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Extra info for Analysis of Complex Networks: From Biology to Linguistics
Recently it has been found that networks exhibiting degree distributions compatible with q-exponentials are not at all limited to growing and preferentially organizing networks. Degree distributions of real-world networks as well as of nongrowing models of various kinds seem to exhibit a universality in this respect [7–9]. A model for non-growing networks which was recently put forward in  also unambiguously exhibit q-exponential degree distributions. This model was motivated by interpreting networks as a certain type of “gas” where upon an (inelastic) collision of two nodes, links get transfered in analogy to the energy-momentum transfer in real gases.
The model corresponding to this limit has been proposed and studied in . • Albert–Barabasi limit. The lim λ¯ → ∞ and lim αA → 0 gets rid of the metric in the Soares et al. model and recovers the original Albert–Barabasi preferential attachment model . • Kim et al. limits. The limit lim λ¯ → 0 allows no preferential growing of the network. If at each timestep after every merger a new node is linked randomly with ¯l links to the network, the model reported in  is recovered. The lim λ¯ → 0 model with lim αM → 0 (lim αM → ∞) recovers the random case (“neighbor” case) in .
If the N involved nodes are indistinguishable, then this 1 , in the argument of the logahas to be taken care of by an additional factor, N! rithm. As soon as networks become subjected to constraints such as speciﬁc linking rules or probabilities, or through the deﬁnition of a Hamiltonian (as will be discussed later), the evaluation of the number of possible states (adjacency matrices) becomes more difﬁcult. Clearly, Sc is a nonextensive quantity, regardless of whether or not nodes are distinguishable.