## 2 Materials and Methods
### Experiments
#### 2.2.1 Experiment 1: The Precision for Driving  
In this experiment, we performed three simulations. In simulation 1, the parameter sigma = 0; in simulation 2 sigma = 0.5; in simulation 3, sigma = 1. At this point, it should be mentioned that sigma = 0.5 (in between perfect driving and the most imperfect driving) is comparable with human driving behavior [41]. The network density vs. network speed pattern (for any simulation) was modeled with a linear equation of the form \(y=ax+b\) and solved with regression analysis, where a and b are parameters, x = network density and y = network speed; the network speed vs. network flow pattern (for any simulation) was modeled with a polynomial equation of the second degree, i.e., \(y=x^{2}a+xb+c\) which was solved with regression analysis, where a, b, and c are parameters, x = network speed and y = network flow. The network density vs. network flow pattern (for any simulation) was modeled with a polynomial equation of the second degree, i.e., \(y=x^{2}a+xb+c\) which was solved with regression analysis, where a, b, and c are parameters, x = network density and y = network flow. To model the three patterns for any simulation, we considered a range of periods [3; 202], since at the beginning of a simulation, links are starting to fill up, whereas they become empty towards the end. Hereafter, simulation 1 is abbreviated as S1, simulation 2 is S2, and simulation 3 is S3. We establish if network density vs. network speed (DS) patterns are similar using Equation (9), which is presented with the example for calculating the distance between the modeled DS patterns of simulation 1 and simulation 2,  
\[D_{\text{speed}}^{\text{S1S2}}=\frac{1}{N}\sum_{i=1}^{i=N}\left|y_{i}^{S2}-y_{i} ^{S1}\right| \tag{9}\]  
where N is the number of points selected in the x-axis (network density values), i is the index for N values, superscripts S1, S2, and S3 stands for simulation 1, simulation 2, and simulation 3, respectively; then \(D_{\text{speed}}^{\text{S1S2}}\) is the speed distance between the modeled DS patterns of S1 and S2. In Equation (9), we can see the absolute value of the difference between the modeled network speed values of S1 and S2 for a certain network density value. As we are comparing S1 and S2, the x-axis values set of S1 is \(R^{\text{S1}}=\left\{x_{1}^{\text{S1}},x_{2}^{\text{S1}},\ldots,x_{N1}^{\text {S1}}\right\}\) and the x-axis values set of S2 is \(R^{\text{S2}}=\left\{x_{1}^{\text{S2}},x_{2}^{\text{S2}},\ldots x_{N2}^{\text {S2}}\right\}\). With N1, for the number of network density values of S1 and N2 the number of network density values of S2, the values of these sets are ordered from the lowest to the highest. The x-axis set of values for comparing the y-axis values of S1 and S2 is \(\left\{x_{1},x_{3},\ldots,x_{N}\right\}=\left\{x:x\in R^{\text{S1}}\cup R^{ \text{S2}}\text{ and }x_{1}\leq x\leq x_{N}\right\}\), where \(x_{1}=\max\left(x_{1}^{\text{S1}},x_{1}^{\text{S2}}\right)\) and \(x_{N}=\min\left(x_{N1}^{\text{S1}},x_{N2}^{\text{S2}}\right)\). Based on Equation (9), we compare the DC patterns of two simulations. We compare S1 with S2, S2 with S3, and S1 with S3. When comparing the DS patterns of two simulations, a condition rule is established that depends on a threshold value. This value is intended to be flexible so it can be set according to the needs of an investigation (providing the rigor needed to distinguish DS patterns). We establish that if \(D_{\text{speed}}\leq 1\) m/s (in this example, the threshold value is 6% of the normal_speed, i.e., \(16.66\frac{m}{s}*0.06\cong 1\frac{m}{s}\)) the following is true: there is no significant difference between the patterns. If \(D_{\text{speed}}>1\) m/s, there is a significant difference between the patterns.