1314:.) The preferential attachment process does not incorporate the taking away part. This point may be moot, however, since the scientific insight behind the Matthew effect is in any case entirely different. Qualitatively it is intended to describe not a mechanical multiplicative effect like preferential attachment but a specific human behavior in which people are more likely to give credit to the famous than to the little known. The classic example of the Matthew effect is a scientific discovery made simultaneously by two different people, one well known and the other little known. It is claimed that under these circumstances people tend more often to credit the discovery to the well-known scientist. Thus the real-world phenomenon the Matthew effect is intended to describe is quite distinct from (though certainly related to) preferential attachment.
741:, meaning a process in which discrete units of wealth, usually called "balls", are added in a random or partly random fashion to a set of objects or containers, usually called "urns". A preferential attachment process is an urn process in which additional balls are added continuously to the system and are distributed among the urns as an increasing function of the number of balls the urns already have. In the most commonly studied examples, the number of urns also increases continuously, although this is not a necessary condition for preferential attachment and examples have been studied with constant or even decreasing numbers of urns.
27:
60:
761:(i.e., split in two) and, assuming that new species belong to the same genus as their parent (except for those that start new genera), the probability that a new species is added to a genus will be proportional to the number of species the genus already has. This process, first studied by British statistician
1364:
in 1999. Barabási and Albert also coined the name "preferential attachment" by which the process is best known today and suggested that the process might apply to the growth of other networks as well. For growing networks, the precise functional form of preferential attachment can be estimated by
1326:
in 1925, who used it to explain the power-law distribution of the number of species per genus of flowering plants. The process is sometimes called a "Yule process" in his honor. Yule was able to show that the process gave rise to a distribution with a power-law tail, but the details of his proof
700:
is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who are already wealthy receive more than those who are not. "Preferential attachment" is
1205:
in its tail. This is the primary reason for the historical interest in preferential attachment: the species distribution and many other phenomena are observed empirically to follow power laws and the preferential attachment process is a leading candidate mechanism to explain this behavior.
1206:
Preferential attachment is considered a possible candidate for, among other things, the distribution of the sizes of cities, the wealth of extremely wealthy individuals, the number of citations received by learned publications, and the number of links to pages on the World Wide Web.
756:
of biotic organisms. New genera ("urns") are added to a taxon whenever a newly appearing species is considered sufficiently different from its predecessors that it does not belong in any of the current genera. New species ("balls") are added as old ones
924:
1044:
1349:. It is in the context of network growth that the process is most frequently studied today. Price also promoted preferential attachment as a possible explanation for power laws in many other phenomena, including
725:
distributions. If preferential attachment is non-linear, measured distributions may deviate from a power law. These mechanisms may generate distributions which are approximately power law over transient periods.
1345:
in 1976. (He referred to the process as a "cumulative advantage" process.) His was also the first application of the process to the growth of a network, producing what would now be called a
1113:
1189:
1209:
The general model described here includes many other specific models as special cases. In the species/genus example above, for instance, each genus starts out with a single species (
1327:
are, by today's standards, contorted and difficult, since the modern tools of stochastic process theory did not yet exist and he was forced to use more cumbersome methods of proof.
30:
Graph generated using preferential attachment. A small number of nodes have a large number of incoming edges, whereas a large number of nodes have a small number of incoming edges.
821:
1384:
956:
774:
Linear preferential attachment processes in which the number of urns increases are known to produce a distribution of balls over the urns following the so-called
1648:
Falkenberg, Max; Lee, Jong-Hyeok; Amano, Shun-ichi; Ogawa, Ken-ichiro; Yano, Kazuo; Miyake, Yoshihiro; Evans, Tim S.; Christensen, Kim (18 June 2020).
576:
1700:
1307:
1303:: "For everyone who has will be given more, and he will have an abundance. Whoever does not have, even what he has will be taken from him." (
1595:
Krapivsky, Paul; Krioukov, Dmitri (21 August 2008). "Scale-free networks as preasymptotic regimes of superlinear preferential attachment".
