4681:
implicitly assumes that each node can get attached to any other node of the network. This assumption is however unreasonable if the network is very large, as the horizon of a node includes a small part of the network, ignoring most of it. Moreover, this implies that the expected number of edges between two groups of nodes decreases if the size of the network increases. So, if a network is large enough, the expected number of edges between two groups of nodes in modularity's null model may be smaller than one. If this happens, a single edge between the two clusters would be interpreted by modularity as a sign of a strong correlation between the two clusters, and optimizing modularity would lead to the merging of the two clusters, independently of the clusters' features. So, even weakly interconnected complete graphs, which have the highest possible density of internal edges, and represent the best identifiable communities, would be merged by modularity optimization if the network were sufficiently large. For this reason, optimizing modularity in large networks would fail to resolve small communities, even when they are well defined. This bias is inevitable for methods like modularity optimization, which rely on a global null model.
726:
and topologically studied to reveal some unexpected structural features. Most of these networks possess a certain community structure that has substantial importance in building an understanding regarding the dynamics of the network. For instance, a closely connected social community will imply a faster rate of transmission of information or rumor among them than a loosely connected community. Thus, if a network is represented by a number of individual nodes connected by links which signify a certain degree of interaction between the nodes, communities are defined as groups of densely interconnected nodes that are only sparsely connected with the rest of the network. Hence, it may be imperative to identify the communities in networks since the communities may have quite different properties such as node degree, clustering coefficient, betweenness, centrality, etc., from that of the average network. Modularity is one such measure, which when maximized, leads to the appearance of communities in a given network.
689:
52:
3412:
3404:
1245:, is rewired randomly with any other stub in the network, even allowing self-loops (which occur when a stub is rewired to another stub from the same node) and multiple-edges between the same two nodes. Thus, even though the node degree distribution of the graph remains intact, the configuration model results in a completely random network.
778:. It is positive if the number of edges within groups exceeds the number expected on the basis of chance. For a given division of the network's vertices into some modules, modularity reflects the concentration of edges within modules compared with random distribution of links between all nodes regardless of modules.
4671:
Although the method of modularity maximization is motivated by computing a deviation from a null model, this deviation is not computed in a statistically consistent manner. Because of this, the method notoriously finds high-scoring communities in its own null model (the configuration model), which by
716:
in networks. Biological networks, including animal brains, exhibit a high degree of modularity. However, modularity maximization is not statistically consistent, and finds communities in its own null model, i.e. fully random graphs, and therefore it cannot be used to find statistically significant
725:
Many scientifically important problems can be represented and empirically studied using networks. For example, biological and social patterns, the World Wide Web, metabolic networks, food webs, neural networks and pathological networks are real world problems that can be mathematically represented
711:
which measures the strength of division of a network into modules (also called groups, clusters or communities). Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Modularity is often used in optimization
4680:
Modularity compares the number of edges inside a cluster with the expected number of edges that one would find in the cluster if the network were a random network with the same number of nodes and where each node keeps its degree, but edges are otherwise randomly attached. This random null model
4709:
in front of the null-case term in the definition of modularity, which controls the relative importance between internal links of the communities and the null model. Optimizing modularity for values of these parameters in their respective appropriate ranges, it is possible to recover the whole
4225:
2458:
for the expected number of edges between two nodes. Additionally, in a large random network, the number of self-loops and multi-edges is vanishingly small. Ignoring self-loops and multi-edges allows one to assume that there is at most one edge between any two nodes. In that case,
2362:
4596:
3140:
3293:
734:
Modularity is the fraction of the edges that fall within the given groups minus the expected fraction if edges were distributed at random. The value of the modularity for unweighted and undirected graphs lies in the range
2912:
holds good for partitioning into two communities only. Hierarchical partitioning (i.e. partitioning into two communities, then the two sub-communities further partitioned into two smaller sub communities only to maximize
2888:
4026:
1166:
is then defined as the fraction of edges that fall within group 1 or 2, minus the expected number of edges within groups 1 and 2 for a random graph with the same node degree distribution as the given network.
4015:
1919:
3390:
4663:, to maximize the modularity. The general form of the modularity for arbitrary numbers of communities is equivalent to a Potts spin glass and similar algorithms can be developed for this case also.
4384:
2080:
2715:
2598:
2456:
2088:
4474:
4672:
definition cannot be statistically significant. Because of this, the method cannot be used to reliably obtain statistically significant community structure in empirical networks.
1484:
717:
community structures in empirical networks. Furthermore, it has been shown that modularity suffers a resolution limit and, therefore, it is unable to detect small communities.
2929:
1651:
1534:
4710:
mesoscale of the network, from the macroscale in which all nodes belong to the same community, to the microscale in which every node forms its own community, hence the name
1439:
1141:
4446:
1095:
1019:
959:
903:
3844:
2487:
1952:
4646:
4275:
1767:
1581:
1375:
1345:
1318:
1239:
1720:
4619:
4466:
4407:
4295:
4248:
3924:
3904:
3884:
3864:
2737:
2646:
2626:
2547:
2527:
2507:
2405:
2385:
1992:
1972:
1787:
1740:
1691:
1671:
1601:
1554:
1395:
1291:
1271:
1212:
1192:
1164:
1059:
1039:
983:
923:
870:
850:
826:
803:
781:
There are different methods for calculating modularity. In the most common version of the concept, the randomization of the edges is done so as to preserve the
776:
3808:
The communities in the graph are represented by the red, green and blue node clusters in Fig 1. The optimal community partitions are depicted in Fig 2.
3181:
568:
5273:
J.M. Kumpula; J. Saramäki; K. Kaski & J. Kertész (2007). "Limited resolution in complex network community detection with Potts model approach".
5043:
2747:
4220:{\displaystyle Q={\frac {1}{2m}}\sum _{vw}\sum _{r}\leftS_{vr}S_{wr}={\frac {1}{2m}}\mathrm {Tr} (\mathbf {S} ^{\mathrm {T} }\mathbf {BS} ),}
2917:) is a possible approach to identify multiple communities in a network. Additionally, (3) can be generalized for partitioning a network into
5326:
Alex Arenas, Alberto Fernández and Sergio Gómez (2008). "Analysis of the structure of complex networks at different resolution levels".
1347:
respectively, from a randomly rewired network as described above. We calculate the expected number of full edges between these nodes.
675:
4929:
4689:
There are two main approaches which try to solve the resolution limit within the modularity context: the addition of a resistance
4389:
All rows and columns of the modularity matrix sum to zero, which means that the modularity of an undivided network is also always
3932:
1795:
3317:
3816:
An alternative formulation of the modularity, useful particularly in spectral optimization algorithms, is as follows. Define
4303:
1997:
5446:
5142:
Guimera, Roger; Sales-Pardo, Marta (August 19, 2004), "Modularity from fluctuations in random graphs and complex networks",
558:
287:
2653:
4722:
There are a couple of software tools available that are able to compute clusterings in graphs with good modularity.
1143:. (It is important to note that multiple edges may exist between two nodes, but here we assess the simplest case).
632:
215:
4714:. However, it has been shown that these methods have limitations when communities are very heterogeneous in size.
5530:
5472:
4652:
2357:{\displaystyle E=E\left=\sum _{i=1}^{k_{v}}E=\sum _{i=1}^{k_{v}}{\frac {k_{w}}{2m-1}}={\frac {k_{v}k_{w}}{2m-1}}}
528:
5499:
5379:
Andrea
Lancichinetti & Santo Fortunato (2011). "Limits of modularity maximization in community detection".
518:
5121:
Peixoto, Tiago P. (2021). "Descriptive vs. inferential community detection: pitfalls, myths and half-truths".
4591:{\displaystyle Q={1 \over 4m}\sum _{vw}B_{vw}s_{v}s_{w}={1 \over 4m}\mathbf {s} ^{\mathrm {T} }\mathbf {Bs} ,}
2552:
2410:
513:
668:
627:
144:
4957:
708:
473:
317:
264:
79:
2367:
Many texts then make the following approximations, for random networks with a large number of edges. When
688:
508:
3135:{\displaystyle Q={\frac {1}{(2m)}}\sum _{vw}\left\delta (c_{v},c_{w})=\sum _{i=1}^{c}(e_{ii}-a_{i}^{2})}
1444:
637:
543:
538:
503:
302:
200:
139:
1606:
1489:
225:
1174:. The configuration model is a randomized realization of a particular network. Given a network with
1097:
means there is an edge between the two. Also for simplicity we consider an undirected network. Thus
1400:
1100:
661:
563:
523:
30:
2489:
becomes a binary indicator variable, so its expected value is also the probability that it equals
421:
5525:
4415:
3400:
We consider an undirected network with 10 nodes and 12 edges and the following adjacency matrix.
