INCT2012 Complex Networks, Long-Range Interactions and Nonextensive Statistics

Transcription

1 Complex Networks, Long-Range Interactions and Nonextensive Statistics L. R. da Silva UFRN DFTE Natal Brazil 04/05/12 1

2 OUR GOALS Growth of an asymptotically scale-free network including metrics. Growth of a geographically localized network (around its baricenter). To exhibit effects of competition between metrical neighborhood, connectivity and fitness. To analyze the influences of considering a fitness powerlaw distributed. Last but not least, to exhibit the connection between scale-free networks and nonextensive statistics. 2

3 Random Network Sites have in average the same conectivity. Scale free Network P ( k ) k γ 3

4 Internet 4

5 Swiss road map INCT2012 Physica A (2012) 5

6 Gossip Network Europhysics Letters 78, (2007) 6

7 Apollonian Network INCT2012 7

8 Apollonian Network INCT2012 Apollonius of Perga lived from about 262 BC to about 190 BC Apollonius was known as 'The Great Geometer'. 8

9 Apollonian Network: Connectivity Distribution γ = 1.58±0.01 Sítios k P( k) k γ γ = ln3/ ln 2 = ( n+ 1 ) = N n 9

10 Scale-free Networks 1,2,3 P( k) k γ Barabási and Albert 1 ; γ = 3 Π( k < k i ) = >= j= 1 t i k i i N k β j (01) (02) 1 Science 286, 509 (1999) ; Rev. Mod Phys. 74, 47 (2002) 2 M. Boguñá and R. Pastor-Satorras, Physical Review E 68, (2003) 3 S. Thurner and C. Tsallis, Europhycs Letters 72, 197 (2005) 10

11 Fitness Model Bianconi and Barabási 4 ; Albert and Barabási 5 ; Π( k < k i i ) = >= N j= 1 t i kη i k j i η β ( η ) i j (04) (05) P( k) k γ = γ 4 Europhys. Lett. 54,436 (2001) ; 5 Rev. Mod Phys. 74, 47 (2002) 11

12 Geographic Model INCT2012 Continental Airlines

13 Network Construction: INCT2012 Barabási-Albert Model with Euclidean Distance Power-law Distributed α A ki / ri Π ( ki ) = N k / r j= 1 j α A j P( r) γ G > 1 r γ G where γ α G G 3 + αg = 2 + α 0 G r = (1 ξ') θ = 2πξ' ' (2+ α ) G 13

14 Examples N = 250 nodes (a) (α G, α A ) = (1, 0) and (b) (α G, α A ) = (1, 4). The starting site is at (X, Y ) = (0, 0). Notice the spontaneous emergence of hubs. 14

15 Links omitted 15

16 Barabási-Albert Model with Euclidean Distance Power-law Distributed ki / r Π ( ki ) = N k / j= 1 j α i A r α j A (03) P(k) E-3 1E-4 1E-5 α A =0 α A =0,4 α A =1 α A =2 α A =3 1E-6 1E-7 1E k Soares, Tsallis, Mariz and da Silva, Europhys. Lett. 70, 70 (2005) 16

17 Fitness Model with Euclidean Distance Power-law Distributed Inspired in the works of Soares, Tsallis, Mariz and da Silva 3, and Bianconi and Barabasi 4. α A kiη i / ri Π ( ki ) = N k η / r j= 1 j j α j A (06) P(k) ,1 0,01 1E-3 1E-4 1E-5 1E-6 1E-7 α A =0 α A =0,4 α A =1 α A =2 α A = Meneses, Cunha, Soares, and da Silva, Progress of Theoretical Physics Supplement 162, 131 (2006) 1E-8 k 17

18 Network Construction P( r) α γ G > 1 r γ G where γ α G G 3 + α = 2 + α 0 G G r = (1 ξ') θ = 2πξ'' (2+ α ) G 18

19 q INCT2012 Tsallis Nonextensive Statistical Mechanics 1 dk[ P( k)] s q = ( q R; S = S q 1 P( k) = P(1) k λ ( k 1) /κ e q 1 BG i i p i = 1 and i = U. i i p p q ε q i ) q e x q [1 + (1 q) x] 1/(1 q ) 19

