase transition
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2579:Network science
2573:Complex network
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1307:
1284:
1280:
1278:
1275:
1274:
1254:
1250:
1242:
1230:
1226:
1218:
1210:
1207:
1206:
1189:
1185:
1183:
1180:
1179:
1162:
1158:
1156:
1153:
1152:
1149:
1130:directed graphs
1122:directed graphs
1099:
1097:
1094:
1093:
1068:
1066:
1063:
1062:
1032:
1017:
1013:
1005:
1003:
1000:
999:
983:
980:
979:
968:
929:
926:
925:
893:
890:
889:
870:
865:
862:
861:
857:
853:
834:
829:
826:
825:
793:
786:
782:
780:
777:
776:
751:
748:
747:
730:
726:
724:
721:
720:
694:
690:
688:
685:
684:
667:
663:
655:
652:
651:
628:
624:
582:
578:
576:
573:
572:
555:
551:
549:
546:
545:
528:
524:
522:
519:
518:
501:
497:
495:
492:
491:
467:
462:
453:
449:
447:
444:
443:
421:
418:
417:
389:
386:
385:
361:
357:
353:
351:
348:
347:
330:
326:
324:
321:
320:
299:
290:
286:
278:
275:
274:
266:
239:
237:
229:
226:
225:
217:
201:
198:
197:
171:
168:
167:
140:
138:
130:
127:
126:
122:
118:
111:
83:giant component
43:
32:
29:
28:
17:
12:
11:
5:
2864:
2854:
2853:
2848:
2832:
2831:
2762:
2688:
2650:
2627:
2620:
2596:
2595:
2593:
2590:
2589:
2588:
2582:
2576:
2570:
2564:
2558:
2552:
2546:
2537:
2529:
2526:
2521:
2520:
2509:
2506:
2501:
2497:
2487:
2477:
2473:
2469:
2465:
2462:
2457:
2448:
2444:
2440:
2436:
2431:
2422:
2418:
2414:
2410:
2407:
2402:
2393:
2389:
2385:
2381:
2372:
2368:
2364:
2360:
2357:
2352:
2343:
2339:
2335:
2331:
2322:
2318:
2314:
2310:
2307:
2298:
2288:
2284:
2280:
2276:
2267:
2263:
2259:
2255:
2245:
2238:
2237:
2226:
2223:
2220:
2217:
2214:
2210:
2206:
2202:
2198:
2195:
2192:
2189:
2185:
2181:
2177:
2156:
2153:
2150:
2141:
2137:
2133:
2129:
2126:
2121:
2118:
2115:
2111:
2101:
2097:
2093:
2082:
2075:
2074:
2063:
2060:
2057:
2054:
2051:
2047:
2043:
2039:
2018:
2015:
2012:
2009:
2006:
2002:
1998:
1995:
1992:
1987:
1983:
1979:
1975:
1964:
1960:
1959:
1956:
1936:
1933:
1930:
1926:
1919:
1916:
1909:
1904:
1896:
1893:
1888:
1885:
1882:
1879:
1876:
1854:
1851:
1848:
1844:
1837:
1834:
1827:
1822:
1814:
1811:
1806:
1803:
1800:
1797:
1794:
1769:
1766:
1763:
1759:
1754:
1746:
1743:
1739:
1734:
1730:
1724:
1721:
1718:
1714:
1710:
1705:
1702:
1698:
1693:
1690:
1682:
1679:
1676:
1672:
1668:
1663:
1660:
1656:
1651:
1647:
1644:
1641:
1638:
1635:
1632:
1627:
1605:
1585:
1565:
1560:
1557:
1554:
1550:
1546:
1541:
1538:
1534:
1530:
1527:
1500:
1497:
1494:
1490:
1486:
1483:
1480:
1477:
1471:
1468:
1465:
1461:
1453:
1449:
1444:
1441:
1438:
1435:
1430:
1426:
1405:
1402:
1399:
1394:
1390:
1369:
1366:
1363:
1360:
1333:
1329:
1325:
1322:
1319:
1316:
1310:
1306:
1301:
1298:
1295:
1292:
1287:
1283:
1262:
1253:
1249:
1245:
1241:
1238:
1229:
1225:
1221:
1217:
1214:
1188:
1161:
1148:
1145:
1144:
1143:
1140:
1137:
1108:
1105:
1102:
1081:
1078:
1075:
1071:
1051:
1048:
1045:
1042:
1039:
1035:
1031:
1028:
1025:
1020:
1016:
1012:
1008:
987:
967:
964:
951:
948:
945:
942:
939:
936:
933:
909:
906:
903:
900:
897:
877:
873:
869:
841:
837:
833:
800:
796:
792:
789:
785:
764:
761:
758:
755:
733:
729:
705:
702:
697:
693:
670:
666:
662:
659:
639:
636:
631:
627:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
585:
581:
558:
554:
531:
527:
504:
500:
476:
473:
470:
466:
461:
456:
452:
431:
428:
425:
405:
402:
399:
396:
393:
368:
364:
360:
356:
333:
329:
306:
303:
298:
293:
289:
285:
282:
252:
248:
245:
242:
236:
233:
205:
181:
178:
175:
153:
149:
146:
143:
137:
134:
110:
107:
79:network theory
62:
59:
56:
53:
50:
46:
42:
39:
36:
15:
9:
6:
4:
3:
2:
2863:
2852:
2851:Random graphs
2849:
2847:
2844:
2843:
2841:
2827:
2823:
2819:
2815:
2811:
2807:
2803:
2799:
2795:
2791:
2786:
2781:
2778:(1): 012315.
