Condensation phenomena of conserved-mass aggregation model on weighted complex networks

Abstract

We investigate the condensation phase transitions of conserved-mass aggregation (CA) model on weighted scale-free networks (WSFNs). In WSFNs, the weight $w_{ij}$ is assigned to the link between the nodes $i$ and $j$. We consider the symmetric weight given as $w_{ij}=(k_i{k}_j{)}^{\alpha }$. In CA model, the mass $m_i$ on the randomly chosen node $i$ diffuses to a linked neighbor of $i$,$j$, with the rate $T_{ji}$ or an unit mass chips off from the node $i$ to $j$ with the rate $\omega T_{ji}$. The hopping probability $T_{ji}$ is given as $T_{ji}=w_{ji}/{\sum }_{<l>}{w}_{li}$, where the sum runs over the linked neighbors of the node $i$. On the WSFNs, we numerically show that a certain critical ${\alpha }_c$ exists below which CA model undergoes the same type of the condensation transitions as those of CA model on regular lattices. However for $\alpha \ge {\alpha }_c$, the condensation always occurs for any density $\rho$ and $\omega$. We analytically find ${\alpha }_c=(\gamma -3)/2$ on the WSFN with the degree exponent $\gamma$. To obtain ${\alpha }_c$, we analytically derive the scaling behavior of the stationary distribution $P_k^{\infty }$ of finding a walker at nodes with degree $k$, and the probability $D(k)$ of finding two walkers simultaneously at the same node with degree $k$. We find $P_k^{\infty }\sim k^{\alpha +1-\gamma }$ and $D(k)\sim k^{2(\alpha +1)-\gamma }$ respectively. With $P_k^{\infty }$, we also show analytically and numerically that the average mass $m(k)$ on a node with degree $k$ scales as $k^{\alpha +1}$ without any jumps at the maximal degree of the network for any $\rho$ as in the SFNs with $\alpha =0$.

8 pages, 6 figures

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