20 P( k) = P(1) k λ ( k 1) /κ e q Connectivity distribution P(k) for typical values α A for η 1 and η=1 models. The symbols are numerical results and continuous lines are the best fits according to P(k). 20

21 α A -dependence of q for both η 1 and η=1 models. In this graph we observe some kinds of changements of regimes at α A = 2 (which coincides with the space dimensionality). 21

22 α A -dependence of λ for both η 1 and η=1 models. In this graph we observe some kinds of changements of regimes at α A = 2 (which coincides with the space dimensionality). 22

23 α A -dependence of q for both η 1 and η=1 models. In this graph we observe some kinds of changements of regimes at α A = 2 (which coincides with the space dimensionality). 23

24 Temporal dependence of the average connectivity for η 1, in 2000 samples. 24

25 Average connectivity exponent for α A values relative to measures on node i =

26 Generalized Model: Fitness and Euclidean Distance Power-law Distributed Inspired in the works of Meneses et al; αa kiη i / ri Π ( ki ) = N k η / r with Mendes et al; ρ( η) j= 1 δ 0; α = 2 A j η δ j αa j (08) 26

27 P( k) = P(1) k λ ( k 1) /κ e q Conectivity distribution P(k) for α A =2 for Meneses et al, Mendes et al and Soares et al models. The symbols are numerical results and continuous lines are the best fits in according to P(k). 27

28 The generalized model contains the five previous models: Model CONECTIVITY FITNESS METRIC Barabási-Albert YES NO NO Bianconi et al YES UNIFORM NO Soares et al YES NO POWER-LAW Meneses et al YES UNIFORM POWER-LAW Mendes et al YES POWER-LAW NO Generalized YES POWER-LAW POWER-LAW 28

29 Affinity Model Inspired in the works of Bianconi and Barabási; G.A. Mendes and L.R. da Silva. The links between similar sites are favored. M.L. Almeida, G.A. Mendes, G.M. Viswanathan A.M. Mariz and L.R. da Silva (Preprint) 29

30 blue: big affinity; black: medium affinity; red: small affinity M.L. Almeida, G.A. Mendes, G.M. Viswanathan A.M. Mariz and L.R. da Silva (Preprint) 30

31 Affinity Model M.L. Almeida, G.A. Mendes, G.M. Viswanathan A.M. Mariz and L.R. da Silva (Preprint) 31

32 Affinity Model: Connectivity Time Evolution 32

33 Affinity Model 33

34 Affinity Model 34

35 Affinity Model 35

36 Affinity Model: Tsallis Statistics 36

37 Summary (a) We study the effect of competition between the relevant variables: connectivity k, fitness η and metrics r. The fitness may give the possibility to the younger nodes to compete equally with the older ones, when the younger node gets a high fitness. By including metrics favors the linking between first neighbors. The average connectivity <k i > is appreciably influenced by metrics and by fitness, while the average path length <l> keeps approximatively the same. 37

38 Summary (b) The degree distribution P(k) of the present generalized model appears to be the q-exponential function that emerges naturally within Tsallis nonextensive statistics. We modify the rule of the preferential attachment of the Bianconi-Barabasi model including a factor which represents similarity of the sites. The term that corresponds to this similarity is called the affinity and is obtained by the modulus of the difference between the fitness (or quality) of the sites. This variation in the preferential attachment generate very unusual and interesting results. 38

