By Harold Acton
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5 Scale-Free Graphs 35 and assuming long times, we find pk = k 1 1−k/m e e exp − = m m m . 76) which is identical to half of the average connectivities of the vertices in the system, since k = 2m. Random attachment does not lead to a scale-free degree distribution. Note that pk in Eq. 75) is not properly normalized, nor in Eq. 71), since we used a large-k approximation during the respective derivations. Internal Growth with Preferential Attachment The original preferential attachment model yields a degree distribution pk ∼ k −γ with γ = 3.
15) provides a description of the short time behavior. For a corresponding characterization of the long time dynamics one defines the “maximal Lyapunov exponent” 50 2 Chaos, Bifurcations and Diffusion λ(max) = lim n 1 d f (n) (x) 1 , log n dx f (n) (x) = f ( f (n−1) (x)) . 16) Using Eq. 15) for the short time evolution we can decompose λ(max) into an averaged sum of short time Lyapunov exponents. We leave this as an exercise to the reader, λ(max) is also denoted the “global Lyapunov exponent”. One needs to select advisedly the number of iterations n in Eq.
We are, however, interested only in the first moment, viz the mean component size, see Eq. 32). The component size distribution is generated by H0 (x), Eq. 44) where we have made use of the normalization G 0 (1) = H1 (1) = H0 (1) = 1 . of generating functions, see Eq. 31). The value of H1 (1) can be calculated from Eq. 42) by differentiating: H1 (x) = G 1 (H1 (x)) + x G 1 (H1 (x)) H1 (x), H1 (1) = 1 . 45) 24 1 Graph Theory and Small-World Networks s = 1+ G 0 (1) . 47) k G 1 (1) = k k(k − 1) pk = k kpk k2 − k k = z2 , z1 where we have made use of Eq.
An Indian Ass by Harold Acton