A Study of Memory Effects in a Chess Database - Fig 1

<p>(a) Chess tree corresponding to the main opening-lines up to depth <i>d</i> = 4. The size of the nodes is proportional to their popularity. Here only the main lines are shown. (b) Distribution of popularities of the nodes at depth <i>d</i> = 1, 2, 3 and 4; these distributions are well fitted by power laws <i>P</i>(<i>k</i>) ∝ <i>k</i>^−<i>α</i> with <i>α</i> = 1.10 ± 0.05, 1.29 ± 0.03, 1.47 ± 0.02 and 1.59 ± 0.02 (<i>R</i>² = 0.972, 0.993, 0.996 and 0.997), respectively. Errors estimated by the fitting.</p

(a) Burstiness parameter <i>B</i> and (b) burstiness exponent <i>β</i> as a function of the length of the time series for the database growing by Player Aggregation (black triangles) and by chronological game aggregation (gray diamonds).

<p>The error bars are not shown because they are of the same size as the points.</p

Summary of the results corresponding to the inter-event time distributions of Figs 5 and 6.

<p>Summary of the results corresponding to the inter-event time distributions of Figs <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0168213#pone.0168213.g005" target="_blank">5</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0168213#pone.0168213.g006" target="_blank">6</a>.</p

Cumulative distribution of inter-event times of the most popular opening-line measured in the database (black triangles) and generated with CM for <i>p</i> = 0.005 and <i>τ</i>_<i>c</i> = 96 (green diamonds).

<p>The lines are fits according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0168213#pone.0168213.e020" target="_blank">Eq (7)</a> with <i>β</i> = 0.927 ± 0.003 (<i>R</i>² = 0.999) for the database (black dashed line) and for CM with <i>β</i> = 1.035 ± 0.003 (<i>R</i>² = 0.999) (green full line). Inset: cumulative frequency distribution of inter-event times measured in the database (black triangles), YSM for <i>p</i> = 0.1 (magenta circles) and <i>p</i> = 0.005 (cyan squares).</p

Hurst exponent obtained by the DFA method as a function of the length of the time series in the database (dotted black line with triangles), and generated with CM, <i>p</i> = 0.005 and <i>τ</i>_<i>c</i> = 96 (full green line with diamonds): (a) using the PAR; (b) using the GAR; and (c) using the UAR.

<p>Hurst exponent obtained by the DFA method as a function of the length of the time series in the database (dotted black line with triangles), and generated with CM, <i>p</i> = 0.005 and <i>τ</i>_<i>c</i> = 96 (full green line with diamonds): (a) using the PAR; (b) using the GAR; and (c) using the UAR.</p

Log-log plot of the distribution of popularities: measured in the database (black triangles), fitted with <i>P</i>(<i>k</i>) ∝ <i>k</i>^−<i>α</i> and exponent <i>α</i>_<i>d</i> = 1.59 ± 0.02 (<i>R</i>² = 0.997) (dotted black line); generated with CM, <i>p</i> = 0.005 and <i>τ</i>_<i>c</i> = 96 (green diamonds) fitted with <i>P</i>_<i>CM</i>(<i>k</i>) (see Eq (3)) with parameter Ω = 1.5 ± 0.3 (full green line); and generated with YSM model, <i>p</i> = 0.1 (magenta circles) and fitted with <i>P</i>(<i>k</i>) and exponent <i>α</i>_<i>YSM</i> = 2.12 ± 0.03 (<i>R</i>² = 0.997) (magenta point dashed line).

<p>Log-log plot of the distribution of popularities: measured in the database (black triangles), fitted with <i>P</i>(<i>k</i>) ∝ <i>k</i>^−<i>α</i> and exponent <i>α</i>_<i>d</i> = 1.59 ± 0.02 (<i>R</i>² = 0.997) (dotted black line); generated with CM, <i>p</i> = 0.005 and <i>τ</i>_<i>c</i> = 96 (green diamonds) fitted with <i>P</i>_<i>CM</i>(<i>k</i>) (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0168213#pone.0168213.e006" target="_blank">Eq (3)</a>) with parameter Ω = 1.5 ± 0.3 (full green line); and generated with YSM model, <i>p</i> = 0.1 (magenta circles) and fitted with <i>P</i>(<i>k</i>) and exponent <i>α</i>_<i>YSM</i> = 2.12 ± 0.03 (<i>R</i>² = 0.997) (magenta point dashed line).</p

Cumulated inter-event time distribution for a single player (black triangles).

<p>Also, corresponding fits according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0168213#pone.0168213.e020" target="_blank">Eq (7)</a> with <i>β</i> = 0.583 ± 0.004 (<i>R</i>² = 0.993) for the player (black dashed line).</p