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

Abstract

We investigate condensation phase transitions of the symmetric conserved-mass aggregation (SCA) model on random networks (RNs) and scale-free networks (SFNs) with degree distribution P(k) approximately k(-gamma). In the SCA model, masses diffuse with unit 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 is not equal to infinity, we numerically show that the SCA model undergoes the same type of condensation transitions as those on regular lattices. However, the critical line rho(c)(omega) depends on network structures. On SFNs with gamma < or = 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 nonzero density. For the existence of the condensed phase for gamma < or = 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< or = 3. The finite lifetime of a lamb on SFNs with gamma < or = 3 ensures the existence of the condensation at the zero density limit on SFNs with gamma < or = 3, at which direct numerical simulations are practically impossible. At omega = infinity, 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.

Condensation phenomena of the generalized conserved-mass aggregation model on complex networks

“…Region III-(a) of λ < λ c is the condensed phase, where the condensation with background distribution occurs for any ρ. Since the background distribution is stable in the limit N → ∞, we call the condensation the incomplete condensation to distinguish it from the complete condensation [16]. On the contrary, region III-(b) of λ λ z is the fluid phase where no condensation takes place for any ρ.…”

“…For α = λ = 0 the GSCA model is the ordinary SCA model [6,16]. In the SCA model masses diffuse with unit rate and unit mass is chipped with rate ω.…”

“…The condensation transition arises as a result of the competition between the diffusion and chipping process. It was shown that the existence of the condensation transitions depends on the spatial disorder [8], the symmetry of moving directions [9], the mass-dependent rate of diffusion and fragmentation [10][11][12][13], and also the underlying network structure [14][15][16][17].…”

“…On scale-free networks (SFNs) with degree distribution P (k) ∼ k −γ , it was shown that CA models [14][15][16][17] and ZRP [21,22] also exhibit the anomalous condensation transitions distinct from those on regular lattices. These mass-transporting models may model the transmission of data packets between computers.…”

“…The studies of the SCA model on SFNs showed that the nature of the condensation transitions for γ > 3 is identical to those on regular lattices. However, for γ 3 the condensation phenomena strongly depends on the underlying network structure, which leads to the completely different behavior from those on regular lattices [14][15][16][17]. Therefore, to investigate the general aspects of the condensation phenomena of the SCA model, it is better to study the SCA model on SFNs rather than regular lattices.…”