1843:
710:
683:
1061:
721:. The principal reason for scientific interest in preferential attachment is that it can, under suitable circumstances, generate
1146:
771:
preferential attachment process, since the rate at which genera accrue new species is linear in the number they already have.
1414:
566:
295:
701:
only the most recent of many names that have been given to such processes. They are also referred to under the names
640:
223:
536:
526:
521:
1366:
676:
635:
152:
481:
325:
272:
87:
516:
1542:
Krapivsky, P. L.; Redner, S.; Leyvraz, F. (20 November 2000). "Connectivity of
Growing Random Networks".
1449:
1439:
1404:
1394:
714:
645:
551:
546:
511:
310:
208:
147:
1985:
1311:
233:
1990:
669:
571:
531:
38:
1267:
429:
1342:
652:
471:
238:
127:
1923:"PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks"
778:. In the most general form of the process, balls are added to the system at an overall rate of
1419:
1360:
The application of preferential attachment to the growth of the World Wide Web was proposed by
556:
541:
456:
1820:
657:
476:
446:
335:
290:
1934:
1879:
1748:
1500:
1389:
424:
305:
20:
8:
1696:"A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, F.R.S"
1444:
1216: = 1) and gains new species in direct proportion to the number it already has (
1198:
461:
330:
320:
315:
167:
112:
102:
1938:
1883:
1752:
1504:
1957:
1922:
1903:
1764:
1738:
1661:
1630:
1604:
1577:
1551:
1524:
1490:
1379:
1346:
919:{\displaystyle P(k)={\mathrm {B} (k+a,\gamma ) \over \mathrm {B} (k_{0}+a,\gamma -1)},}
735:
300:
253:
228:
117:
107:
744:
A classic example of a preferential attachment process is the growth in the number of
1962:
1895:
1768:
1676:
1649:
1622:
1569:
1516:
1354:
1304:
1300:
775:
597:
263:
213:
122:
97:
26:
1634:
1581:
1292:, but the two are not precisely equivalent. The Matthew effect, first discussed by
1952:
1942:
1907:
1887:
1867:
1835:
1796:
1756:
1709:
1671:
1614:
1561:
1508:
1424:
1341:
The first application of preferential attachment to learned citations was given by
1335:
1293:
356:
345:
243:
203:
187:
1528:
1947:
1512:
1399:
1331:
718:
592:
373:
248:
157:
92:
46:
1891:
1800:
1565:
1322:
The first rigorous consideration of preferential attachment seems to be that of
1618:
1429:
1350:
1289:
1053:
789:
balls and further balls are added to urns at a rate proportional to the number
602:
408:
383:
378:
352:
341:
218:
182:
177:
137:
75:
1760:
1481:
Barabási, A.-L.; R. Albert (1999). "Emergence of scaling in random networks".
1255:. Similarly the Price model for scientific citations corresponds to the case
1979:
1338:
in 1955, in work on the distribution of sizes of cities and other phenomena.
948:
561:
466:
451:
393:
142:
132:
1729:
Newman, M. E. J. (2005). "Power laws, Pareto distributions and Zipf's law".
1039:{\displaystyle \mathrm {B} (x,y)={\Gamma (x)\Gamma (y) \over \Gamma (x+y)},}
1966:
1899:
1839:
1821:"A general theory of bibliometric and other cumulative advantage processes"
1714:
1695:
1626:
1573:
1520:
1409:
506:
403:
258:
1921:
Pham, Thong; Sheridan, Paul; Shimodaira, Hidetoshi (September 17, 2015).
1743:
1556:
1495:
1434:
738:
1454:
758:
441:
398:
388:
1323:
1202:
1194:
762:
722:
606:
162:
782:
new balls for each new urn. Each newly created urn starts out with
59:
1666:
1361:
1787:
Simon, H. A. (1955). "On a class of skew distribution functions".
