We investigate the kinetics of bimolecular chemical reactions A+A→0 and A+B→0 on weighted scale-free networks (WSFNs) with degree distribution P(k)∼k^{-γ} . On WSFNs, a weight w{ij} is assigned to the link between node i and j . We consider the symmetric weight given as w{ij}=(k{i}k{j})^{μ} , where k{i} and k{j} are the degree of node i and j . The hopping probability T{ij} of a particle from node i to j is then given as T{ij}∝(k{i}k{j})^{μ} . From a mean-field analysis, we analytically show in the thermodynamic limit that the kinetics of A+A→0 and A+B→0 are identical and there exist two crossover μ values, μ{1c}=γ-2 and μ{2c}=(γ-3)/2 . The density of particles ρ(t) algebraically decays in time t as t^{-α} with α=1 for μ<μ{2c} and α=(μ+1)/(γ-μ-2) for μ{2c}≤μ<μ{1c} . For μ≥μ{1c} , ρ decays exponentially. With the mean-field rate equation for ρ(t) , we also analytically show that the kinetics on the WSFNs is mapped onto that on unweighted SFNs with P(k)∼k^{-γ^{'}} with γ^{'}=(μ+γ)/(μ+1) .

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.