Nonextensive Statistical Mechanics

7.8 Scale-Free Networks 291Fig. 7.96 Connectivity distribution for α AN = 10,000 networks (from [790]).= 1 and typical values of α G ; 2000 realizations of7.8.2 Albert–Barabasi ModelAnother growth model, also including preferential attachment, has been introducedand analytically solved in 2000 by Albert and Barabasi [864] as a prototype ofemergence of the ubiquitous scale-free networks. At each time step, m new linksare added with probability p, orm existing links are rewired with probability r, ora new node with m links is added with probability 1 − p − r; all linkings are donewith probability (k i ) = (k i + 1)/ ∑ j (k j + 1), where k i is the number of links ofthe ith node. The exact stationary state distribution of the number k of links at eachsite is given [864] by Eq. (7.32) withand[2m(1 − r)]k 0 = 1 + (p − r) 1 + > 0 . (7.34)1 − p − rμ =m(3 − 2r) + 1 − p − rm> 0 . (7.35)With the notation change (7.33), this degree distribution can be rewritten in theform of Eq. (7.31) withq =2m(2 − r) + 1 − p − rm(3 − 2r) + 1 − p − r≥ 1, (7.36)with κ>0 given by Eqs. (7.34) and (7.35) replaced into κ = k 0 (q − 1).