39 References INCT D. J. B. Soares, C. Tsallis, A. M..Mariz, and L. R. da Silva. Preferential Attachment Growth Model and Noextensive Statistical Mechanics Europhysics Letters 70, 70 (2005) 2. J. S. Andrade Jr., H. J. Herrmann, R.F Andrade and L. R. da Silva. Apollonian Networks: Simultaneously Scale-free, Small World, Euclidean, Space Filling and with Matching Graphs. Physical Review Letters 94, (2005). 3. M. D. de Meneses, Sharon D. da Cunha, D.J.B. Soares and L. R. da Silva. Preferential Attachment Scale-free Growth Model with Random Fitness and Connection with Tsallis Statistics Progress of Theoretical Physics Supplement (2006) 4. D. J. B. Soares, J. S. Andrade Jr., H. J. Herrmann and L. R. da Silva Three Dimension Apollonian Networks International Journal of Modern Physics (2006) 5. P. G. Lind, L. R. da Silva, J. S. Andrade Jr. and H. J. Herrmann. The Spread of Gossip in American Schools Europhysics Letters 78, (2007) 39

40 References 6. P. G. Lind, L. R. da Silva, J. S. Andrade Jr. and H. J. Herrmann. Spreading Gossip in Social Networks Physical Review E 76, (2007) 7. G. A. Mendes and L.R. da Silva Generating more realistic complex networks from power-law distribution of fitness Brazilian Journal of Physics (2009) 8. S. S. B. Jácome, L. R. da Silva, A. A. Moreira, J. S. Andrade Jr. and H. J. Herrmann Iterative Decomposition of the Barabasi-Albert Scale-free Networks Physica A (2010) 9. G. A. Mendes, L. R. da Silva and Hans J. Herrmann Traffic Gridlock on Complex Networks Physica A (2012) 10. M.L Almeida, G.A. Mendes, G. M. Viswanathan A.M. Mariz and L. R. da Silva (Preprint) Affinity model in complex networks 40

41 41

42 THANK YOU VERY MUCH 42

43 Rede de Apolônio INCT

44 Apollonius of Perga lived from about 262 BC to about 190 BC Apollonius was known as 'The Great Geometer'. 44

45 Rede de Apolônio : Distribuição de Conectividade γ = 1.58±0.01 Sítio k k 4 3s s s P( k) k γ γ = ln3/ ln 2 = ( n+ 1 ) = N n 45

46 Menor Caminho Médio ( log ( )) l N β β 3/ 4 46

47 Coeficiente de Agregação Médio N 47

48 Danyel J. B. Soares, J. S. Andrade Jr, Hans J. Herrmann, L. R. da Silva 7 7 International Journal of Modern Physics C (2006) 48

49 Movie Star 49

50 6. How gossip propagates Europhysics Letters 78, (2007) 50

51 Spread Factor (f=n f /k) U.S Friend Schools Inset graph: Barabási- Albert 51

52 o Friendship τ = A+ B log ( k) 52

53 o Friendship * BA; m=7 Apollonian 53

54 GAS-LIKE (NODE COLLAPSING) NETWORK: S. Thurner and C. T., Europhys Lett 72, 197 (2005) Number N of nodes fixed (chemostat); i=1, 2,, N 1 Merging probability p ( α 0) d ij ij d α ij shortest path ( chemical distance) connecting nodes i and j on the network α = 0 and α recover the random and the neighbor schemes respectively (Kim, Trusina, Minnhagen and Sneppen, Eur. Phys. J. B 43 (2005) 369) α 7 ( N = 2 ; = 0; r = 2) Degree of the most connected node Degree of a randomly chosen node 54

55 ( α ; < r >= 8) Z q [ P k ] ( k) ln ( > ) q ( α ; ( optimal q = 1.84) c [ P( k) ] 1 > 1 1 q S. Thurner and C. T., Europhys Lett 72, 197 (2005) q 55

56 9 ( N = 2 ; r = 2) e - ( k-2)/ κ P( k) = ( k = 2, 3, 4,...) q c linear correlation [ , ] S. Thurner and C. T., Europhys Lett 72, 197 (2005) 56

57 ( r = 2) [ ] q ( α) = q ( ) + q (0) q ( ) e α c c c c 9 ( N = 2 ) S. Thurner and C. T., Europhys Lett 72, 197 (2005) 57