644:
463:
230:
164:
119:
4659:, a connection that has been exploited to create simple computer algorithms, for instance using
5060:
Joerg
Reichardt & Stefan Bornholdt (2006). "Statistical mechanics of community detection".
4878:; Delling, D.; Gaertler, M.; Gorke, R.; Hoefer, M.; Nikoloski, Z.; Wagner, D. (February 2008).
829:
548:
533:
448:
1064:
988:
928:
5037:
1241:, the configuration model cuts each edge into two halves, and then each half edge, called a
875:
805:
782:
649:
438:
327:
282:
3819:
2462:
1927:
5398:
5345:
5292:
5227:
5161:
5079:
5001:
4860:
Newman, M. E. J. (2007). Palgrave
Macmillan, Basingstoke (ed.). "Mathematics of networks".
4814:
4624:
4253:
1745:
1559:
1353:
1323:
1296:
1217:
416:
297:
1696:
832:) such that the graph can be partitioned into two communities using a membership variable
8:
5535:
4977:
4918:
4753:
4660:
1171:
713:
453:
322:
312:
307:
159:
104:
94:
5402:
5357:
5349:
5296:
5231:
5165:
5083:
5005:
4879:
4818:
5422:
5388:
5361:
5335:
5308:
5282:
5250:
5217:
5205:
5182:
5151:
5122:
5103:
5069:
5025:
4991:
4899:
4837:
4804:
4792:
4763:
4604:
4451:
4392:
4280:
4233:
3909:
3889:
3869:
3849:
2722:
2631:
2611:
2532:
2512:
2492:
2390:
2370:
1977:
1957:
1772:
1725:
1676:
1656:
1586:
1539:
1380:
1276:
1256:
1197:
1177:
1149:
1044:
1024:
968:
908:
855:
835:
811:
788:
693:
292:
245:
220:
109:
99:
738:
5414:
5255:
5187:
5095:
5017:
4842:
589:
255:
205:
114:
89:
5426:
5365:
3407:
Fig 1. Sample
Network corresponding to the Adjacency matrix with 10 nodes, 12 edges.
2509:, which means one can approximate the probability of an edge existing between nodes
5406:
5353:
5312:
5300:
5245:
5235:
5177:
5169:
5087:
5029:
5009:
4891:
4832:
4822:
4733:
962:
348:
337:
235:
195:
179:
5107:
4903:
5440:
5304:
4748:
3288:{\displaystyle e_{ij}=\sum _{vw}{\frac {A_{vw}}{2m}}1_{v\in c_{i}}1_{w\in c_{j}}}
704:
584:
365:
240:
149:
84:
38:
5410:
5325:
5210:
Proceedings of the
National Academy of Sciences of the United States of America
5173:
5091:
5013:
4797:
Proceedings of the
National Academy of Sciences of the United States of America
4739:
The Vienna Graph
Clustering (VieClus) algorithm, a parallel memetic algorithm.
4726:
594:
400:
375:
370:
344:
333:
210:
174:
169:
129:
67:
5519:
4895:
4875:
4412:
For networks divided into just two communities, one can alternatively define
553:
458:
443:
385:
134:
124:
5240:
4827:
4705:) the aversion of nodes to form communities; or the addition of a parameter
3307:
is the fraction of ends of edges that are attached to vertices in community
5418:
5259:
5191:
5099:
5021:
4846:
498:
395:
250:
5287:
5272:
5156:
5074:
4996:
2883:{\displaystyle Q={\frac {1}{2m}}\sum _{vw}\left{\frac {s_{v}s_{w}+1}{2}}}
5340:
5222:
4809:
4979:
4758:
4656:
433:
390:
380:
20:
5466:
3411:
2608:
Hence, the difference between the actual number of edges between node
1170:
The expected number of edges shall be computed using the concept of a
5493:
4694:
598:
154:
51:
5378:
5127:
2407:
in the denominator above and simply use the approximate expression
1742:
is the number of edges in the original graph), and since there are
5393:
4953:
3403:
5059:
2719:
Summing over all node pairs gives the equation for modularity,
5203:
4982:(2004). "Finding community structure in very large networks".
3167:
is the fraction of edges with one end vertices in community
4874:
3415:
Fig 2. Network partitions that maximize Q. Maximum Q=0.4896
4010:{\displaystyle \delta (c_{v},c_{w})=\sum _{r}S_{vr}S_{wr}}
3395:
1914:{\displaystyle p(I_{i}^{(v,w)}=1)=E={\frac {k_{w}}{2m-1}}}
3385:{\displaystyle a_{i}={\frac {k_{i}}{2m}}=\sum _{j}e_{ij}}
1248:
4379:{\displaystyle B_{vw}=A_{vw}-{\frac {k_{v}k_{w}}{2m}}.}
4297:
is the so-called modularity matrix, which has elements
2075:{\displaystyle J_{vw}=\sum _{i=1}^{k_{v}}I_{i}^{(v,w)}}
1603:
in this particular random graph. If it does not, then
4627:
4607:
4477:
4454:
4418:
4395:
4306:
4283:
4256:
4236:
4029:
3935:
3912:
3892:
3872:
3852:
3822:
3320:
3184:
2932:
2750:
2725:
2656:
2634:
2614:
2555:
2535:
2515:
2495:
2465:
2413:
2393:
2373:
2091:
2000:
1980:
1960:
1930:
1798:
1775:
1748:
1728:
1699:
1679:
1659:
1609:
1589:
1562:
1542:
1492:
1447:
1403:
1383:
1356:
1326:
1299:
1279:
1259:
1220:
1200:
1180:
1152:
1103:
1067:
1047:
1027:
1021:
means there's no edge (no interaction) between nodes
991:
971:
931:
911:
878:
858:
838:
814:
791:
741:
692:
Example of modularity measurement and colouring on a
4884:
IEEE Transactions on
Knowledge and Data Engineering
4736:which additionally avoids unconnected communities.
5141:
4640:
4613:
4590:
4460:
4440:
4401:
4378:
4289:
4269:
4242:
4219:
4009:
3918:
3898:
3878:
3858:
3838:
3384:
3287:
3134:
2882:
2731:
2709:
2640:
2620:
2592:
2541:
2521:
2501:
2481:
2450:
2399:
2379:
2356:
2074:
1986:
1966:
1946:
1913:
1781:
1761:
1734:
1714:
1685:
1665:
1645:
1595:
1575:
1548:
1528:
1478:
1433:
1389:
1369:
1339:
1312:
1285:
1265:
1233:
1206:
1186:
1158:
1135:
1089:
1053:
1033:
1013:
977:
953:
917:
897:
864:
844:
820:
797:
770:
2648:and the expected number of edges between them is
5517:
4859:
4793:"Modularity and community structure in networks"
4790:
5459:
4916:
2710:{\displaystyle A_{vw}-{\frac {k_{v}k_{w}}{2m}}}
5204:Santo Fortunato & Marc Barthelemy (2007).
1722:remaining stubs with equal probability (while
5433:
1769:stubs it can connect to associated with node
669:
5042:: CS1 maint: multiple names: authors list (
2082:, so the expected value of this quantity is
4725:Original implementation of the multi-level
4250:is the (non-square) matrix having elements
4862:The New Palgrave Encyclopedia of Economics
4684:
1556:-th stub happens to connect to one of the
1397:and create associated indicator variables
676:
662:
5442:First implementation of Louvain algorithm
5392:
5339:
5286:
5249:
5239:
5221:
5206:"Resolution limit in community detection"
5181:
5155:
5126:
5073:
4995:
4836:
4826:
4808:
4978:Clauset, Aaron and Newman, M. E. J. and
4971:
4448:to indicate the community to which node
3410:
3402:
2593:{\displaystyle {\frac {k_{v}k_{w}}{2m}}}
2451:{\displaystyle {\frac {k_{v}k_{w}}{2m}}}
687:
5486:
5120:
4786:
4784:
4782:
4780:
4778:
4651:This function has the same form as the
3396:Example of multiple community detection
2387:is large, they drop the subtraction of
5518:
5055:
5053:
4853:
1249:Expected Number of Edges Between Nodes
785:of each vertex. Consider a graph with
16:Measure of network community structure
5475:from the original on 26 November 2020
4910:
3811:
5502:from the original on 21 October 2020
4775:
2923:
2741:
5050:
4675:
4621:is the column vector with elements
13:
5495:Vienna graph clustering repository
4926:Random Graphs and Complex Networks
4571:
4197:
4182:
4179:
965:for the network be represented by
14:
5547:
4717:
1479:{\displaystyle i=1,\ldots ,k_{v}}
703:is a measure of the structure of
4693:to every node, in the form of a
4581:
4578:
4565:
4207:
4204:
4191:
50:
5449:from the original on 2021-03-17
5372:
5319:
5266:
5197:
4960:from the original on 2020-03-05
4935:from the original on 2013-12-18
4917:van der Hofstad, Remco (2013).