1609:
1330:
Most modern treatments of preferential attachment make use of the
19:"Yule process" redirects here. For the type of birth process, see
1193:
In other words, the preferential attachment process generates a "
745:
767:
1297:
753:
749:
1288:
Preferential attachment is sometimes referred to as the
1647:
1920:
1541:
1385:
Bose–Einstein condensation: a network theory approach
1149:
1064:
959:
824:
1334:method, whose use in this context was pioneered by
16:
Stochastic process formalizing cumulative advantage
1183:
1107:
1038:
918:
1701:Philosophical Transactions of the Royal Society B
1594:
1480:
1977:
1689:
1687:
1650:"Identifying time dependence in network growth"
1117:The beta function behaves asymptotically as B(
1782:
1780:
1778:
677:
1814:
1812:
1810:
1684:
1108:{\displaystyle \gamma =2+{k_{0}+a \over m}.}
816:balls in the limit of long time is given by
1476:
1474:
1472:
1470:
1775:
684:
670:
1956:
1946:
1870:(1968). "The Matthew effect in science".
1807:
1742:
1713:
1675:
1665:
1608:
1555:
1494:
1184:{\displaystyle P(k)\propto k^{-\gamma }.}
1137:, which implies that for large values of
1467:
804:. With these definitions, the fraction
25:
793:that they already have plus a constant
734:A preferential attachment process is a
1978:
1866:
1728:
1266: = 1 and the widely studied
1818:
1786:
1693:
1415:Link-centric preferential attachment
13:
1012:
998:
986:
961:
872:
844:
14:
2002:
1677:10.1103/PhysRevResearch.2.023352
1296:, is named for a passage in the
58:
1914:
1849:from the original on 2020-12-01
1353:of scientific productivity and
698:preferential attachment process
1860:
1722:
1641:
1588:
1535:
1159:
1153:
1027:
1015:
1007:
1001:
995:
989:
977:
965:
939:(and zero otherwise), where B(
907:
876:
866:
848:
834:
828:
1:
1461:
1367:maximum likelihood estimation
729:
1948:10.1371/journal.pone.0137796
1513:10.1126/science.286.5439.509
7:
1892:10.1126/science.159.3810.56
1819:Price, D. J. de S. (1976).
1566:10.1103/PhysRevLett.85.4629
1405:Double jeopardy (marketing)
1372:
1220: = 0), and hence
1197:" distribution following a
717:. They are also related to
10:
2007:
1619:10.1103/PhysRevE.78.026114
1395:Chinese restaurant process
1317:
18:
1828:J. Amer. Soc. Inform. Sci
1801:10.1093/biomet/42.3-4.425
1761:10.1080/00107510500052444
1440:Success to the successful
1312:New International Version
537:Exponential random (ERGM)
204:Informational (computing)
1654:Physical Review Research
224:Scientific collaboration
1544:Physical Review Letters
1450:Yule–Simon distribution
653:Category:Network theory
173:Preferential attachment
1840:10.1002/asi.4630270505
1715:10.1098/rstb.1925.0002
1185:
1109:
1040:
920:
542:Random geometric (RGG)
31:
1268:Barabási-Albert model
1247: − 1) with
1186:
1110:
1052:) being the standard
1041:
921:
658:Category:Graph theory
29:
1731:Contemporary Physics
1694:Yule, G. U. (1925).
1390:Capital accumulation
1147:
1062:
957:
822:
707:cumulative advantage
21:Simple birth process
1939:2015PLoSO..1037796P
1884:1968Sci...159...56M
1753:2005ConPh..46..323N
1505:1999Sci...286..509B
1445:Wealth condensation
1362:Barabási and Albert
1199:Pareto distribution
711:the rich get richer
462:Degree distribution
113:Community structure
1708:(402–410): 21–87.
