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The exponent β is bounded in (0, 1) since the degree of a node in general increases, but always less than linearly since it cannot acquire new edges at a rate faster than the one at which new nodes enter the system. In order to solve Eq. s.. The determination of such a formula is not amenable for a specific realization of the fitness. Therefore we compute its average over many realizations assignments of the η variables: t ηj kj j = dt0 kη (t, t0 ) = m dηρ(η)η 1 dηρ(η)η t − tβ(η) . 1 − β(η) (27) where we labeled nodes according to their entrance time t0 , rather than to their index j.

In particular, the three vertices conditional probability can be written as P (k , k |k) = P (k |k)P (k |k) and the same applies to higher order correlation functions. The meaning of the term Markovian network that we use in this chapter is borrowed from the theory of Stochastic Processes. In this field, a stochastic process X(t) is called Markovian if the probability to find the process at the position X(t) = x at time t only depends on its position at the previous time t < t. Then, the process is completely characterized by the probability density function p(x, t) of being at x at time t and the transition probability density p(x, t|x , t ) of being at x at time t, provided that the process was at x at time t .

The degree distribution for fixed η can then be found using the usual relation P (k, η)dk = dt0 , which gives P (k, η) ∝ C η m k 1+ C η (31) from which the global degree distribution P (k) can be obtained as P (k) = ρ(η)P (k, η)dη. A choice of a constant η, that is ρ(η) = δ(η − η0 ) clearly corresponds to the BA model and gives P (k) ∼ k −3 , as already derived. A powerlaw degree distribution can also be obtained from a uniform distribution ρ(η) = 1, 0 < η < 1. Indeed Eq. 255 (and C > 1, as needed to avoid singularities in Eq.