Fig 1.
Diagram illustrating the different topologies on a subset of the network.
(a) Dashed black lines: two-dimensional directed lattice with coordination number K = 2. (b) Black solid lines: layered network with a randomised degree distribution (e.g. Gaussian or Scale-free) with randomly chosen neighbours in the adjacent layer below. (c) Black solid lines and red solid lines: networks created by adding links connecting non adjacent layers (red) in both directions to the layered network with randomised degree distribution (black).
Fig 2.
The avalanche-size pdf P(y) versus the avalanche size y obtained using the inter-firm Japanese network (solid black line).
The grey dashed lines are guides to the eyes for the different universality classes’ avalanche-size exponents.
Fig 3.
For all panels, the inset displays the avalanche-size pdf P (y; L) vs. the avalanche size y.
The large figures show the data collapse obtained by plotting the transformed avalanche-size pdf yτ P (y; L) vs. the rescaled avalanche size y/LD using the estimates of the avalanche-size scaling exponents τ and D obtained from moment scaling analysis, see Tables 1, 2, 3 and 4. For all figures, including insets, the line style indicates the system size, dashed-dotted: L = 100; dotted line: L = 200; dashed line: L = 400; solid line: L = 600 (a) Regular lattice; grey: K = 2, red: K = 4, blue: K = 6, black: K = 8, L = 200, 400, 600 (b) Gaussian out-degree distribution; red: σ = 0, blue: σ = 1, black: σ = 2, L = 200, 400, 600 (c) Truncated scale-free out-degree distribution; red: γ = 2.5, blue: = 3.0, black: = 3.5, L = 200, 400, 600 (d) Truncated scale-free in- and out-degree distribution; red: γ = 2.5, L = 100, 200, 400, black: γ = 3.5, L = 200, 400, 600.
Table 1.
The avalanche-size exponent, τ, and the avalanche-dimension, D, for regular lattice structures with coordination numbers, K = 2, 4, 6, 8 and circumference C, see Fig 3(a) for the data collapse.
The scaling relation D(2 − τ) = 1 is fulfilled and, within error bars, both scaling exponents (apart from K = 8 and C = 2000) are consistent with the universality class of the two-dimensional directed sandpile model τ = 4/3 and D = 3/2. The numerical result for K = 8, C = 2000 and 4000 suggests that the apparent drift is due to finite size effects.
Table 2.
The avalanche-size exponent, τ, and the avalanche-dimension, D, for networks with nodes’ out-degrees drawn from a Gausssian distribution with a fixed mean coordination number μ = 4, and standard deviations σ = 0, 1, 2.
Note that the case of σ = 0 is just a randomly rewired version of the regular lattice with coordination number K = 4, see Fig 3(b) for the data collapse. Within error bars, both scaling exponents are consistent with the mean-field model τ = 3/2 and D = 2.
Table 3.
The avalanche-size exponent, τ, and the avalanche-dimension, D, for networks with nodes’ out-degrees drawn from a truncated scale free distribution with exponent γ = 2.5, 3.0, 3.5, see Fig 3(c) for the data collapse.
The central limit theorem ensures the distribution of in-degrees is Gaussian. Within error bars, both scaling exponents are consistent with the mean-field model τ = 3/2 and D = 2.
Table 4.
The avalanche-size exponent, τ, and the avalanche-dimension, D, for networks with nodes’ in- and out-degrees were both drawn from a truncated scale free distribution with exponents γ = 2.5, 3.5, see Fig 3(d) for the data collapse.
Within error bars, both scaling exponents are consistent with the mean field-model τ = 3/2 and D = 2.
Fig 4.
The avalanche-size pdf P(y) versus the avalanche-size y obtained using the inter-firm Japanese network (solid black line).
With 25% of long range connections across layers in an otherwise layered network with nodes in- and out- degree drawn from a truncated scale-free distribution with exponent γ = 2.5 and system size L = 400 (dashed red line). The grey dashed lines are guides to the eyes for the different universality classes’ avalanche-size exponents.