1924:The total number of full edges
1646:{\displaystyle I_{i}^{(v,w)}=0}
1529:{\displaystyle I_{i}^{(v,w)}=1}
5135:
5114:
4946:
4868:
4666:
4211:
4186:
3965:
3939:
3129:
3095:
3068:
3042:
3028:
3019:
2954:
2945:
2249:
2244:
2232:
2219:
2175:
2163:
2111:
2095:
2067:
2055:
1877:
1872:
1860:
1847:
1838:
1827:
1815:
1802:
1632:
1620:
1515:
1503:
1426:
1414:
765:
742:
1:
5358:10.1088/1367-2630/10/5/053039
4769:
4468:belongs, which then leads to
2906:It is important to note that
2603:
1434:{\displaystyle I_{i}^{(v,w)}}
1136:{\displaystyle A_{vw}=A_{wv}}
729:
720:
1350:Let us consider each of the
7:
5468:Leiden algorithm repository
5275:European Physical Journal B
4742:
4441:{\displaystyle s_{v}=\pm 1}
3171:and the other in community
3148:
2908:
2896:
10:
5552:
5411:10.1103/PhysRevE.84.066122
5305:10.1140/epjb/e2007-00088-4
5174:10.1103/PhysRevE.70.025101
5092:10.1103/PhysRevE.74.016110
5014:10.1103/PhysRevE.70.066111
4956:. Albert-László Barabási.
4880:"On Modularity Clustering"
1693:can connect to any of the
18:
4791:Newman, M. E. J. (2006).
529:Exponential random (ERGM)
196:Informational (computing)
4896:10.1109/TKDE.2007.190689
1090:{\displaystyle A_{vw}=1}
1014:{\displaystyle A_{vw}=0}
954:{\displaystyle s_{v}=-1}
925:belongs to community 2,
872:belongs to community 1,
216:Scientific collaboration
5241:10.1073/pnas.0605965104
4828:10.1073/pnas.0601602103
4712:multiresolution methods
4685:Multiresolution methods
1253:Now consider two nodes
1194:nodes, where each node
898:{\displaystyle s_{v}=1}
645:Category:Network theory
165:Preferential attachment
5531:Algebraic graph theory
5328:New Journal of Physics
4642:
4615:
4592:
4462:
4442:
4403:
4380:
4291:
4271:
4244:
4221:
4011:
3920:
3900:
3880:
3860:
3840:
3839:{\displaystyle S_{vr}}
3416:
3408:
3386:
3289:
3136:
3094:
2884:
2733:
2711:
2642:
2622:
2594:
2543:
2523:
2503:
2483:
2482:{\displaystyle J_{vw}}
2452:
2401:
2381:
2358:
2282:
2215:
2152:
2076:
2044:
1988:
1968:
1948:
1947:{\displaystyle J_{vw}}
1915:
1783:
1763:
1736:
1716:
1687:
1667:
1647:
1597:
1577:
1550:
1530:
1480:
1435:
1391:
1371:
1341:
1314:
1287:
1267:
1235:
1208:
1188:
1160:
1137:
1091:
1055:
1035:
1015:
979:
955:
919:
899:
866:
846:
822:
799:
772:
712:methods for detecting
697:
534:Random geometric (RGG)
4643:
4641:{\displaystyle s_{v}}
4616:
4593:
4463:
4443:
4404:
4381:
4292:
4272:
4270:{\displaystyle S_{v}}
4245:
4222:
4012:
3921:
3901:
3881:
3861:
3841:
3414:
3406:
3387:
3290:
3137:
3074:
2885:
2734:
2712:
2643:
2623:
2595:
2544:
2524:
2504:
2484:
2453:
2402:
2382:
2359:
2255:
2188:
2125:
2077:
2017:
1989:
1969:
1949:
1916:
1784:
1764:
1762:{\displaystyle k_{w}}
1737:
1717:
1688:
1668:
1648:
1598:
1578:
1576:{\displaystyle k_{w}}
1551:
1531:
1481:
1436:
1392:
1372:
1370:{\displaystyle k_{v}}
1342:
1340:{\displaystyle k_{w}}
1315:
1313:{\displaystyle k_{v}}
1288:
1268:
1236:
1234:{\displaystyle k_{v}}
1209:
1189:
1161:
1138:
1092:
1056:
1036:
1016:
980:
956:
920:
900:
867:
847:
823:
800:
773:
691:
650:Category:Graph theory
5471:, 15 December 2021,
4625:
4605:
4475:
4452:
4416:
4393:
4304:
4281:
4254:
4234:
4027:
3933:
3910:
3890:
3870:
3850:
3820:
3318:
3182:
2930:
2748:
2723:
2654:
2632:
2612:
2553:
2533:
2513:
2493:
2463:
2411:
2391:
2371:
2089:
1998:
1978:
1958:
1928:
1796:
1773:
1746:
1726:
1715:{\displaystyle 2m-1}
1697:
1677:
1657:
1607:
1587:
1560:
1540:
1490:
1445:
1401:
1381:
1354:
1324:
1297:
1293:, with node degrees
1277:
1257:
1218:
1198:
1178:
1150:
1101:
1065:
1045:
1025:
989:
969:
929:
909:
876:
856:
836:
812:
789:
739:
19:For other uses, see
5403:2011PhRvE..84f6122L
5350:2008NJPh...10e3039A
5297:2007EPJB...56...41K
5232:2007PNAS..104...36F
5166:2004PhRvE..70b5101G
5084:2006PhRvE..74a6110R
5006:2004PhRvE..70f6111C
4819:2006PNAS..103.8577N
4754:Community structure
4697:, which increases (
4661:simulated annealing
3128:
2248:
2179:
2071:
1876:
1831:
1636:
1519:
1430:
1172:configuration model
714:community structure
454:Degree distribution
105:Community structure
4764:Percolation theory
4638:
4611:
4588:
4511:
4458:
4438:
4399:
4376:
4287:
4267:
4240:
4217:
4073:
4063:
4007:
3980:
3916:
3896:
3876:
3856:
3836:
3812:Matrix formulation
3417:
3409:
3382:
3368:
3285:
3213:
3132:
3114:
2972:
2880:
2784:
2729:
2707:
2638:
2618:
2590:
2539:
2519:
2499:
2479:
2448:
2397:
2377:
2354:
2222:
2153:
2072:
2045:
1984:
1964:
1944:
1911:
1850:
1805:
1779:
1759:
1732:
1712:
1683:
1663:
1643:
1610:
1593:
1573:
1546:
1526:
1493:
1476:
1431:
1404:
1387:
1367:
1337:
1310:
1283:
1263:
1231:
1214:has a node degree
1204:
1184:
1156:
1133:
1087:
1051:
1031:
1011:
975:
951:
915:
895:
862:
842:
818:
795:
768:
698:
694:scale-free network
638:Network scientists
564:Soft configuration
5498:, 13 April 2021,
5381:Physical Review E
5062:Physical Review E
4980:Moore, Cristopher
4803:(23): 8577–8696.
