1340:
1314:. Many simple parameters exist to describe a static network (number of nodes, edges, path length, connected components), or to describe specific nodes in the graph such as the number of links or the clustering coefficient. These properties can then individually be studied as a time series using signal processing notions. For example, we can track the number of links established to a server per minute by looking at the successive snapshots of the network and counting these links in each snapshot.
53:
690:
1302:
display varying levels of rationality, improving the overall system rationality might be an evolutionary reason for the emergence of scale-free networks. They demonstrated this by applying evolutionary pressure on an initially random network which simulates a range of classic games, so that the network converges towards Nash equilibria while being allowed to re-wire. The networks become increasingly scale-free during this process.
769:
1326:
It may be important to look at properties which cannot be directly observed by treating evolving networks as a sequence of snapshots, such as the duration of contacts between nodes Other similar properties can be defined and then it is possible to instead track these properties through the evolution
1301:
In networked systems where competitive decision making takes place, game theory is often used to model system dynamics, and convergence towards equilibria can be considered as a driver of topological evolution. For example, Kasthurirathna and
Piraveenan have shown that when individuals in a system
1347:
Almost all real world networks are evolving networks since they are constructed over time. By varying the respective probabilities described above, it is possible to use the expanded BA model to construct a network with nearly identical properties as many observed networks. Moreover, the concept of
1317:
Unfortunately, the analogy of snapshots to a motion picture also reveals the main difficulty with this approach: the time steps employed are very rarely suggested by the network and are instead arbitrary. Using extremely small time steps between each snapshot preserves resolution, but may actually
889:
and growth, where nodes are added to the network over time and are more likely to link to other nodes with high degree distributions. The BA model was first applied to degree distributions on the web, where both of these effects can be clearly seen. New web pages are added over time, and each new
1330:
Another issue with using successive snapshots is that only slight changes in network topology can have large effects on the outcome of algorithms designed to find communities. Therefore, it is necessary to use a non classical definition of communities which permits following the evolution of the
1272:
Further complications arise because nodes may be removed from the network with some probability. Additionally, existing links may be destroyed and new links between existing nodes may be created. The probability of these actions occurring may depend on time and may also be related to the node's
974:
The BA model was the first model to derive the network topology from the way the network was constructed with nodes and links being added over time. However, the model makes only the simplest assumptions necessary for a scale-free network to emerge, namely that there is linear growth and linear
997:
whereby the earliest nodes with high degree distributions continue to dominate the network indefinitely. However, this can be alleviated by introducing a fitness for each node, which modifies the probability of new links being created with that node or even of links to that node being removed.
1318:
obscure wider trends which only become visible over longer timescales. Conversely, using larger timescales loses the temporal order of events within each snapshot. Therefore, it may be difficult to find the appropriate timescale for dividing the evolution of a network into static snapshots.
1348:
scale free networks shows us that time evolution is a necessary part of understanding the network's properties, and that it is difficult to model an existing network as having been created instantaneously. Real evolving networks which are currently being studied include
1273:
fitness. Probabilities can be assigned to these events by studying the characteristics of the network in question in order to grow a model network with identical properties. This growth would take place with one of the following actions occurring at each time step:
716:
where people make and lose friends over time, thereby creating and destroying edges, and some people become part of new social networks or leave their networks, changing the nodes in the network. Evolving network concepts build on established
1107:
1001:
In order to preserve the preferential attachment from the BA model, this fitness is then multiplied by the preferential attachment based on degree distribution to give the true probability that a link is created which connects to node
1558:
964:
799:
Despite this achievement, both the ER and the Watts and
Storgatz models fail to account for the formulation of hubs as observed in many real world networks. The degree distribution in the ER model follows a
1217:
1816:
1293:
In addition to growing network models as described above, there may be times when other methods are more useful or convenient for characterizing certain properties of evolving networks.
862:
776:
While the ER model's simplicity has helped it find many applications, it does not accurately describe many real world networks. The ER model fails to generate local clustering and
1491:
1310:
The most common way to view evolving networks is by considering them as successive static networks. This could be conceptualized as the individual still images which compose a
975:
preferential attachment. This minimal model does not capture variations in the shape of the degree distribution, variations in the degree exponent, or the size independent
867:
This exponent turns out to be approximately 3 for many real world networks, however, it is not a universal constant and depends continuously on the network's parameters
1240:
1263:
1130:
1012:
1867:
1595:
Kasthurirathna, Dharshana; Piraveenan, Mahendra. (2015). "Emergence of scale-free characteristics in socioecological systems with bounded rationality".
979:. Therefore, the original model has since been modified to more fully capture the properties of evolving networks by introducing a few new properties.
