  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 $\omega \neq \infty$, 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 $\gamma \leq 3$, 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 $\gamma \leq 3$ 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 $\gamma \leq 3$. The finite life time
of a lamb on SFNs with $\gamma \leq 3$ ensures the existence of the
condensation at the zero density limit on SFNs with $\gamma \leq 3$ at which
direct numerical simulations are practically impossible. At $\omega = \infty$,
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.