1420:Pitman–Yor process
1380:Assortative mixing
1347:scale-free network
1181:
1105:
1036:
916:
646:Network scientists
572:Soft configuration
32:
1868:Merton, Robert K.
1597:Physical Review E
1550:(21): 4629–4632.
1489:(5439): 509–512.
1301:Gospel of Matthew
1251:=2 + 1/
1100:
1031:
911:
812:) of urns having
797: > −
776:Yule distribution
694:
693:
614:
613:
522:Bianconi–Barabási
416:
415:
234:Artificial neural
209:Telecommunication
1998:
1986:Social phenomena
1971:
1970:
1960:
1950:
1918:
1912:
1911:
1864:
1858:
1857:
1855:
1854:
1848:
1825:
1816:
1805:
1804:
1795:(3–4): 425–440.
1784:
1773:
1772:
1746:
1744:cond-mat/0412004
1726:
1720:
1719:
1717:
1691:
1682:
1681:
1679:
1669:
1645:
1639:
1638:
1612:
1592:
1586:
1585:
1559:
1557:cond-mat/0005139
1539:
1533:
1532:
1498:
1496:cond-mat/9910332
1478:
1357:of journal use.
1294:Robert K. Merton
1285: = 0.
1262: = 0,
1228:) = B(
1190:
1188:
1187:
1182:
1177:
1176:
1114:
1112:
1111:
1106:
1101:
1096:
1089:
1088:
1078:
1045:
1043:
1042:
1037:
1032:
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1010:
984:
964:
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923:
922:
917:
912:
910:
888:
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875:
869:
847:
841:
686:
679:
672:
557:Stochastic block
547:Hyperbolic (HGN)
496:
495:
359:
348:
280:
279:
188:Social influence
62:
34:
33:
2006:
2005:
2001:
2000:
1999:
1997:
1996:
1995:
1991:Network science
1976:
1975:
1974:
1933:(9): e0137796.
1919:
1915:
1878:(3810): 56–63.
1865:
1861:
1852:
1850:
1846:
1823:
1817:
1808:
1785:
1776:
1727:
1723:
1692:
1685:
1646:
1642:
1593:
1589:
1540:
1536:
1479:
1468:
1464:
1459:
1400:Complex network
1375:
1332:master equation
1320:
1276:
1270:corresponds to
1261:
1242:
1215:
1169:
1165:
1148:
1145:
1144:
1084:
1080:
1079:
1077:
1063:
1060:
1059:
1011:
985:
983:
960:
958:
955:
954:
947:) is the Euler
938:
883:
879:
871:
870:
843:
842:
840:
823:
820:
819:
803:
788:
752:in some higher
732:
690:
628:
593:Boolean network
567:Maximum entropy
517:Barabási–Albert
434:
351:
340:
128:Controllability
93:Complex network
80:
67:
66:
65:
64:
63:
47:Network science
24:
17:
12:
11:
5:
2004:
1994:
1993:
1988:
1973:
1972:
1913:
1859:
1834:(5): 292–306.
1806:
1774:
1737:(5): 323–351.
