Bak-Tang-Wiesenfeld model in the finite range random link lattice

We consider the BTW model in random link lattices with finite range interaction (RLFRI). The degree distribution of nodes is considered to be uniform in the interval (0,n0). We tune the topology of the lattices by two parameters (n0,R) in which R is the range of interactions. We numerically calculate the exponents of the statistical distribution functions in terms of these parameters. Dijkstra radius is utilized to calculate the fractal dimension of the avalanches. Our analysis shows that for a fixed n0 value there are two intervals of R, namely (1,R0) and (R0,L) each of which has a distinct behavior. In the first interval the fractal dimension monotonically grows from Df(R=1)≃DfBTW=1.25, up to Df≃5.0±0.4. We found however that in the second interval there is a length scale r0(n0,R) in which the behaviors are changed. For the scales smaller than r0(n0,R), which is typically one decade, the fractal dimension is nearly independent of n0 and R and is nearly equal to 2.0±0.2. We retrieve the BTW-type behaviors in the limit R→1 and find some new behaviors in the random scaleless lattice limit, i.e. R→L. We also numerically calculate the explicit form of the number of unstable nodes (NUN) as a time-dependent process and show that for regular lattice, it is (up to a normalization) proportional to a one-dimensional Weiner process and for RLFRI it acquires a drift term. Our analytical analysis shows that the relaxation time (exit time) for NUN process for RLFRI is related to a fitting parameter of NUN and is shorter than the regular one.