Condensation phase transitions of symmetric conserved-mass aggregation model on complex networks

We investigate condensation phase transitions of symmetric conserved-mass aggregation (SCA) model on random networks (RNs) and scale-free networks (SFNs) with degree distribution $P(k)\sim k^{-\gamma }$. In SCA model, masses diffuse with unite rate, and unit mass chips off from mass with rate $\omega$. The dynamics conserves total mass density $\rho$. In the steady state, on RNs and SFNs with $\gamma >3$ for $\omega \mathrm{\ne }\mathrm{\infty }$, we numerically show that SCA model undergoes the same type condensation transitions as those on regular lattices. However the critical line ${\rho }_c(\omega )$ depends on network structures. On SFNs with $\gamma \le 3$, the fluid phase of exponential mass distribution completely disappears and no phase transitions occurs. Instead, the condensation with exponentially decaying background mass distribution always takes place for any non-zero density. For the existence of the condensed phase for $\gamma \le 3$ at the zero density limit, we investigate one lamb-lion problem on RNs and SFNs. We numerically show that a lamb survives indefinitely with finite survival probability on RNs and SFNs with $\gamma >3$, and dies out exponentially on SFNs with $\gamma \le 3$. The finite life time of a lamb on SFNs with $\gamma \le 3$ ensures the existence of the condensation at the zero density limit on SFNs with $\gamma \le 3$ at which direct numerical simulations are practically impossible. At $\omega =\mathrm{\infty }$, we numerically confirm that complete condensation takes place for any $\rho > 0$ on RNs. Together with the recent study on SFNs, the complete condensation always occurs on both RNs and SFNs in zero range process with constant hopping rate.