Condensation phase transition in nonlinear fitness networks

We analyze the condensation phase transitions in out-of-equilibrium complex networks in a unifying framework which includes the nonlinear model and the fitness model as its appropriate limits. We show a novel phase structure which depends on both the fitness parameter and the nonlinear exponent. The occurrence of the condensation phase transitions in the dynamical evolution of the network is demonstrated by using Bianconi-Barabási method. We find that the nonlinear and the fitness preferential attachment mechanisms play important roles in formation of an interesting phase structure. Introduction. – Condensation phenomena emerge in various physical contexts, to name a few, the wellknown Bose-Einstein condensation (BEC) in dilute atomic gases [1–3], jamming in traffic flow [4, 5], wealth condensation in macroeconomies [6], and condensation in zerorange process (ZRP, see e.g., recent review [7] and references therein). Since the pioneered research on complex networks [8–16], in the last decade, condensation phenomena, i.e., condensation of links (or edges) in complex networks has also been widely discussed [17–35]. In the context of complex networks, the condensation phase corresponds to the situation that a single node captures a macroscopic finite fraction of total links/edges. It has been found that condensation phenomena can occur in both growing and non-growing complex networks. The condensation phase transitions occurring in non-growing networks [20,22,23,25–28] are formally equivalent to that in balls-in-boxes model [36], and hence has been well studied to a large extent via methods of equilibrium statistical mechanics. While for growing complex networks, the appearance of the condensation phase transitions during the dynamical evolution of the network are particularly interesting due to its out-of-equilibrium characteristics. The tasks in this paper are two folds: first, we merge two important models on this regard, the growing network with nonlinear preferential attachment (we will refer to “nonlinear model” [18] from now on), and the fitness model [37] into a unifying framework–the nonlinear fitness model. We then argue that the condensation phase transition appearing in fitness model and that in nonlinear model stems from different mechanisms; Second, particularly interesting, we reveal a novel phase structure in the model and this may increase our understanding on the non-equilibrium phase transitions in dynamical evolution of complex networks. The nonlinear model is defined as follows. At each time step t, the newly-added node created m directed links to ones of the earlier existing nodes with k-link, according to a probability, say, Π, that is proportional to some “connection kernel” k , Π ∝ k . Here the exponent γ ≥ 0 reflects the tendency of preferential linking to a popular node and hence controls the preferential attachment. In Ref. [18], P. L. Krapivsky et al. had discussed the cases of different choices on exponent γ, i.e., γ = 1, γ < 1 and γ > 1, for growing complex networks with connection probability, Π = k i Σjk γ j . (1) They proved that the number of nodes with k links, Nk, follows a power law distribution in the case that γ closes to unity. While in the case of γ < 1, the distribution shows a stretched exponential form. For γ > 1, that is so called the super-linear case, the model exhibits a condensation phase transition. Especially when γ > 2, there exists a limiting