Article

# Random walk and Pair-Annihilation Processes on Scale-Free Networks

10/2005;
Source: arXiv

ABSTRACT We investigate the dynamic scaling properties of stochastic particle systems on a non-deterministic scale-free network. It has been known that the dynamic scaling behavior depends on the degree distribution exponent of the underlying scale-free network. Our study shows that it also depends on the global structure of the underlying network. In random walks on the tree structure scale-free network, we find that the relaxation time follows a power-law scaling $\tau\sim N$ with the network size $N$. And the random walker return probability decays algebraically with the decay exponent which varies from node to node. On the other hand, in random walks on the looped scale-free network, they do not show the power-law scaling. We also study a pair-annihilation process on the scale-free network with the tree and the looped structure, respectively. We find that the particle density decays algebraically in time both cases, but with the different exponent.

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### Keywords

decay exponent

degree distribution exponent

different exponent

dynamic scaling behavior

dynamic scaling properties

network size $N$

particle density decays algebraically

power-law scaling

power-law scaling $\tau\sim N$

random walker return probability decays algebraically

random walks

relaxation time

stochastic particle systems

tree structure scale-free network

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