Bimolecular chemical reactions on weighted complex networks

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)similar to k(-gamma). 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))(mu), 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 proportional to(k(i)k(j))(mu). 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 mu values, mu(1c)=gamma-2 and mu(2c)=(gamma-3)/2. The density of particles rho(t) algebraically decays in time t as t(-alpha) with alpha=1 for mu<mu(2c) and alpha=(mu+1)/(gamma-mu-2) for mu(2c) <=mu<mu(1c). For mu >=mu(1c), rho decays exponentially. With the mean-field rate equation for rho(t), we also analytically show that the kinetics on the WSFNs is mapped onto that on unweighted SFNs with P(k)similar to k(-gamma)' with gamma'=(mu+gamma)/(mu+1).