Figure 1

Evolution of scores $S\left(t\right)$ for the three types of cricket. To make the notation easier, we have denoted the event-by-event evolution as time evolution. The main difference between these types is the maximum length of the game: 20 time steps for T20, 50 for ODI, and 200 for test. (a) Evolution of the scores of a hundred games selected at random from our database; (b) this evolution after removing the mean tendency to increase, that is, $S\left(t\right)-\langle S\left(t\right)\rangle$.Reuse & Permissions

Figure 2

Anomalous diffusion of the scores. (a) Mean value of the scores as a function of time, $\langle S\left(t\right)\rangle$, for the three types of cricket. Dashed lines are linear fits to each data set. We find the mean values to grow linearly with time. (b) Spreading of the score trajectories measured as the variance ${\sigma }^2\left(t\right)=\langle {[S\left(t\right)-\langle S\left(t\right)\rangle ]}^2\rangle$ versus time. Dashed lines are power-law fits to the variance, where we find the exponents $\alpha =1.32\pm 0.02$ for T20, $\alpha =1.31\pm 0.02$ for ODI, and $\alpha =1.30\pm 0.02$ for Test cricket. Since $\alpha >1$, the diffusive process underlying the evolution of scores is superdiffusive. The error bars are $95\%$ confidence intervals obtained via bootstrapping [13].Reuse & Permissions

Figure 3

Scale invariance of the scores. (a) Evolution of the cumulative distribution function (CDF) for the three types of cricket and for different values of time $t$. Note that the distributions shift towards positive values of the score and that the width of the CDFs increases. (b) Scale invariance of the scores. We evaluate the CDF using normalized scores $\xi \left(t\right)=\frac{S\left(t\right)-\langle S\left(t\right)\rangle }{\sigma \left(t\right)}$, where $\langle S\left(t\right)\rangle$ is the mean value of the scores and $\sigma \left(t\right)$ is the square root of the variance of the scores. Note that the good collapse of the distributions indicates that the scores present scaling properties, i.e., after normalization they follow the same universal distribution. In these plots, the solid lines are the CDFs for each value of $t$ and the symbols are the averaged values of these CDFs. The error bars are $95\%$ confidence intervals obtained via bootstrapping [13]. We note, further, that these distributions are very close to a normalized Gaussian distribution (dashed lines). Insets: $p$-values for the Pearson chi-square test [14] as a function of time. The dashed line is the threshold $0.1$ for rejecting the Gaussian hypothesis. Note that the normality is rejected for small values of $t$ because of the discrete values of $S\left(t\right)$ and also the asymmetric initial condition of the diffusive process. After enough time ($t\sim 90$), we cannot reject the Gaussian hypothesis in Test cricket.Reuse & Permissions

Figure 4

Long-range correlations in scores. (a) Detrended fluctuation analysis of score increments $\Delta S\left(t\right)=S(t+1)-S\left(t\right)$. We show the fluctuation functions $F\left(n\right)$ versus the scale $n$ (solid lines) for all games of Test cricket that are longer than 120 units of time (431 games). Note that $F\left(n\right)$ follows a power law, where the exponent $h$ is the Hurst exponent. We estimate the mean value of the Hurst exponent to be $h=0.63\pm 0.01$, and the dashed line is a power law with this exponent. We find the average value of the Pearson linear correlation coefficient to be $0.89\pm 0.02$, which enhances the quality of the power-law relationships. Symbols represent average values of the fluctuation functions, and error bars are standard errors of the means. (b) Comparison of the model prediction, that is, $\alpha =2h$, for Test cricket. The left bar shows the empirical value of $2h$ and the right bar shows the value of $\alpha$. Error bars are $95\%$ confidence intervals and the horizontal line is the upper limit of the confidence interval for $2h$. We note the existence of overlap in the confidence intervals, indicating that the relation $\alpha =2h$ holds.Reuse & Permissions