4614:{\displaystyle s}
4561:
4499:
4497:
4461:{\displaystyle v}
4402:{\displaystyle 0}
4371:
4290:{\displaystyle B}
4243:{\displaystyle S}
4176:
4127:
4064:
4051:
4049:
3971:
3926:otherwise. Then
3919:{\displaystyle 0}
3899:{\displaystyle r}
3886:belongs to group
3879:{\displaystyle v}
3859:{\displaystyle 1}
3806:
3805:
3359:
3354:
3237:
3201:
3156:
3155:
3032:
2960:
2958:
2904:
2903:
2878:
2838:
2772:
2770:
2732:{\displaystyle Q}
2705:
2641:{\displaystyle w}
2621:{\displaystyle v}
2588:
2542:{\displaystyle w}
2522:{\displaystyle v}
2502:{\displaystyle 1}
2446:
2400:{\displaystyle 1}
2380:{\displaystyle m}
2352:
2309:
1987:{\displaystyle w}
1967:{\displaystyle v}
1909:
1782:{\displaystyle w}
1735:{\displaystyle m}
1686:{\displaystyle v}
1673:-th stub of node
1666:{\displaystyle i}
1596:{\displaystyle w}
1549:{\displaystyle i}
1390:{\displaystyle v}
1286:{\displaystyle w}
1266:{\displaystyle v}
1207:{\displaystyle v}
1187:{\displaystyle n}
1159:{\displaystyle Q}
1054:{\displaystyle w}
1034:{\displaystyle v}
978:{\displaystyle A}
918:{\displaystyle v}
865:{\displaystyle v}
845:{\displaystyle s}
821:{\displaystyle m}
798:{\displaystyle n}
686:
685:
606:
605:
514:Bianconi–Barabási
408:
407:
226:Artificial neural
201:Telecommunication
5543:
5511:
5510:
5509:
5507:
5490:
5484:
5483:
5482:
5480:
5463:
5457:
5456:
5455:
5454:
5437:
5431:
5430:
5396:
5376:
5370:
5369:
5343:
5323:
5317:
5316:
5290:
5288:cond-mat/0610370
5270:
5264:
5263:
5253:
5243:
5225:
5201:
5195:
5194:
5185:
5159:
5157:cond-mat/0403660
5139:
5133:
5132:
5130:
5118:
5112:
5111:
5077:
5075:cond-mat/0603718
5057:
5048:
5047:
5041:
5033:
4999:
4997:cond-mat/0408187
4975:
4969:
4968:
4966:
4965:
4954:"NetworkScience"
4950:
4944:
4943:
4941:
4940:
4934:
4923:
4914:
4908:
4907:
4872:
4866:
4865:
4857:
4851:
4850:
4840:
4830:
4812:
4788:
4734:Leiden algorithm
4701:) or decreases (
4676:Resolution limit
4647:
4645:
4644:
4639:
4637:
4636:
4620:
4618:
4617:
4612:
4597:
4595:
4594:
4589:
4584:
4576:
4575:
4574:
4568:
4562:
4560:
4549:
4544:
4543:
4534:
4533:
4524:
4523:
4510:
4498:
4496:
4485:
4467:
4465:
4464:
4459:
4447:
4445:
4444:
4439:
4428:
4427:
4408:
4406:
4405:
4400:
4385:
4383:
4382:
4377:
4372:
4370:
4362:
4361:
4360:
4351:
4350:
4340:
4335:
4334:
4319:
4318:
4296:
4294:
4293:
4288:
4276:
4274:
4273:
4268:
4266:
4265:
4249:
4247:
4246:
4241:
4226:
4224:
4223:
4218:
4210:
4202:
4201:
4200:
4194:
4185:
4177:
4175:
4164:
4159:
4158:
4146:
4145:
4133:
4129:
4128:
4126:
4118:
4117:
4116:
4107:
4106:
4096:
4091:
4090:
4072:
4062:
4050:
4048:
4037:
4016:
4014:
4013:
4008:
4006:
4005:
3993:
3992:
3979:
3964:
3963:
3951:
3950:
3925:
3923:
3922:
3917:
3905:
3903:
3902:
3897:
3885:
3883:
3882:
3877:
3865:
3863:
3862:
3857:
3845:
3843:
3842:
3837:
3835:
3834:
3419:
3418:
3391:
3389:
3388:
3383:
3381:
3380:
3367:
3355:
3353:
3345:
3344:
3335:
3330:
3329:
3294:
3292:
3291:
3286:
3284:
3283:
3282:
3281:
3261:
3260:
3259:
3258:
3238:
3236:
3228:
3227:
3215:
3212:
3197:
3196:
3150:
3141:
3139:
3138:
3133:
3127:
3122:
3110:
3109:
3093:
3088:
3067:
3066:
3054:
3053:
3038:
3034:
3033:
3031:
3017:
3016:
3015:
3006:
3005:
2995:
2990:
2989:
2971:
2959:
2957:
2940:
2924:
2898:
2889:
2887:
2886:
2881:
2879:
2874:
2867:
2866:
2857:
2856:
2846:
2844:
2840:
2839:
2837:
2829:
2828:
2827:
2818:
2817:
2807:
2802:
2801:
2783:
2771:
2769:
2758:
2742:
2738:
2736:
2735:
2730:
2716:
2714:
2713:
2708:
2706:
2704:
2696:
2695:
2694:
2685:
2684:
2674:
2669:
2668:
2647:
2645:
2644:
2639:
2627:
2625:
2624:
2619:
2599:
2597:
2596:
2591:
2589:
2587:
2579:
2578:
2577:
2568:
2567:
2557:
2548:
2546:
2545:
2540:
2528:
2526:
2525:
2520:
2508:
2506:
2505:
2500:
2488:
2486:
2485:
2480:
2478:
2477:
2457:
2455:
2454:
2449:
2447:
2445:
2437:
2436:
2435:
2426:
2425:
2415:
2406:
2404:
2403:
2398:
2386:
2384:
2383:
2378:
2363:
2361:
2360:
2355:
2353:
2351:
2337:
2336:
2335:
2326:
2325:
2315:
2310:
2308:
2294:
2293:
2284:
2281:
2280:
2279:
2269:
2247:
2230:
2214:
2213:
2212:
2202:
2184:
2180:
2178:
2161:
2151:
2150:
2149:
2139:
2110:
2109:
2081:
2079:
2078:
2073:
2070:
2053:
2043:
2042:
2041:
2031:
2013:
2012:
1993:
1991:
1990:
1985:
1973:
1971:
1970:
1965:
1953:
1951:
1950:
1945:
1943:
1942:
1920:
1918:
1917:
1912:
1910:
1908:
1894:
1893:
1884:
1875:
1858:
1830:
1813:
1788:
1786:
1785:
1780:
1768:
1766:
1765:
1760:
1758:
1757:
1741:
1739:
1738:
1733:
1721:
1719:
1718:
1713:
1692:
1690:
1689:
1684:
1672:
1670:
1669:
1664:
1652:
1650:
1649:
1644:
1635:
1618:
1602:
1600:
1599:
1594:
1582:
1580:
1579:
1574:
1572:
1571:
1555:
1553:
1552:
1547:
1535:
1533:
1532:
1527:
1518:
1501:
1485:
1483:
1482:
1477:
1475:
1474:
1440:
1438:
1437:
1432:
1429:
1412:
1396:
1394:
1393:
1388:
1376:
1374:
1373:
1368:
1366:
1365:
1346:
1344:
1343:
1338:
1336:
1335:
1319:
1317:
1316:
1311:
1309:
1308:
1292:
1290:
1289:
1284:
1272:
1270:
1269:
1264:
1240:
1238:
1237:
1232:
1230:
1229:
1213:
1211:
1210:
1205:
1193:
1191:
1190:
1185:
1165:
1163:
1162:
1157:
1142:
1140:
1139:
1134:
1132:
1131:
1116:
1115:
1096:
1094:
1093:
1088:
1080:
1079:
1060:
1058:
1057:
1052:
1040:
1038:
1037:
1032:
1020:
1018:
1017:
1012:
1004:
1003:
984:
982:
981:
976:
963:adjacency matrix
960:
958:
957:
952:
941:
940:
924:
922:
921:
916:
904:
902:
901:
896:
888:
887:
871:
869:
868:
863:
851:
849:
848:
843:
827:
825:
824:
819:
804:
802:
801:
796:
777:
775:
774:
771:{\displaystyle }
769:
755:
678:
671:
664:
549:Stochastic block
539:Hyperbolic (HGN)
488:
487:
351:
340:
272:
271:
180:Social influence
54:
26:
25:
5551:
5550:
5546:
5545:
5544:
5542:
5541:
5540:
5516:
5515:
5514:
5505:
5503:
5492:
5491:
5487:
5478:
5476:
5465:
5464:
5460:
5452:
5450:
5439:
5438:
5434:
5377:
5373:
5341:physics/0703218
5324:
5320:
5271:
5267:
5223:physics/0607100
5202:
5198:
5144:Physical Review
5140:
5136:
5119:
5115:
5058:
5051:
