Numerical World Color Survey with artificial JNDs.

<p>Top: the dispersion of the JND case, as defined in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0125019#pone.0125019.e001" target="_blank">Eq 1</a>, normalised by the neutral one obtained with a flat JND is plotted for populations of size <i>N</i> = 100. The horizontal line indicates the experimental value 1.14 obtained by the analysis of WCS data in [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0125019#pone.0125019.ref015" target="_blank">15</a>]. Bottom: the relative error <i>e</i>(%) between the average dispersions and the experimental result, <i>e</i> = ∣<i>sim</i> − <i>exp</i>∣/<i>exp</i> is plotted (in %). This measure quantifies the distance of the dots from the horizontal line in the Top panel. Different JNDs are named after <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0125019#pone.0125019.g002" target="_blank">Fig 2</a>. Vertical bars refer to the variation of values in the late stage of the simulation, in the range 1.5 × 10⁶ − 2 × 10⁶ games per agent.</p

The human Just Noticeable Difference (JND) function describes the wavelength change in a monochromatic stimulus needed to elicit a particular JND in the hue space.

<p>Both as a function of the wavelength of the incident light (measured in nanometers) and on the rescaled interval [0, 1). For convenience we display also the spectrum of the visible light. For the purpose of the Category Game we rescale the monochromatic stimulus in visible spectrum (measured in nanometers) in the range [0, 1) for the Topic <i>x</i>. In the same way the JND function (in nanometers in the left y-axis) is rescaled into a <i>JND</i>(<i>x</i>) function (right y-axis).</p

Numerical World Color Survey with human JNDs: weakness of the bias.

<p>The histogram of the squared distance </p><p></p><p><mi>d</mi><mo>=</mo></p><p><mo>∑</mo></p><p><mi>i</mi><mo>=</mo><mn>1</mn></p><mn>14</mn><p></p><p></p><p><mo stretchy="false">(</mo></p><p><mi>x</mi><mi>i</mi></p><mo>−</mo><p></p><p><mi>x</mi><mi>i</mi></p><mo>¯</mo><p></p><mo stretchy="false">)</mo><p></p><mn>2</mn><p></p><p></p><p></p> between the position of the <i>i</i>-th centroid and its average (“typical”) value. Black data represent the statistics of those populations displaying 14 linguistic categories at the end of Category Game simulations with the human JND, which were roughly ∼ 1300 of the total 5000 considered populations. The inset displays the actual values of <i>d</i> for each population. Green and red data come from a “Random” model where each “language” is produced by a uniform random distribution of 14 category centroids. A random case with a very large number of languages (red curve) represents the “ideal” statistics of such a model. The distances <i>d</i> from the average pattern (shown in the inset) appear to have a larger average <p></p><p></p><p></p><p><mi>d</mi><mo>¯</mo></p><p><mi>r</mi><mi>a</mi><mi>n</mi><mi>d</mi></p><p></p><p></p><p></p> than the Category Game model <p></p><p></p><p></p><p><mi>d</mi><mo>¯</mo></p><p><mi>C</mi><mi>G</mi></p><p></p><p></p><p></p> (where by <p></p><p></p><p><mi>d</mi><mo>¯</mo></p><p></p><p></p> we refer to the mean value of the random and the CG cases, respectively), but since we are interested in the fluctuations we have rescaled the random model data, dividing them by <p></p><p></p><p></p><p><mi>d</mi><mo>¯</mo></p><p><mi>r</mi><mi>a</mi><mi>n</mi><mi>d</mi></p><p></p><mo>/</mo><p></p><p><mi>d</mi><mo>¯</mo></p><p><mi>C</mi><mi>G</mi></p><p></p><p></p><p></p>, in order to compare the histograms. A power law fit ∼ <i>x</i>^{− 3} is also shown as a guide for the eye.<p></p

Artificial JNDs.

<p>Each panel depicts one of the artificial JNDs used in the experiments (continuous lines). The empirical fit of the human JND (dashed line, all panels) is obtained according to the expression JND(<i>x</i>) = <i>c</i>₁ + <i>c</i>₂cos(<i>ax</i> + <i>c</i>₃) + <i>c</i>₄(<i>x</i> − 0.5)², where <i>c</i>₁, <i>c</i>₂, <i>c</i>₃, <i>c</i>₄ and <i>a</i> are fitting constants (case <i>α</i>). By constant increases of the <i>a</i> parameter one finds increasingly irregular artificial JNDs (<i>β</i>, <i>γ</i> and <i>δ</i>). The last two panels present a Gauss-like JND (<i>ϵ</i>), and a reflection of the JND around the average value (with a further prescription avoiding negative values of the JND) (<i>ζ</i>).</p

Velocity distributions.

<p>PDF of the rotator’s angular velocity rescaled by for low (black circles, rad/s) and high (blue squares, rad/s) densities. The red dashed line shows a Gaussian fit for comparison.</p

Coupling with the gas.

<p>Correlation between the angular velocity of the probe and the average angular velocity of the fluid (see text for definition) for the most dilute and the most dense experiments.</p

Experimental setup.

<p>A sketch of the setup illustrates the essential components. A wheel rotating around a fixed axis is suspended in a cylindrical cell containing steel spheres. The cell is shaken in order to fluidize the material and obtain a granular gas. The wheel performs a Brownian-like dynamics, randomly excited by collisions with the spheres. A small motor is coupled to the wheel axis, in order to apply an external impulsive perturbation. An angular encoder reads the angular velocity of the wheel. Statistical properties of the velocities of the spheres are collected through a fast camera, placed above the system. A detailed description is presented in Methods section.</p

Response and autocorrelation.

<p>Response function (black circles), rescaled velocity autocorrelation (red squares), and GFDT response with the factorization assumption, Eq. (6), (green diamonds) for (a), (b) and (c), that is packing fractions , and , respectively. In the inset the parametric plot <i>vs </i>, in the region where is positive and monotonously decreasing, is plotted in log-log scale. In the densest cases, and behave very differently and Einstein’s relation is significantly violated.</p

Typical long-time configuration of five representative agents in the population.

<p>For each agent perceptual and linguistic categories (separated by short and long bars, respectively) are shown. The highlighted portion of two agents illustrates an instance of a successful game in a so-called mismatch region between the linguistic categories of the two agents associated with the words “a” and “b” (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0016677#s4" target="_blank">Materials and Methods</a> for details). The hearer - in a previous game - learned the word “a” as a synonym for the perceptual category at the leftmost boundary of the linguistic category “b”. During the game the speaker utters “a” for the topic; as a result the hearer deletes “b” from her inventory, keeping “a” as the name for that perceptual category, moving <i>de facto</i> the linguistic boundary.</p

Words per perceptual category.

<p>The average number of words per perceptual category across the population of  = 300, 500 agents versus the number of games per player. The inset is a zoom showing after games per player. Clearly, does not settle to one even after a very long time. The value of here is equal to which is the average of human JND (when projected on the interval) <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0016677#pone.0016677-Long1" target="_blank">[38]</a>.</p