Complete condensation in a zero range process on scale-free networks

We study a zero range process on scale-free networks in order to investigate how network structure influences particle dynamics. The zero range process is defined with the rate p(n)=nδ at which particles hop out of nodes with n particles. We show analytically that a complete condensation occurs when δ≤δc 1/(γ-1) where γ is the degree distribution exponent of the underlying networks. In the complete condensation, those nodes whose degree is higher than a threshold are occupied by macroscopic numbers of particles, while the other nodes are occupied by negligible numbers of particles. We also show numerically that the relaxation time follows a power-law scaling τ∼Lz with the network size L and a dynamic exponent z in the condensed phase.