5035:
5034:
4976:
4972:
4963:
4961:
4952:
4951:
4947:
4938:
4936:
4932:
4921:
4915:
4911:
4873:
4869:
4858:
4854:
4810:physics/0602124
4789:
4776:
4772:
4749:Complex network
4745:
4720:
4687:
4678:
4669:
4632:
4628:
4626:
4623:
4622:
4606:
4603:
4602:
4577:
4570:
4569:
4564:
4563:
4553:
4548:
4539:
4535:
4529:
4525:
4516:
4512:
4503:
4489:
4484:
4476:
4473:
4472:
4453:
4450:
4449:
4423:
4419:
4417:
4414:
4413:
4394:
4391:
4390:
4363:
4356:
4352:
4346:
4342:
4341:
4339:
4327:
4323:
4311:
4307:
4305:
4302:
4301:
4282:
4279:
4278:
4261:
4257:
4255:
4252:
4251:
4235:
4232:
4231:
4203:
4196:
4195:
4190:
4189:
4178:
4168:
4163:
4151:
4147:
4138:
4134:
4119:
4112:
4108:
4102:
4098:
4097:
4095:
4083:
4079:
4078:
4074:
4068:
4055:
4041:
4036:
4028:
4025:
4024:
3998:
3994:
3985:
3981:
3975:
3959:
3955:
3946:
3942:
3934:
3931:
3930:
3911:
3908:
3907:
3891:
3888:
3887:
3871:
3868:
3867:
3851:
3848:
3847:
3827:
3823:
3821:
3818:
3817:
3814:
3398:
3373:
3369:
3363:
3346:
3340:
3336:
3334:
3325:
3321:
3319:
3316:
3315:
3306:
3277:
3273:
3266:
3262:
3254:
3250:
3243:
3239:
3229:
3220:
3216:
3214:
3205:
3189:
3185:
3183:
3180:
3179:
3166:
3123:
3118:
3102:
3098:
3089:
3078:
3062:
3058:
3049:
3045:
3018:
3011:
3007:
3001:
2997:
2996:
2994:
2982:
2978:
2977:
2973:
2964:
2944:
2939:
2931:
2928:
2927:
2862:
2858:
2852:
2848:
2847:
2845:
2830:
2823:
2819:
2813:
2809:
2808:
2806:
2794:
2790:
2789:
2785:
2776:
2762:
2757:
2749:
2746:
2745:
2724:
2721:
2720:
2697:
2690:
2686:
2680:
2676:
2675:
2673:
2661:
2657:
2655:
2652:
2651:
2633:
2630:
2629:
2613:
2610:
2609:
2606:
2580:
2573:
2569:
2563:
2559:
2558:
2556:
2554:
2551:
2550:
2534:
2531:
2530:
2514:
2511:
2510:
2494:
2491:
2490:
2470:
2466:
2464:
2461:
2460:
2438:
2431:
2427:
2421:
2417:
2416:
2414:
2412:
2409:
2408:
2392:
2389:
2388:
2372:
2369:
2368:
2338:
2331:
2327:
2321:
2317:
2316:
2314:
2295:
2289:
2285:
2283:
2275:
2271:
2270:
2259:
2231:
2226:
2208:
2204:
2203:
2192:
2162:
2157:
2145:
2141:
2140:
2129:
2124:
2120:
2102:
2098:
2090:
2087:
2086:
2054:
2049:
2037:
2033:
2032:
2021:
2005:
2001:
1999:
1996:
1995:
1979:
1976:
1975:
1959:
1956:
1955:
1935:
1931:
1929:
1926:
1925:
1895:
1889:
1885:
1883:
1859:
1854:
1814:
1809:
1797:
1794:
1793:
1774:
1771:
1770:
1753:
1749:
1747:
1744:
1743:
1727:
1724:
1723:
1698:
1695:
1694:
1678:
1675:
1674:
1658:
1655:
1654:
1619:
1614:
1608:
1605:
1604:
1588:
1585:
1584:
1567:
1563:
1561:
1558:
1557:
1541:
1538:
1537:
1502:
1497:
1491:
1488:
1487:
1470:
1466:
1446:
1443:
1442:
1413:
1408:
1402:
1399:
1398:
1382:
1379:
1378:
1361:
1357:
1355:
1352:
1351:
1331:
1327:
1325:
1322:
1321:
1304:
1300:
1298:
1295:
1294:
1278:
1275:
1274:
1258:
1255:
1254:
1251:
1225:
1221:
1219:
1216:
1215:
1199:
1196:
1195:
1179:
1176:
1175:
1151:
1148:
1147:
1124:
1120:
1108:
1104:
1102:
1099:
1098:
1072:
1068:
1066:
1063:
1062:
1046:
1043:
1042:
1026:
1023:
1022:
996:
992:
990:
987:
986:
970:
967:
966:
936:
932:
930:
927:
926:
910:
907:
906:
883:
879:
877:
874:
873:
857:
854:
853:
837:
834:
833:
813:
810:
809:
790:
787:
786:
751:
740:
737:
736:
732:
723:
682:
620:
585:Boolean network
559:Maximum entropy
509:Barabási–Albert
426:
343:
332:
120:Controllability
85:Complex network
72:
59:
58:
57:
56:
55:
39:Network science
24:
17:
12:
11:
5:
5549:
5539:
5538:
5533:
5528:
5526:Network theory
5513:
5512:
5485:
5458:
5432:
5371:
5318:
5265:
5196:
5134:
5113:
5049:
4970:
4945:
4909:
4890:(2): 172–188.
4867:
4852:
4773:
4771:
4768:
4767:
4766:
4761:
4756:
4751:
4744:
4741:
4727:Louvain method
4719:
4718:Software Tools
4716:
4686:
4683:
4677:
4674:
4668:
4665:
4635:
4631:
4610:
4599:
4598:
4587:
4583:
4580:
4573:
4567:
4559:
4556:
4552:
4547:
4542:
4538:
4532:
4528:
4522:
4519:
4515:
4509:
4506:
4502:
4495:
4492:
4488:
4483:
4480:
4457:
4437:
4434:
4431:
4426:
4422:
4398:
4387:
4386:
4375:
4369:
4366:
4359:
4355:
4349:
4345:
4338:
4333:
4330:
4326:
4322:
4317:
4314:
4310:
4286:
4264:
4260:
4239:
4228:
4227:
4216:
4213:
4209:
4206:
4199:
4193:
4188:
4184:
4181:
4174:
4171:
4167:
4162:
4157:
4154:
4150:
4144:
4141:
4137:
4132:
4125:
4122:
4115:
4111:
4105:
4101:
4094:
4089:
4086:
4082:
4077:
4071:
4067:
4061:
4058:
4054:
4047:
4044:
4040:
4035:
4032:
4018:
4017:
4004:
4001:
3997:
3991:
3988:
3984:
3978:
3974:
3970:
3967:
3962:
3958:
3954:
3949:
3945:
3941:
3938:
3915:
3895:
3875:
3855:
3833:
3830:
3826:
3813:
3810:
3804:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3769:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3734:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3703:
3699:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3664:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3629:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3594:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3559:
3558:
3555:
3552:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3524:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3496:
3493:
3489:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3454:
3453:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3397:
3394:
3393:
3392:
3379:
3376:
3372:
3366:
3362:
3358:
3352:
3349:
3343:
3339:
3333:
3328:
3324:
3302:
3296:
3295:
3280:
3276:
3272:
3269:
3265:
3257:
3253:
3249:
3246:
3242:
3235:
3232:
3226:
3223:
3219:
3211:
3208:
3204:
3200:
3195:
3192:
3188:
3162:
3154:
3153:
3144:
3142:
3131:
3126:
3121:
3117:
3113:
3108:
3105:
3101:
3097:
3092:
3087:
3084:
3081:
3077:
3073:
3070:
3065:
3061:
3057:
3052:
3048:
3044:
3041:
3037:
3030:
3027:
3024:
3021:
3014:
3010:
3004:
3000:
2993:
2988:
2985:
2981:
2976:
2970:
2967:
2963:
2956:
2953:
2950:
2947:
2943:
2938:
2935:
2902:
2901:
2892:
2890:
2877:
2873:
2870:
2865:
2861:
2855:
2851:
2843:
2836:
2833:
2826:
2822:
2816:
2812:
2805:
2800:
2797:
2793:
2788:
2782:
2779:
2775:
2768:
2765:
2761:
2756:
2753:
2728:
