Dynamics of Symmetric Conserved Mass Aggregation Model on Complex Networks

We investigate the dynamical behaviour of the aggregation process in the symmetric conserved mass aggregation model under three different topological structures. The dispersion σ(t, L) = (σ_i(m_i – ρ₀)²/L)^½ is defined to describe the dynamical behaviour where ρ_o is the density of particle and m_i is the particle number on a site. It is found numerically that for a regular lattice and a scale-free network, σ(t, L) follows a power-law scaling σ(t, L) ∼ t^δ₁ and σ(t, L) ∼ ^δ₄ from a random initial condition to the stationary states, respectively. However, for a small-world network, there are two power-law scaling regimes, σ(t, L) ∼ t^δ₂ when t < T and σ(t, L) ∼ t^δ₃ when t > T. Moreover, it is found numerically that δ₂ is near to δ₁ for small rewiring probability q, and δ₃ hardly changes with varying q and it is almost the same as δ₄. We speculate that the aggregation of the connection degree accelerates the mass aggregation in the initial relaxation stage and the existence of the long-distance interactions in the complex networks results in the acceleration of the mass aggregation when t > T for the small-world networks. We also show that the relaxation time τ follows a power-law scaling τ ∼ L^z and σ(t, T) in the stationary state follows a power-law σ_s(L) ∼ L^α for three different structures.