2777:
2773:
2766:
2758:
2754:
2750:
2746:
2741:
2736:
2732:
2728:
2723:
2718:
2715:(2): 026118.
2714:
2710:
2706:
2699:
2697:
2695:
2693:
2684:
2680:
2676:
2672:
2668:
2664:
2657:
2655:
2646:
2642:
2638:
2631:
2623:
2617:
2613:
2612:Random Graphs
2606:
2604:
2602:
2597:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2541:
2538:
2535:
2532:
2531:
2507:
2504:
2499:
2485:
2475:
2463:
2455:
2446:
2429:
2420:
2408:
2400:
2391:
2370:
2358:
2350:
2341:
2320:
2308:
2296:
2286:
2265:
2253:
2244:
2239:
2224:
2221:
2215:
2208:
2204:
2200:
2196:
2190:
2183:
2179:
2175:
2154:
2151:
2139:
2127:
2119:
2116:
2113:
2109:
2099:
2081:
2076:
2061:
2058:
2052:
2045:
2041:
2037:
2016:
2013:
2007:
1996:
1993:
1985:
1981:
1961:
1953:
1950:
1934:
1931:
1928:
1917:
1894:
1891:
1886:
1880:
1874:
1852:
1849:
1846:
1835:
1812:
1809:
1804:
1798:
1792:
1767:
1764:
1761:
1757:
1752:
1744:
1741:
1737:
1732:
1722:
1719:
1716:
1712:
1708:
1703:
1700:
1696:
1688:
1680:
1677:
1674:
1670:
1666:
1661:
1658:
1654:
1649:
1645:
1639:
1636:
1633:
1603:
1583:
1558:
1555:
1552:
1548:
1544:
1539:
1536:
1532:
1525:
1518:
1498:
1495:
1492:
1488:
1481:
1475:
1466:
1459:
1451:
1447:
1442:
1436:
1428:
1424:
1400:
1392:
1388:
1364:
1358:
1351:
1331:
1327:
1320:
1314:
1308:
1304:
1299:
1293:
1285:
1281:
1251:
1239:
1227:
1215:
1212:
1186:
1178:in-edges and
1159:
1141:
1138:
1135:
1134:
1133:
1131:
1127:
1123:
1103:
1076:
1049:
1046:
1040:
1029:
1026:
1018:
1014:
985:
977:
973:
963:
946:
943:
940:
937:
923:
907:
904:
901:
898:
895:
875:
871:
867:
839:
835:
831:
823:
818:
816:
798:
794:
790:
787:
783:
759:
753:
731:
727:
717:
703:
700:
691:
668:
664:
660:
657:
625:
618:
612:
606:
603:
600:
597:
591:
588:
579:
552:
525:
502:
498:
471:
464:
459:
454:
450:
426:
403:
400:
397:
394:
391:
382:
366:
362:
358:
354:
327:
304:
301:
296:
291:
287:
283:
280:
270:
250:
246:
243:
240:
234:
231:
221:
203:
195:
179:
176:
173:
151:
147:
144:
141:
135:
132:
116:
106:
104:
98:
96:
92:
88:
84:
80:
57:
54:
51:
44:
40:
37:
34:
26:
21:
2775:
2771:
2765:
2712:
2708:
2666:
2662:
2636:
2630:
2611:
2549:Graph theory
2242:
2079:
1150:
969:
819:
718:
383:
268:
219:
112:
99:
91:random graph
82:
76:
2567:Percolation
822:empty graph
89:of a given
2840:Categories
2785:1607.03793
2592:References
2241:directed:
2078:directed:
571:fulfills,
2826:206251373
2810:2470-0045
2749:1063-651X
2683:1042-9832
2645:456837194
2464:−
2359:−
2309:−
2128:−
1994:−
1958:Criteria
1915:∂
1903:∂
1833:∂
1821:∂
1650:∑
1496:−
1470:⟩
1464:⟨
1448:∑
1305:∑
1107:⟩
1101:⟨
1027:−
944:
932:Θ
902:−
788:−
622:⟩
616:⟨
613:−
607:
601:−
475:⟩
469:⟨
430:⟩
424:⟨
401:−
247:ϵ
235:≥
174:ϵ
148:ϵ
145:−
136:≤
55:−
2818:27575156
2757:11497662
2540:Fractals
2528:See also
2209:′
2184:′
2046:′
490:. Above
95:vertices
2790:Bibcode
2727:Bibcode
976:moments
650:. For
192:, then
2824:
2816:
2808:
2755:
2747:
2681:
2643:
2618:
273:. For
267:O(log
218:O(log
2822:S2CID
2780:arXiv
2717:arXiv
2167:, or
2029:, or
1955:Type
1273:. If
85:is a
2814:PMID
2806:ISSN
2753:PMID
2745:ISSN
2679:ISSN
2641:OCLC
2616:ISBN
2505:>
2152:>
2014:>
1867:and
1616:as:
1596:and
1047:>
661:<
177:>
81:, a
2798:doi
2735:doi
2671:doi
2490:out
2451:out
2346:out
2301:out
1256:out
1191:out
941:log
719:At
696:inf
630:inf
604:exp
584:inf
557:inf
530:inf
332:inf
77:In
23:An
2842::
2820:.
2812:.
2804:.
2796:.
2788:.
2776:94
2774:.
2751:.
2743:.
2733:.
2725:.
2713:64
2711:.
2707:.
2691:^
2677:.
2665:.
2653:^
2600:^
2480:in
2425:in
2396:in
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817:.
544:.
381:.
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2647:.
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2476:k
2472:[
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2100:k
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2005:[
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