2703:
2700:
2693:
2689:
2683:
2679:
2672:
2667:
2664:
2660:
2637:
2617:
2605:
2602:
2586:
2583:
2576:
2572:
2566:
2562:
2538:
2518:
2498:
2476:
2473:
2469:
2444:
2441:
2434:
2430:
2424:
2420:
2396:
2376:
2365:
2364:
2350:
2347:
2344:
2341:
2334:
2330:
2324:
2320:
2313:
2307:
2304:
2301:
2298:
2292:
2288:
2278:
2274:
2268:
2265:
2262:
2258:
2254:
2251:
2246:
2243:
2240:
2237:
2234:
2229:
2225:
2221:
2218:
2211:
2207:
2201:
2198:
2195:
2191:
2187:
2183:
2177:
2174:
2171:
2168:
2165:
2160:
2156:
2148:
2144:
2138:
2135:
2132:
2128:
2123:
2119:
2116:
2113:
2108:
2105:
2101:
2097:
2094:
2069:
2066:
2063:
2060:
2057:
2052:
2048:
2040:
2036:
2030:
2027:
2024:
2020:
2016:
2011:
2008:
2004:
1983:
1963:
1941:
1938:
1934:
1922:
1921:
1907:
1904:
1901:
1898:
1892:
1888:
1882:
1879:
1874:
1871:
1868:
1865:
1862:
1857:
1853:
1849:
1846:
1843:
1840:
1837:
1834:
1829:
1826:
1823:
1820:
1817:
1812:
1808:
1804:
1801:
1778:
1756:
1752:
1731:
1711:
1708:
1705:
1702:
1682:
1662:
1642:
1639:
1634:
1631:
1628:
1625:
1622:
1617:
1613:
1592:
1583:stubs of node
1570:
1566:
1545:
1525:
1522:
1517:
1514:
1511:
1508:
1505:
1500:
1496:
1473:
1469:
1465:
1462:
1459:
1456:
1453:
1450:
1428:
1425:
1422:
1419:
1416:
1411:
1407:
1386:
1377:stubs of node
1364:
1360:
1334:
1330:
1307:
1303:
1282:
1262:
1250:
1247:
1228:
1224:
1203:
1183:
1155:
1130:
1127:
1123:
1119:
1114:
1111:
1107:
1086:
1083:
1078:
1075:
1071:
1050:
1030:
1010:
1007:
1002:
999:
995:
974:
950:
947:
944:
939:
935:
914:
894:
891:
886:
882:
861:
841:
817:
794:
767:
764:
761:
758:
754:
750:
747:
744:
731:
728:
722:
719:
684:
683:
681:
680:
673:
666:
658:
655:
654:
653:
652:
647:
641:
640:
635:
630:
622:
621:
619:
618:
615:
611:
608:
607:
604:
603:
602:
601:
592:
587:
579:
578:
574:
573:
572:
571:
566:
561:
556:
551:
546:
541:
536:
531:
526:
524:Watts–Strogatz
521:
516:
511:
506:
501:
493:
492:
484:
483:
479:
478:
477:
476:
471:
466:
461:
456:
451:
446:
441:
436:
428:
427:
425:
424:
419:
413:
410:
409:
406:
405:
404:
403:
398:
393:
388:
383:
378:
373:
368:
360:
359:
355:
354:
353:
352:
345:Incidence list
341:
334:Adjacency list
330:
325:
320:
315:
310:
305:
303:Data structure
300:
295:
290:
285:
277:
276:
268:
267:
261:
260:
259:
258:
253:
248:
243:
238:
233:
231:Interdependent
228:
223:
218:
213:
208:
203:
198:
190:
189:
185:
184:
183:
182:
177:
175:Network effect
172:
170:Balance theory
167:
162:
157:
152:
147:
142:
137:
132:
130:Social capital
127:
122:
117:
112:
107:
102:
97:
92:
87:
82:
74:
73:
71:
70:
64:
61:
60:
49:
48:
47:
46:
45:
42:
41:
35:
34:
15:
9:
6:
4:
3:
2:
5548:
5537:
5534:
5532:
5529:
5527:
5524:
5523:
5521:
5501:
5497:
5496:
5489:
5474:
5470:
5469:
5462:
5448:
5444:
5443:
5436:
5428:
5424:
5420:
5416:
5412:
5408:
5404:
5400:
5395:
5390:
5387:(6): 066122.
5386:
5382:
5375:
5367:
5363:
5359:
5355:
5351:
5347:
5342:
5337:
5334:(5): 053039.
5333:
5329:
5322:
5314:
5310:
5306:
5302:
5298:
5294:
5289:
5284:
5280:
5276:
5269:
5261:
5257:
5252:
5247:
5242:
5237:
5233:
5229:
5224:
5219:
5215:
5211:
5207:
5200:
5193:
5189:
5184:
5179:
5175:
5171:
5167:
5163:
5158:
5153:
5150:(2): 025101,
5149:
5145:
5138:
5129:
5124:
5117:
5109:
5105:
5101:
5097:
5093:
5089:
5085:
5081:
5076:
5071:
5068:(1): 016110.
5067:
5063:
5056:
5054:
5045:
5039:
5031:
5027:
5023:
5019:
5015:
5011:
5007:
5003:
4998:
4993:
4990:(6): 066111.
4989:
4985:
4981:
4974:
4959:
4955:
4949:
4931:
4927:
4920:
4913:
4905:
4901:
4897:
4893:
4889:
4885:
4881:
4877:
4871:
4864:(2 ed.).
4863:
4856:
4848:
4844:
4839:
4834:
4829:
4824:
4820:
4816:
4811:
4806:
4802:
4798:
4794:
4787:
4785:
4783:
4781:
4779:
4774:
4765:
4762:
4760:
4757:
4755:
4752:
4750:
4747:
4746:
4740:
4737:
4735:
4730:
4728:
4723:
4715:
4713:
4708:
4704:
4700:
4696:
4692:
4682:
4673:
4664:
4662:
4658:
4654:
4649:
4633:
4629:
4608:
4585:
4557:
4554:
4550:
4545:
4540:
4536:
4530:
4526:
4520:
4517:
4513:
4507:
4504:
4500:
4493:
4490:
4486:
4481:
4478:
4471:
4470:
4469:
4455:
4435:
4432:
4429:
4424:
4420:
4410:
4396:
4373:
4367:
4364:
4357:
4353:
4347:
4343:
4336:
4331:
4328:
4324:
4320:
4315:
4312:
4308:
4300:
4299:
4298:
4284:
4262:
4258:
4237:
4214:
4172:
4169:
4165:
4160:
4155:
4152:
4148:
4142:
4139:
4135:
4130:
4123:
4120:
4113:
4109:
4103:
4099:
4092:
4087:
4084:
4080:
4075:
4069:
4065:
4059:
4056:
4052:
4045:
4042:
4038:
4033:
4030:
4023:
4022:
4021:
4002:
3999:
3995:
3989:
3986:
3982:
3976:
3972:
3968:
3960:
3956:
3952:
3947:
3943:
3936:
3929:
3928:
3927:
3913:
3893:
3873:
3853:
3831:
3828:
3824:
3809:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3770:
3766:
3763:
3760:
3757:
3754:
3751:
3748:
3745:
3742:
3739:
3736:
3735:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3700:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3669:
3666:
3665:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3631:
3630:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3595:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3560:
3556:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3525:
3521:
3518:
3515:
3512:
3509:
3506:
3503:
3500:
3497:
3494:
3491:
3490:
3486:
3483:
3480:
3477:
3474:
3471:
3468:
3465:
3462:
3459:
3456:
3455:
3451:
3448:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3420:
3413:
3405:
3401:
3377:
3374:
3370:
3364:
3360:
3356:
3350:
3347:
3341:
3337:
3331:
3326:
3322:
3314:
3313:
3312:
3310:
3305:
3301:
3278:
3274:
3270:
3267:
3263:
3255:
3251:
3247:
3244:
3240:
3233:
3230:
3224:
3221:
3217:
3209:
3206:
3202:
3198:
3193:
3190:
3186:
3178:
3177:
3176:
3174:
3170:
3165:
3161:
3152:
3145:
3143:
3124:
3119:
3115:
3111:
3106:
3103:
3099:
3090:
3085:
3082:
3079:
3075:
3071:
3063:
3059:
3055:
3050:
3046:
3039:
3035:
3025:
3022:
3012:
3008:
3002:
2998:
2991:
2986:
2983:
2979:
2974:
2968:
2965:
2961:
2951:
2948:
2941:
2936:
2933:
2926:
2925:
2922:
2921:communities.