569:
1748:
1762:
Y. Chi, S. Zhu; X. Song; J. Tatemura; B.L. Tseng (2007). "Structural and temporal analysis of the blogosphere through community factorization".
1343:
Route map of the world's scheduled commercial airline traffic, 2009. This network evolves continuously as new routes are scheduled or cancelled.
900:
1687:
G. Palla; A. Barabasi; T. Vicsek; Y. Chi, S. Zhu, X. Song, J. Tatemura, and B.L. Tseng (2007). "Quantifying social group evolution".
1382:
676:
784:
was proposed, whereby a network is constructed as a regular ring lattice, and then nodes are rewired according to some probability
1138:
1620:
1132:
is the fitness, which may also depend on time. A decay of fitness with respect to time may occur and can be formalized by
559:
288:
1791:
17:
894:
which have high degree distributions than to nodes with only a few links. Formally this preferential attachment is:
633:
216:
1755:
529:
1686:
988:
519:
738:
514:
669:
628:
145:
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80:
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509:
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31:
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Proceedings of the 13th ACM SIGKDD international conference on
Knowledge discovery and data mining
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https://web.archive.org/web/20110718151116/http://www.zangani.com/blog/2007-1030-networkingscience
422:
1901:
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886:
741:
paper. Probabilistic network theory then developed with the help of eight famous papers studying
645:
464:
231:
165:
120:
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One concern with the BA model is that the degree distributions of each nodes experience strong
976:
549:
534:
449:
1331:
community through a set of rules such as birth, death, merge, split, growth, and contraction.
1742:
1638:
1455:
Travers
Jeffrey; Milgram Stanley (1969). "An Experimental Study of the Small World Problem".
1225:
709:
650:
469:
439:
328:
283:
1666:
1102:{\displaystyle \Pi (k_{i})={\frac {\eta _{i}k_{i}}{\displaystyle \sum _{j}\eta _{j}k_{j}}},}
1843:
1808:
1706:
1518:
1413:
1245:
1115:
801:
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417:
298:
8:
812:. Many networks are instead scale free, meaning that their degree distribution follows a
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Watts, D.J.; Strogatz, S.H. (1998). "Collective dynamics of 'small-world' networks".
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994:
750:
590:
256:
206:
115:
90:
1801:
1387:"Linked: The New Science of Networks", A.-L. Barabási Perseus Publishing, Cambridge.
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since almost all real world networks evolve over time, either by adding or removing
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or links over time. Often all of these processes occur simultaneously, such as in
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777:
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705:
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366:
241:
150:
85:
39:
1613:
1530:
1667:"Impact of human mobility on the design of opportunistic forwarding algorithms"
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718:
693:
Animation of an evolving network according to the initial
Barabasi–Albert model
595:
401:
376:
371:
345:
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211:
175:
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130:
68:
1339:
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The Barabási–Albert (BA) model was the first widely accepted model to produce
1890:
742:
554:
459:
444:
386:
135:
125:
1783:
761:
labeled nodes where each pair of nodes is connected by a preset probability
746:
721:
and are now being introduced into studying networks in many diverse fields.
1726:
1581:
Albert R. and Barabási A.-L., "Statistical mechanics of complex networks",
1546:
730:
499:
396:
251:
1761:
1538:
1433:
1838:
1513:
805:
788:. This produces a locally clustered network and dramatically reduces the
1718:
1665:
A. Chaintreau; P. Hui; J. Crowcroft; C. Diot; R. Gass; J. Scott (2006).
689:
1476:
434:
391:
381:
1664:
1824:
870:
813:
599:
155:
1468:
959:{\displaystyle p_{i}={\frac {k_{i}}{\displaystyle \sum _{j}k_{j}}},}
52:
1357:
1306:
Treat evolving networks as successive snapshots of a static network
780:
as often as they are found in real world networks. Therefore, the
729:
The study of networks traces its foundations to the development of
704:
that change as a function of time. They are a natural extension of
1701:
1425:
1680:
1288:
768:
891:
1492:"Topology of Evolving Networks: Local Events and Universality"
1454:
804:, while the Watts and Strogatz model produces graphs that are
1817:"Networks in life: scaling properties and eigenvalue spectra"
1311:
1814:
1212:{\displaystyle \Pi (k_{i})\propto k_{i}(t-t_{i})^{-\nu },}
1403:
1594:
890:
page is more likely to link to highly visible hubs like
1489:
1815:
I. Farkas; I. Derenyi; H. Heong; et al. (2002).
1611:
1248:
1228:
1141:
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871:First evolving network model – scale-free networks
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1888:
1289:Other ways of characterizing evolving networks
670:
1747:: CS1 maint: multiple names: authors list (
1321:
724:
1483:
1327:of a network and visualize them directly.
