Condensation phenomena of a conserved-mass aggregation model on weighted complex networks

We investigate the condensation phase transitions of the conserved-mass aggregation (CA) model on weighted scale-free networks (WSFNs). In WSFNs, the weight w_{ij} is assigned to the link between the nodes i and j . We consider the symmetric weight given by w_{ij}=(k_{i}k_{j});{alpha} . On WSFNs, we numerically show that a certain critical alpha_{c} exists below which the CA model undergoes the same type of condensation transitions as those of the CA model on regular lattices. However, for alpha > or = alpha_{c} , the condensation always occurs for any density rho and omega . We analytically find alpha_{c}=(gamma-3)/2 on the WSFN with the degree exponent gamma . To obtain alpha_{c} , we analytically derive the scaling behavior of the stationary probability distribution P_{k};{infinity} of finding a walker at nodes with degree k , and the probability D(k) of finding two walkers simultaneously at the same node with degree k . We find P_{k};{infinity} approximately k;{alpha+1-gamma} and D(k) approximately k;{2(alpha+1)-gamma} , respectively. With P_{k};{infinity} , we also show analytically and numerically that the average mass m(k) on a node with degree k scales as k;{alpha+1} without any jumps at the maximal degree of the network for any rho as in SFNs with alpha=0 .