2920:
2916:
2911:
2910:
2900:
2893:
2891:
2875:
2871:
2868:
2863:
2859:
2853:
2849:
2841:
2834:
2831:
2824:
2820:
2814:
2810:
2803:
2798:
2795:
2791:
2786:
2780:
2777:
2773:
2766:
2763:
2759:
2754:
2751:
2744:
2743:
2740:
2726:
2717:
2701:
2698:
2691:
2687:
2681:
2677:
2670:
2665:
2662:
2658:
2649:
2635:
2615:
2601:
2584:
2581:
2574:
2570:
2564:
2560:
2536:
2516:
2496:
2474:
2471:
2467:
2442:
2439:
2432:
2428:
2422:
2418:
2394:
2374:
2348:
2345:
2342:
2339:
2332:
2328:
2322:
2318:
2311:
2305:
2302:
2299:
2296:
2290:
2286:
2276:
2272:
2266:
2263:
2260:
2256:
2252:
2241:
2238:
2235:
2227:
2223:
2216:
2209:
2205:
2199:
2196:
2193:
2189:
2185:
2181:
2172:
2169:
2166:
2158:
2154:
2146:
2142:
2136:
2133:
2130:
2126:
2121:
2117:
2114:
2106:
2103:
2099:
2092:
2085:
2084:
2083:
2064:
2061:
2058:
2050:
2046:
2038:
2034:
2028:
2025:
2022:
2018:
2014:
2009:
2006:
2002:
1981:
1961:
1939:
1936:
1932:
1905:
1902:
1899:
1896:
1890:
1886:
1880:
1869:
1866:
1863:
1855:
1851:
1844:
1841:
1835:
1832:
1824:
1821:
1818:
1810:
1806:
1799:
1792:
1791:
1790:
1776:
1754:
1750:
1729:
1709:
1706:
1703:
1700:
1680:
1660:
1640:
1637:
1629:
1626:
1623:
1615:
1611:
1590:
1568:
1564:
1543:
1523:
1520:
1512:
1509:
1506:
1498:
1494:
1471:
1467:
1463:
1460:
1457:
1454:
1451:
1448:
1423:
1420:
1417:
1409:
1405:
1384:
1362:
1358:
1348:
1332:
1328:
1305:
1301:
1280:
1260:
1246:
1244:
1226:
1222:
1201:
1181:
1173:
1168:
1153:
1144:
1128:
1125:
1121:
1117:
1112:
1109:
1105:
1084:
1081:
1076:
1073:
1069:
1048:
1028:
1008:
1005:
1000:
997:
993:
972:
964:
948:
945:
942:
937:
933:
912:
892:
889:
884:
880:
859:
839:
831:
815:
807:
792:
784:
779:
762:
759:
756:
752:
748:
745:
727:
718:
715:
710:
706:
702:
695:
690:
679:
674:
672:
667:
665:
660:
659:
657:
656:
651:
648:
646:
643:
642:
639:
636:
634:
631:
629:
626:
625:
624:
623:
616:
613:
612:
610:
609:
600:
596:
593:
591:
588:
586:
583:
582:
581:
580:
576:
575:
570:
569:LFR Benchmark
567:
565:
562:
560:
557:
555:
554:Blockmodeling
552:
550:
547:
545:
542:
540:
537:
535:
532:
530:
527:
525:
522:
520:
519:Fitness model
517:
515:
512:
510:
507:
505:
502:
500:
497:
496:
495:
494:
490:
489:
486:
485:
481:
480:
475:
472:
470:
467:
465:
462:
460:
459:Assortativity
457:
455:
452:
450:
447:
445:
442:
440:
437:
435:
432:
431:
430:
429:
423:
420:
418:
415:
414:
412:
411:
402:
399:
397:
394:
392:
389:
387:
384:
382:
379:
377:
374:
372:
369:
367:
364:
363:
362:
361:
357:
356:
350:
346:
342:
339:
335:
331:
329:
326:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
280:
279:
278:
274:
273:
270:
269:
266:
263:
262:
257:
254:
252:
249:
247:
244:
242:
239:
237:
234:
232:
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
193:
192:
191:
188:Network types
187:
186:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
136:
135:Link analysis
133:
131:
128:
126:
125:Graph drawing
123:
121:
118:
116:
113:
111:
108:
106:
103:
101:
98:
96:
93:
91:
88:
86:
83:
81:
78:
77:
76:
75:
69:
66:
65:
63:
62:
53:
44:
43:
40:
37:
36:
32:
28:
27:
22:
5504:, retrieved
5494:
5488:
5477:, retrieved
5467:
5461:
5451:, retrieved
5441:
5435:
5384:
5380:
5374:
5331:
5327:
5321:
5281:(1): 41–45.
5278:
5274:
5268:
5216:(1): 36–41.
5213:
5209:
5199:
5147:
5143:
5137:
5116:
5065:
5061:
5038:cite journal
4987:
4984:Phys. Rev. E
4983:
4973:
4962:. Retrieved
4948:
4937:. Retrieved
4925:
4912:
4887:
4883:
4870:
4861:
4855:
4800:
4796:
4738:
4731:
4724:
4721:
4711:
4706:
4702:
4698:
4690:
4688:
4679:
4670:
4655:of an Ising
4650:
4600:
4411:
4388:
4229:
4019:
3815:
3807:
3399:
3308:
3303:
3299:
3297:
3172:
3168:
3163:
3159:
3157:
3146:
2918:
2914:
2907:
2905:
2894:
2718:
2650:
2607:
2366:
1923:
1789:, evidently
1653:. Since the
1349:
1252:
1242:
1169:
1145:
852:. If a node
780:
733:
724:
700:
699:
544:Hierarchical
499:Random graph
468:
347: /
336: /
318:Neighborhood
160:Transitivity
140:Optimization
5506:30 November
5479:30 November
4919:"Chapter 7"
4876:Brandes, U.
4667:Overfitting
4653:Hamiltonian
1146:Modularity
590:agent based
504:Erdős–Rényi
145:Reciprocity
110:Percolation
95:Small-world
5536:Modularity
5520:Categories
5453:2020-11-30
5128:2112.00183
4964:2020-03-20
4939:2013-12-08
4770:References
4759:Null model
4657:spin glass
4020:and hence
3866:if vertex
2604:Modularity
1441:for them,
961:. Let the
730:Definition
721:Motivation
701:Modularity
617:Categories
474:Efficiency
469:Modularity
449:Clustering
434:Centrality
422:Algorithms
246:Dependency
221:Biological
100:Scale-free
21:Modularity
5394:1107.1155
4695:self-loop
4501:∑
4433:±
4337:−
4093:−
4066:∑
4053:∑
3973:∑
3937:δ
3361:∑
3271:∈
3248:∈
3203:∑
3112:−
3076:∑
3040:δ
2992:−
2962:∑
2804:−
2774:∑
2671:−
2346:−
2303:−
2257:∑
2190:∑
2127:∑
2019:∑
1903:−
1707:−
1461:…
946:−
746:−
366:Bipartite
288:Component
206:Transport
155:Homophily
115:Evolution
90:Contagion
5500:archived
5473:archived
5447:archived
5427:16180375
5419:22304170
5366:11544197
5260:17190818
5192:15447530
5100:16907154
5022:15697438
4958:Archived
4930:Archived
4847:16723398
4743:See also
1994:is just
1954:between
985:, where
905:, or if
705:networks
633:Software
595:Epidemic
577:Dynamics
491:Topology
464:Distance
401:Weighted
376:Directed
371:Complete
275:Features
236:Semantic
31:a series
29:Part of
5399:Bibcode
5346:Bibcode
5313:4411525
5293:Bibcode
5251:1765466
5228:Bibcode
5183:2441765
5162:Bibcode
5080:Bibcode
5030:8977721
5002:Bibcode
4838:1482622
4815:Bibcode
3422:Node ID
1536:if the
1486:, with
828:links (
417:Metrics
386:Labeled
256:on-Chip
241:Spatial
150:Closure
5425:
5417:
5364:
5311:
5258:
5248:
5190:
5180:
5108:792965
5106:
5098:
5028:
5020:
4904:150684
4902:
4845:
4835:
4707:Îł>0
4703:r<0
4699:r>0
4601:where
4230:where
3846:to be
3158:where
783:degree
709:graphs
628:Topics
482:Models
439:Degree
396:Random
349:matrix
338:matrix
328:Vertex
283:Clique
265:Graphs
211:Social
68:Theory
5423:S2CID
5389:arXiv
5362:S2CID
5336:arXiv
5309:S2CID
5283:arXiv
5218:arXiv
5152:arXiv
5123:arXiv
5104:S2CID
5070:arXiv
5026:S2CID
4992:arXiv
4933:(PDF)
4922:(PDF)
4900:S2CID
4805:arXiv
2909:Eq. 3
830:edges
806:nodes
614:Lists
444:Motif
391:Multi
381:Hyper
358:Types
298:Cycle
80:Graph
5508:2020
5481:2020
5415:PMID
5256:PMID
5188:PMID
5096:PMID
5044:link
5018:PMID
4843:PMID
4732:The
4277:and
3906:and
3298:and
2628:and
2529:and
1974:and
1320:and
1273:and
1243:stub
1061:and
1041:and
808:and
323:Path
313:Loop
308:Edge
251:Flow
5407:doi
5354:doi
5301:doi
5246:PMC
5236:doi
5214:104
5178:PMC
5170:doi
5088:doi
5010:doi
4892:doi
4833:PMC
4823:doi
4801:103
3452:10
2549:as
707:or
599:SIR
293:Cut
5522::
5445:,
5421:.
