Collective decisions on a fully-connected network: comparison between the micro and the macro dynamics and scaling properties.

<p>(<i>A</i>,<i>C</i>) Comparison between the stochastic finite-size macroscopic model (black lines) and the multiagent implementation with both the homogeneous strategy (red lines) and the heterogenous strategy (green lines). Results are displayed for varying system size <i>N</i>. For each possible configuration (<i>v</i>_<i>A</i>, <i>v</i>_<i>B</i>), 500 independent runs are performed. We show results for configurations with effectivity <math><mrow><mi>E</mi><mo>></mo><mn>0</mn><mo>.</mo><mn>7</mn></mrow></math>. The plot is divided in two parts: in the bottom-right triangle, we consider the success rate <math><mi>S</mi></math> for each configuration (<i>v</i>_<i>A</i>, <i>v</i>_<i>B</i>), where <i>v</i>_<i>A</i> ≥ <i>v</i>_<i>B</i>. For each group size <i>N</i>, we show the isolines at <math><mrow><mi>S</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>9</mn></mrow></math>. The gray triangle indicates quality value pairs below the target resolution <i>R</i> = 0.15 (i.e., configurations in which the two options are considered equivalent). In the top-left half of the plot, we consider the convergence time <math><mi>C</mi></math> for each configuration (<i>v</i>_<i>A</i>, <i>v</i>_<i>B</i>), where <i>v</i>_<i>B</i> ≥ <i>v</i>_<i>A</i> (i.e., the symmetric problems with respect to the bottom-right plot). For each group size <i>N</i>, we show the isolines at <math><mrow><mi>C</mi><mo>=</mo><mi>C</mi><mo>^</mo></mrow></math>. (<i>B</i>, <i>D</i>) Scaling of the convergence time <math><mi>C</mi></math> with the system size <i>N</i>. For each configuration (<i>v</i>_<i>A</i>, <i>v</i>_<i>B</i>), we fit the curve <math><mrow><mi>C</mi><mo>=</mo><mi>b</mi><mi>N</mi><mi>a</mi></mrow></math> and we show the heat-map for the fitted coefficient <i>a</i> (see the bottom-right triangle showing the coefficient value for each configuration (<i>v</i>_<i>A</i>, <i>v</i>_<i>B</i>), <i>v</i>_<i>A</i> ≥ <i>v</i>_<i>B</i>) and <i>b</i> (see the top-left triangle showing the coefficient value for symmetric configurations (<i>v</i>_<i>A</i>, <i>v</i>_<i>B</i>), <i>v</i>_<i>B</i> ≥ <i>v</i>_<i>A</i>) across the decision space. Also in this case we show only configurations where <math><mrow><mi>E</mi><mo>></mo><mn>0</mn><mo>.</mo><mn>7</mn></mrow></math> for all <i>N</i>, and the white space indicates configurations with low effectivity. (<i>A</i>,<i>B</i>) Results for case-study 1A with <i>v</i>_<i>i</i> ∈ [0, 1], <i>γ</i>_<i>i</i> = 0.6<i>v</i>_<i>i</i>, <i>α</i>_<i>i</i> = 0, <i>ρ</i>_<i>i</i> = 0.1<i>v</i>_<i>i</i>, <i>σ</i>_<i>i</i> = 1 and <i>i</i> ∈ {A, B}, <math><mrow><mi>C</mi><mo>^</mo><mo>=</mo><mrow><mn>50</mn></mrow><mi>s</mi></mrow></math>. (<i>C</i>,<i>D</i>) Results for case-study 1B with <i>v</i>_<i>i</i> ∈ [1, 10], <i>γ</i>_<i>i</i> = <i>ρ</i>_<i>i</i> = <i>v</i>_<i>i</i>, <i>α</i>_<i>i</i> = 1/<i>v</i>_<i>i</i>, <i>σ</i>_<i>i</i> = 10 and <i>i</i> ∈ {A, B}, <math><mrow><mi>C</mi><mo>^</mo><mo>=</mo><mrow><mn>1</mn></mrow><mi>s</mi></mrow></math>.