By Ladislav NebeskyÌ

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Our model allows loops and multiple edges; there seems no reason to exclude them. However, there will not be very many, so excluding them would not significantly affect our conclusions. 12 Clustering coefficient and small subgraphs 17 Note also that our model includes (a precise version of) the m = 1 case of the original model of Barabási and Albert as a special case, taking β = γ = δout = 0 and α = δin = 1. We could introduce more parameters, adding m edges for each new vertex, or (as in [24]) a random number with a certain distribution, but one of our aims is to keep the model simple, and the main effect, of varying the overall average degree, can be achieved by varying β.

Comput. and Applied Math. 41 (1992), 237–245. M. T. Smythe, A survey of recursive trees, Th. of Probability and Math. Statistics 51 (1995), 1–27. M. T. Smythe and J. Szyma´nski, On the structure of random planeoriented recursive trees and their branches, Random Struct. Alg. 4 (1993), 151–176. , The degree sequence of a random graph. I. The models, Random Structures and Algorithms 11 (1997), 97–117. , The small world phenomenon, Psychol. Today 2 (1967), 60–67. J. H. J. Watts, Random graphs with arbitrary degree distribution and their applications, Physical Review E 64 (2001), 026118.

Let Yi = Xmi , and let Dm = max{Yi − Yi−1 , 1 ≤ i < ∞}, noting that this maximum exists with probability one. Note that the distribution of Dm depends on m only. (n) (n) Let ∆(Gm ) denote the maximum degree of Gm . √ (n) Theorem 17. Let m ≥ 1 be fixed. Then ∆(Gm )/(2m n) converges in distribution to Dm as n → ∞. Proof. As before, here we√ can only give a sketch. Note that if U is a random variable which √ is uniform on [0, 1], then U has a M2 (0, 1) distribution, since Pr( U ≤ t) = Pr(U ≤ t2 ) = t2 .