1721:
1683:
1640:
1587:
1534:
1465:
1463:
1460:
1458:
1457:
1452:
1447:
1442:
1437:
1432:
1430:Proof of stake
1427:
1422:
1417:
1412:
1407:
1402:
1397:
1392:
1387:
1382:
1376:
1374:
1371:
1355:Bradford's law
1319:
1316:
1290:Matthew effect
1274:
1259:
1240:
1213:
1180:
1175:
1172:
1168:
1164:
1161:
1158:
1155:
1152:
1125:) ~
1104:
1099:
1095:
1092:
1087:
1083:
1076:
1073:
1070:
1067:
1054:gamma function
1035:
1029:
1026:
1023:
1020:
1017:
1014:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
982:
979:
976:
973:
970:
967:
963:
936:
915:
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903:
900:
897:
894:
891:
886:
882:
878:
874:
868:
865:
862:
859:
856:
853:
850:
846:
839:
836:
833:
830:
827:
801:
786:
731:
728:
715:Matthew effect
692:
691:
689:
688:
681:
674:
666:
663:
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661:
660:
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643:
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629:
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623:
619:
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612:
611:
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582:
581:
580:
579:
574:
569:
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559:
554:
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532:Watts–Strogatz
529:
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509:
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401:
396:
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386:
381:
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368:
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362:
361:
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353:Incidence list
349:
342:Adjacency list
338:
333:
328:
323:
318:
313:
311:Data structure
308:
303:
298:
293:
285:
284:
276:
275:
269:
268:
267:
266:
261:
256:
251:
246:
241:
239:Interdependent
236:
231:
226:
221:
216:
211:
206:
198:
197:
193:
192:
191:
190:
185:
183:Network effect
180:
178:Balance theory
175:
170:
165:
160:
155:
150:
145:
140:
138:Social capital
135:
130:
125:
120:
115:
110:
105:
100:
95:
90:
82:
81:
79:
78:
72:
69:
68:
57:
56:
55:
54:
53:
50:
49:
43:
42:
15:
9:
6:
4:
3:
2:
2003:
1992:
1989:
1987:
1984:
1983:
1981:
1968:
1964:
1959:
1954:
1949:
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1940:
1936:
1932:
1928:
1924:
1917:
1909:
1905:
1901:
1897:
1893:
1889:
1885:
1881:
1877:
1873:
1869:
1863:
1845:
1841:
1837:
1833:
1829:
1822:
1815:
1813:
1811:
1802:
1798:
1794:
1790:
1783:
1781:
1779:
1770:
1766:
1762:
1758:
1754:
1750:
1745:
1740:
1736:
1732:
1725:
1716:
1711:
1707:
1703:
1702:
1697:
1690:
1688:
1678:
1673:
1668:
1663:
1660:(2): 023352.
1659:
1655:
1651:
1644:
1636:
1632:
1628:
1624:
1620:
1616:
1611:
1606:
1603:(2): 026114.
1602:
1598:
1591:
1583:
1579:
1575:
1571:
1567:
1563:
1558:
1553:
1549:
1545:
1538:
1530:
1526:
1522:
1518:
1514:
1510:
1506:
1502:
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1471:
1466:
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1453:
1451:
1448:
1446:
1443:
1441:
1438:
1436:
1433:
1431:
1428:
1426:
1425:Price's model
1423:
1421:
1418:
1416:
1413:
1411:
1408:
1406:
1403:
1401:
1398:
1396:
1393:
1391:
1388:
1386:
1383:
1381:
1378:
1377:
1370:
1368:
1363:
1358:
1356:
1352:
1348:
1344:
1339:
1337:
1333:
1328:
1325:
1315:
1313:
1309:
1306:
1302:
1299:
1295:
1291:
1286:
1284:
1280:
1277: =
1273:
1269:
1265:
1258:
1254:
1250:
1246:
1239:
1235:
1231:
1227:
1223:
1219:
1212:
1207:
1204:
1200:
1196:
1191:
1178:
1173:
1170:
1166:
1162:
1156:
1150:
1142:
1140:
1136:
1132:
1128:
1124:
1120:
1115:
1102:
1097:
1093:
1090:
1085:
1081:
1074:
1071:
1068:
1065:
1057:
1055:
1051:
1046:
1033:
1024:
1021:
1018:
1004:
992:
980:
974:
971:
968:
952:
950:
949:beta function
946:
942:
935:
932: ≥
931:
926:
913:
904:
901:
898:
895:
892:
889:
884:
880:
863:
860:
857:
854:
851:
837:
831:
825:
817:
815:
811:
807:
800:
796:
792:
785:
781:
777:
772:
770:
769:
764:
760:
755:
751:
747:
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