Irreversible bimolecular chemical reactions on directed scale-free networks

Kinetics of irreversible bimolecular chemical reactions A+A→0 and A+B→0 on directed scale-free networks with the in-degree distribution P(in)(k)∼k(-γ)(in) and the out-degree distribution P(out)(ℓ)∼ℓ(-γ)(out) are investigated. Since the correlation between k and ℓ of each node generally exists in directed networks, we control the correlation (kℓ) with the probability r∈[0,1] by two different algorithms for the construction of the directed networks, i.e., the so-called k and ℓ algorithms. For r=1, the k algorithm gives (kℓ)=(k(2)), whereas the ℓ algorithm gives (kℓ)=(ℓ(2). For r=0, (kℓ)=(k)(ℓ) for both algorithms. The kinetics of both reactions are analyzed using heterogeneous mean-field (HMF) theory and Monte Carlo simulations. The density of particles (ρ) algebraically decays in time t as ρ(t)∼t(-α). The kinetics of both reactions are determined by the same rate equation, dρ/dt=aρ(2)+bρ(θ-1), apart from coefficients. The exponent θ is determined by the algorithm: θ=γ(in) for the k algorithm (r≥0) and θ=γ(min) for the ℓ algorithm (r>0), where γ(min) is the smaller exponent between γ(in) and γ(out). For θ>3, one observes the ordinary mean-field kinetics, ρ∼1/t (α=1). In contrast, for θ<3, ρ(t) anomalously decays with α=1/(θ-2). The HMF predictions are confirmed by the simulations on quenched directed networks.