The network growth model that considered community information

Transcription

1 CNN The network growth model that considered community information EIJI MIYOSHI, 1 IKUO SUZUKI, 1 MASAHITO YAMAMOTO 1 and MASASHI FURUKAWA 1 There is a study of much network model, but, as for the most, a network grows up only by the state of the node that is the component of the network. However, the real network growth takes the influence by the community. In this study, We suggest the growth model that added community information to a rule of the growth of the network. In addition, with a change of the community structure by the network generation that applied suggestion technique. We observe a change by the progress about quantity of characteristic to numbers of increase and decrease of the number of the community, clustering origin in time and we compare the suggestion model with CNN model about the growth process of the network. 1. web CNN 1) Connecting Nearest Neighbor CNN 1) DEB 2) CNN V = v i, E = e ij ( 1 ) u(0 u 1) ( 2 ) 1 Graduate School of Information Science and Technology,Hokkaido University 1 c 2009 Information Processing Society of Japan

2 ( a ) 1 u v j V v i v i V v i v j v i ( b ) u 2.2 Modularity Aaron Clauset 3) Modularity Q Modularity V l a l V l V m M e lm Q = Q i = (e ii a 2 i ) (1) i 1...I i 1...I 2.3 Clauset Newman Moore (CNM) Clauset 3) Modularity Q ( 1 ) 1 Q Q (i+j) = Q i + Q j + 2e ij 2a i a j (2) Q ij = 2(e ij a ia j) ( 2 ) i, j i, j k Q Q ik + Q jk : if k connected with i, j Q jk = Q ik 2a ja k : if k connected with i Q jk 2a i a k : if k connected with j (3) ( 3 ) Q CNM CNM Modularity 2.4 Clauset LocalModul- arity:r 4) R R G = (N, E) C G C C U G C U B G ( 1 ) B 1 : v i, v j B ij = v i, v j B 0 : B LocalModularity R Bijδ(i, j) ij R = = I T ij B ij δ(i, j) v i B v j C 1 0 T B I B U Clauset 4) 1 LocalModularity C ( 1 ) v 0 C B v 0 v 0 U ( 2 ) C ( 3 ) v j U R j (4) (5) 2 c 2009 Information Processing Society of Japan

3 1 4) 2 ( 4 ) R j v j C v j U ( 5 ) R B R j 5 6 R j = x Ry z(1 = R) T z + y x v j B y v j C T (k j x) z v j C T 3. CNN CNN 3 CNN Modularity GN, 5) NF 6) GN (6) Clauset LocalModularity CNN ( 1 ) 3 c 2009 Information Processing Society of Japan

4 ( 2 ) ( a ) 1 u v i ( i ) v i (LC) 2.4 ( ii ) v j ( iii ) v j ( 2.4 ) ( b ) u CNN R < Modularrity : Q 4.2 CNN, 3 µ Modularity CNN Modularity CNM 3 Modularity :CNN : Modularity CNN 2 Modularity Modularity CNM CNN Modularity Modularity CNN 3 Modularity 5 CNN CNN 4 c 2009 Information Processing Society of Japan

5 4 :CNN : (c) 6 µ = 0.1 µ = 0.5 (c)µ = :CNN : 6 7 µ = c 2009 Information Processing Society of Japan

6 7 µ = 0.9 Localmodularity 6, 1) Vázquez, A.: Growing network with local rules: Preferential attachment, clustering hierarchy, and degree correlations, Phys. Rev. E, Vol.67, No.5, p (2003). 2) Davidsen, J., Ebel, H. and Bornholdt, S.: Emergence of a Small World from Local Interactions: Modeling Acquaintance Networks, Phys. Rev. Lett., Vol.88, No.12, p (2002). 3) Clauset, A., Newman, M. E.J. and Moore, C.: Finding community structure in very large networks, Phys. Rev. E, Vol.70, No.6, p (2004). 4) Clauset, A.: Finding local community structure in networks, Phys. Rev. E, Vol.72, No.2, p (2005). 5) Girvan, M. and Newman, M. E.: Community structure in social and biological networks., Proc Natl Acad Sci U S A, Vol.99, No.12, pp (2002). 6) Newman, M. E.J.: Fast algorithm for detecting community structure in networks, Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), Vol.69, No.6 (2004). 5., 6 c 2009 Information Processing Society of Japan