5413:.
5405:.
5397:.
5385:84
5383:.
5360:.
5352:.
5344:.
5332:10
5330:.
5307:.
5299:.
5291:.
5279:56
5277:.
5254:.
5244:.
5234:.
5226:.
5212:.
5208:.
5186:,
5176:,
5168:,
5160:,
5148:70
5146:,
5102:.
5094:.
5086:.
5078:.
5066:74
5064:.
5052:^
5040:}}
5036:{{
5024:.
5016:.
5008:.
5000:.
4988:70
4986:.
4928:.
4924:.
4898:.
4888:20
4886:.
4882:.
4841:.
4831:.
4821:.
4813:.
4799:.
4795:.
4777:^
4729:.
4648:.
4409:.
3802:0
3772:10
3767:0
3732:0
3697:1
3662:0
3627:0
3592:1
3557:0
3522:0
3487:1
3311::
3175::
3164:ij
2739:.
2600:.
33:on
5429:.
5409::
5401::
5391::
5368:.
5356::
5348::
5338::
5315:.
5303::
5295::
5285::
5262:.
5238::
5230::
5220::
5172::
5164::
5154::
5131:.
5125::
5110:.
5090::
5082::
5072::
5046:)
5032:.
5012::
5004::
4994::
4967:.
4942:.
4906:.
4894::
4849:.
4825::
4817::
4807::
4691:r
4634:v
4630:s
4609:s
4586:,
4582:s
4579:B
4572:T
4566:s
4558:m
4555:4
4551:1
4546:=
4541:w
4537:s
4531:v
4527:s
4521:w
4518:v
4514:B
4508:w
4505:v
4494:m
4491:4
4487:1
4482:=
4479:Q
4456:v
4436:1
4430:=
4425:v
4421:s
4397:0
4374:.
4368:m
4365:2
4358:w
4354:k
4348:v
4344:k
4332:w
4329:v
4325:A
4321:=
4316:w
4313:v
4309:B
4285:B
4263:v
4259:S
4238:S
4215:,
4212:)
4208:S
4205:B
4198:T
4192:S
4187:(
4183:r
4180:T
4173:m
4170:2
4166:1
4161:=
4156:r
4153:w
4149:S
4143:r
4140:v
4136:S
4131:]
4124:m
4121:2
4114:w
4110:k
4104:v
4100:k
4088:w
4085:v
4081:A
4076:[
4070:r
4060:w
4057:v
4046:m
4043:2
4039:1
4034:=
4031:Q
4003:r
4000:w
3996:S
3990:r
3987:v
3983:S
3977:r
3969:=
3966:)
3961:w
3957:c
3953:,
3948:v
3944:c
3940:(
3914:0
3894:r
3874:v
3854:1
3832:r
3829:v
3825:S
3799:0
3796:0
3793:1
3790:0
3787:0
3784:1
3781:0
3778:0
3775:1
3764:0
3761:1
3758:1
3755:0
3752:0
3749:0
3746:0
3743:0
3740:0
3737:9
3729:1
3726:0
3723:1
3720:0
3717:0
3714:0
3711:0
3708:0
3705:0
3702:8
3694:1
3691:1
3688:0
3685:0
3682:0
3679:0
3676:0
3673:0
3670:0
3667:7
3659:0
3656:0
3653:0
3650:0
3647:1
3644:1
3641:0
3638:0
3635:0
3632:6
3624:0
3621:0
3618:0
3615:1
3612:0
3609:1
3606:0
3603:0
3600:0
3597:5
3589:0
3586:0
3583:0
3580:1
3577:1
3574:0
3571:0
3568:0
3565:0
3562:4
3554:0
3551:0
3548:0
3545:0
3542:0
3539:0
3536:0
3533:1
3530:1
3527:3
3519:0
3516:0
3513:0
3510:0
3507:0
3504:0
3501:1
3498:0
3495:1
3492:2
3484:0
3481:0
3478:0
3475:0
3472:0
3469:0
3466:1
3463:1
3460:0
3457:1
3449:9
3446:8
3443:7
3440:6
3437:5
3434:4
3431:3
3428:2
3425:1
3378:j
3375:i
3371:e
3365:j
3357:=
3351:m
3348:2
3342:i
3338:k
3332:=
3327:i
3323:a
3309:i
3304:i
3300:a
3279:j
3275:c
3268:w
3264:1
3256:i
3252:c
3245:v
3241:1
3234:m
3231:2
3225:w
3222:v
3218:A
3210:w
3207:v
3199:=
3194:j
3191:i
3187:e
3173:j
3169:i
3160:e
3151:)
3149:4
3147:(
3130:)
3125:2
3120:i
3116:a
3107:i
3104:i
3100:e
3096:(
3091:c
3086:1
3083:=
3080:i
3072:=
3069:)
3064:w
3060:c
3056:,
3051:v
3047:c
3043:(
3036:]
3029:)
3026:m
3023:2
3020:(
3013:w
3009:k
3003:v
2999:k
2987:w
2984:v
2980:A
2975:[
2969:w
2966:v
2955:)
2952:m
2949:2
2946:(
2942:1
2937:=
2934:Q
2919:c
2915:Q
2899:)
2897:3
2895:(
2876:2
2872:1
2869:+
2864:w
2860:s
2854:v
2850:s
2842:]
2835:m
2832:2
2825:w
2821:k
2815:v
2811:k
2799:w
2796:v
2792:A
2787:[
2781:w
2778:v
2767:m
2764:2
2760:1
2755:=
2752:Q
2727:Q
2702:m
2699:2
2692:w
2688:k
2682:v
2678:k
2666:w
2663:v
2659:A
2636:w
2616:v
2585:m
2582:2
2575:w
2571:k
2565:v
2561:k
2537:w
2517:v
2497:1
2475:w
2472:v
2468:J
2443:m
2440:2
2433:w
2429:k
2423:v
2419:k
2395:1
2375:m
2349:1
2343:m
2340:2
2333:w
2329:k
2323:v
2319:k
2312:=
2306:1
2300:m
2297:2
2291:w
2287:k
2277:v
2273:k
2267:1
2264:=
2261:i
2253:=
2250:]
2245:)
2242:w
2239:,
2236:v
2233:(
2228:i
2224:I
2220:[
2217:E
2210:v
2206:k
2200:1
2197:=
2194:i
2186:=
2182:]
2176:)
2173:w
2170:,
2167:v
2164:(
2159:i
2155:I
2147:v
2143:k
2137:1
2134:=
2131:i
2122:[
2118:E
2115:=
2112:]
2107:w
2104:v
2100:J
2096:[
2093:E
2068:)
2065:w
2062:,
2059:v
2056:(
2051:i
2047:I
2039:v
2035:k
2029:1
2026:=
2023:i
2015:=
2010:w
2007:v
2003:J
1982:w
1962:v
1940:w
1937:v
1933:J
1906:1
1900:m
1897:2
1891:w
1887:k
1881:=
1878:]
1873:)
1870:w
1867:,
1864:v
1861:(
1856:i
1852:I
1848:[
1845:E
1842:=
1839:)
1836:1
1833:=
1828:)
1825:w
1822:,
1819:v
1816:(
1811:i
1807:I
1803:(
1800:p
1777:w
1755:w
1751:k
1730:m
1710:1
1704:m
1701:2
1681:v
1661:i
1641:0
1638:=
1633:)
1630:w
1627:,
1624:v
1621:(
1616:i
1612:I
1591:w
1569:w
1565:k
1544:i
1524:1
1521:=
1516:)
1513:w
1510:,
1507:v
1504:(
1499:i
1495:I
1472:v
1468:k
1464:,
1458:,
1455:1
1452:=
1449:i
1427:)
1424:w
1421:,
1418:v
1415:(
1410:i
1406:I
1385:v
1363:v
1359:k
1333:w
1329:k
1306:v
1302:k
1281:w
1261:v
1227:v
1223:k
1202:v
1182:n
1154:Q
1129:v
1126:w
1122:A
1118:=
1113:w
1110:v
1106:A
1085:1
1082:=
1077:w
1074:v
1070:A
1049:w
1029:v
1009:0
1006:=
1001:w
998:v
994:A
973:A
949:1
943:=
938:v
934:s
913:v
893:1
890:=
885:v
881:s
860:v
840:s
816:m
793:n
766:]
763:1
760:,
757:2
753:/
749:1
743:[
696:.
677:e
670:t
663:v
597:/
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
