The avalanche-size exponent, <i>τ</i>, and the avalanche-dimension, <i>D</i>, for regular lattice structures with coordination numbers, <i>K</i> = 2, 4, 6, 8 and circumference <i>C</i>, see Fig 3(a) for the data collapse.

<p>The scaling relation <i>D</i>(2 − <i>τ</i>) = 1 is fulfilled and, within error bars, both scaling exponents (apart from <i>K</i> = 8 and <i>C</i> = 2000) are consistent with the universality class of the two-dimensional directed sandpile model <i>τ</i> = 4/3 and <i>D</i> = 3/2. The numerical result for <i>K</i> = 8, <i>C</i> = 2000 and 4000 suggests that the apparent drift is due to finite size effects.</p

The avalanche-size pdf <i>P</i>(<i>y</i>) versus the avalanche size <i>y</i> obtained using the inter-firm Japanese network (solid black line).

<p>The grey dashed lines are guides to the eyes for the different universality classes’ avalanche-size exponents.</p

The avalanche-size pdf <i>P</i>(<i>y</i>) versus the avalanche-size <i>y</i> obtained using the inter-firm Japanese network (solid black line).

<p>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 <i>L</i> = 400 (dashed red line). The grey dashed lines are guides to the eyes for the different universality classes’ avalanche-size exponents.</p

The avalanche-size exponent, <i>τ</i>, and the avalanche-dimension, <i>D</i>, for networks with nodes’ out-degrees drawn from a truncated scale free distribution with exponent <i>γ</i> = 2.5, 3.0, 3.5, see Fig 3(c) for the data collapse.

<p>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 <i>τ</i> = 3/2 and <i>D</i> = 2.</p

For all panels, the inset displays the avalanche-size pdf <i>P</i> (<i>y</i>; <i>L</i>) vs. the avalanche size <i>y</i>.

<p>The large figures show the data collapse obtained by plotting the transformed avalanche-size pdf <i>y</i>^τ<i>P</i> (<i>y</i>; <i>L</i>) vs. the rescaled avalanche size <i>y</i>/<i>L</i>^<i>D</i> using the estimates of the avalanche-size scaling exponents τ and <i>D</i> obtained from moment scaling analysis, see Tables <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142685#pone.0142685.t001" target="_blank">1</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142685#pone.0142685.t002" target="_blank">2</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142685#pone.0142685.t003" target="_blank">3</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0142685#pone.0142685.t004" target="_blank">4</a>. 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, <i>L</i> = 200, 400, 600 (b) Gaussian out-degree distribution; red: σ = 0, blue: σ = 1, black: σ = 2, <i>L</i> = 200, 400, 600 (c) Truncated scale-free out-degree distribution; red: γ = 2.5, blue: = 3.0, black: = 3.5, <i>L</i> = 200, 400, 600 (d) Truncated scale-free in- and out-degree distribution; red: γ = 2.5, <i>L</i> = 100, 200, 400, black: γ = 3.5, <i>L</i> = 200, 400, 600.</p

Diagram illustrating the different topologies on a subset of the network.

<p>(a) Dashed black lines: two-dimensional directed lattice with coordination number <i>K</i> = 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).</p

RSOS172434_RAWDATA

Raw data corresponding to Table I in submission

The skewed activity distribution.

<p>Individual ant activity is measured on a per-ant basis, as the work done per time-step. Main panel; an activity-rank plot. A rank of 1^st indicates the ant was the most active, and a rank of 20^th indicates the ant was the least active. Panel insert: the same data as the main panel plotted as the survivorship of the individual ant activity. The distribution is exponential-like. Model parameters as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0018416#pone-0018416-g002" target="_blank">Figure 2</a> legend. All realisations were run for 50000 time-steps after reaching the steady-state.</p

The distance separating neighbouring ants depends upon the difference in their response-thresholds.

<p>a) Map of ant locations. Symbol sizes are proportional to the threshold of the ant, so sensitive ants have small symbols. b) Mean distance between an active ant and its nearest neighbour (NND), as a function of the <i>difference</i> in sensitivity between the two (<i>θ_i</i> - <i>θ</i>_NN). Error bars are one standard deviation. Drive, <i>δ</i>S = 10, N simulations = 100, N time-steps per simulation = 5×10⁴. The horizontal line shows the expected NND under conditions of complete spatial randomness (Expected NND = 2.56, <i>σ</i> = 1.25, N simulations = 2000).</p

Stimulus input and ant activity update rules for the two-dimensional model.

<p>Stimulus input and ant activity update rules for the two-dimensional model.</p