Initial flux fluctuations of random walks on regular lattices Sooyeon YOON, Byoung-sun AHN, Yup KIM (Dept. of Phys.,Kyung Hee Univ.)

Abstract We study a characteristic coupling between average flux and dispersion σi at each site i on 1 and 2-dimensional lattices when random fluctuated incoming walkers from the outside undergoes random diffusion. We assume that variations of the number of walkers, W, define a dynamic variable chosen from an uniform distribution in the interval [W-ΔW, W+Δ W]. Each walker does M-step random walks from a location which they are initially placed at random on 1, 2-dimensional lattices. After M steps, we measured the average flux vs dispersion at each lattice site. We study the dispersion depends on the average flux as σ~ α. The two scaling regimes are found. one is the regime with α ≃ 0.5 and the other is one with α=1. Disorder System Research Group Kyung Hee University 1

1. Motivation of Study (1) Fluctuation in Complex Network Structure.  M. Atgollo de Menezes and A.-L. Barabasi, Phys. Rev. Lett. 92, (2004) The coupling of the flux fluctuations with the Total flux on individual nodes obeys a unique scaling law. - Internet, Microchip, WWW, River Network, Highway (2) How about on Regular Lattice? (3) How does dispersion depend on the time? σ~ α Disorder System Research Group Kyung Hee University 2

2. Model We assume that variations of the number of walkers, W, define a dynamic variable chosen from an uniform distribution in the interval [W- ΔW, W+Δ W]. We start with 1D, 2D lattice of size N. (3) Each walker does T-step random walks from a location which they are initially placed at random. (4) After T-steps, we measured the average flux vs dispersion σi at each lattice site. Disorder System Research Group Kyung Hee University 3

3. Simulation Results (1) 1D N=10000, W=10000, R=100, T=100 Disorder System Research Group Kyung Hee University 4

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(2) 2D N=100, W=10000, R=100 Disorder System Research Group Kyung Hee University 6

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(3) Scale-free Network (  =3) N  10000, W=10000, R=100 Disorder System Research Group Kyung Hee University 8

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(1)  W=0 : only internal fluctuations which comes from the random diffusion (2)  W  0 :The total incoming flux is changed at each experiment.The external driving force  dr (  W) contributes to the fluctuations with a dispersion. The total fluctuations for node i,  W=0, For,, the fluctuation are dominated by the changes in the external driving force. 10

Disorder System Research Group Kyung Hee University Assume, Therefore, increasing the fluctuations  W induce a change from the  =1/2 to the  =1. 11

4. Summary and Discussion Disorder System Research Group Kyung Hee University We study a characteristic coupling between average flux and dispersion σi at each site i on 1 and 2-dimensional lattices when random fluctuated incoming walkers from the outside undergoes random diffusion. 1.We found the coupling of the average flux and the dispersion satisfies the scaling relation σ~ α. (1) When the small initial flux comes in the system,  1/2. (2) But the initial incoming flux becomes larger,  1. These results do not depend on the network structures. 2. We also found that the effect of the incoming flux through the relation of dispersion and time. The time dependence of the dispersion changes from to as increasing the incoming flux. 12