757:(ER) supposes that a graph is composed of
677:
663:
1837:
1773:
1700:
1612:Pierre Borgnat; Eric Fleury; et al.
1512:
885:. This was accomplished by incorporating
1338:
969:
792:, creating networks which represent the
767:
688:
14:
1889:
796:observed in many real world networks.
857:{\displaystyle P(k)\sim k^{-\gamma }}
1448:
24:
1490:R. Albert; A.-L. Barabási (2000).
1375:
1142:
1016:
25:
1913:
1397:
1381:"Understanding Network Science,"
1268:Removing nodes and rewiring links
737:in 1736 when he wrote the famous
51:
1626:from the original on 2011-08-12
1564:from the original on 2010-12-24
1334:
1588:
1575:
1297:Convergence towards equilibria
1276:Prob p: add an internal link.
1194:
1174:
1158:
1145:
1032:
1019:
989:Fitness model (network theory)
835:
829:
733:, which was first analyzed by
13:
1:
1856:10.1016/S0378-4371(02)01181-0
1391:
7:
1531:10.1103/PhysRevLett.85.5234
739:Seven Bridges of Königsberg
10:
1918:
1285:Prob 1-p-q-r: add a node.
986:
982:
874:
1583:Reviews of Modern Physics
1322:Define dynamic properties
725:Network theory background
530:Exponential random (ERGM)
197:Informational (computing)
782:Watts and Strogatz model
217:Scientific collaboration
1784:10.1145/1281192.1281213
1500:Physical Review Letters
1370:transportation networks
1354:communications networks
1282:Prob r: delete a node.
1279:Prob q: delete a link.
1235:{\displaystyle \gamma }
887:preferential attachment
646:Category:Network theory
166:Preferential attachment
1646:Cite journal requires
1344:
1259:
1236:
1213:
1126:
1103:
977:clustering coefficient
960:
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794:small world phenomenon
773:
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535:Random geometric (RGG)
1342:
1260:
1258:{\displaystyle \nu .}
1237:
1214:
1127:
1125:{\displaystyle \eta }
1104:
970:Additions to BA model
961:
877:Barabási–Albert model
859:
771:
692:
651:Category:Graph theory
1768:. pp. 163–172.
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1226:
1139:
1116:
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901:
823:
802:Poisson distribution
772:Watts–Strogatz graph
1848:2002PhyA..314...25F
1719:10.1038/nature05670
1711:2007Natur.446..664P
1614:"Evolving Networks"
1523:2000PhRvL..85.5234A
1418:1998Natur.393..440W
1362:movie actor network
883:scale-free networks
790:average path length
455:Degree distribution
106:Community structure
1598:Scientific Reports
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1209:
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639:Network scientists
565:Soft configuration
1695:(7136): 664–667.
1507:(24): 5234–5237.
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995:positive feedback
951:
930:
755:Erdős–Rényi model
698:Evolving networks
687:
686:
607:
606:
515:Bianconi–Barabási
409:
408:
227:Artificial neural
202:Telecommunication
18:Evolving networks
16:(Redirected from
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1866:. Archived from
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1839:cond-mat/0303106
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1514:cond-mat/0005085
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1412:(6684): 409–10.
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550:Stochastic block
540:Hyperbolic (HGN)
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181:Social influence
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26:
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1469:10.2307/2786545
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1376:Further reading
1350:social networks
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706:network science
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586:Boolean network
560:Maximum entropy
510:Barabási–Albert
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121:Controllability
86:Complex network
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40:Network science
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1832:(1–4): 25–34.
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1775:10.1.1.69.6959
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1648:|journal=
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1463:(4): 425–443.
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987:Main article:
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735:Leonhard Euler
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1873:on 2011-10-04
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500:Random graph
348: /
337: /
319:Neighborhood
161:Transitivity
141:Optimization
1601:. In Press.
806:homogeneous
745:written by
591:agent based
505:Erdős–Rényi
146:Reciprocity
111:Percolation
96:Small-world
1891:Categories
1877:2011-04-21
1630:2011-10-26
1568:2011-10-26
1457:Sociometry
1392:References
747:Paul Erdős
618:Categories
475:Efficiency
470:Modularity
450:Clustering
435:Centrality
423:Algorithms
247:Dependency
222:Biological
101:Scale-free
1770:CiteSeerX
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367:Bipartite
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116:Evolution
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492:Topology
465:Distance
402:Weighted
377:Directed
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418:Metrics
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350:matrix
339:matrix
329:Vertex
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266:Graphs
212:Social
69:Theory
1871:(PDF)
1860:S2CID
1834:arXiv
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710:nodes
615:Lists
445:Motif
392:Multi
382:Hyper
359:Types
299:Cycle
81:Graph
1788:ISBN
1749:link
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