## 4 Experimental Evaluation
### Results
A total of 500 users of the application represent 0.8% of the 62,500 people considered to be living in the area (\(2.5\times 25,000\)), down to 0.7% of the 68,350 people, if using the mobile operator's density estimation. This means that for the characteristics of the centre of Athens, the user density required for effective investigations should be more than around 200 users/km\({}^{2}\), representing a little less than 0.7-0.8% of the population for that area. This estimation cannot be accurately extended to the entire city, for example, by calculating the average population density of the entire city, because this will lead to unfounded generalisations. The morphology and street layout greatly affect the citizens' movement and, thus, the circumstances and chances for a missing child to be found. Other parameters, such as if the area is a commercial or residential one, need also to be taken into consideration, and affect human mobility. Nevertheless, it can still be used for crude estimates and ballpark figures in the absence of more accurate information. Figure 4c reveals much variation in the data for each number of users examined, though reduced as we test for more users. This is expected since there is more uncertainty, because a successful encounter of the missing child is less likely to happen at any given moment. This variation is reduced the more agents are added to the experiment, as is the average time until an encounter with the missing child takes place. Using ten (10) test runs for each scenario reduces this uncertainty in the results, but at the same time, emphasises the fact that this greater upwards variation for low numbers of users translates into even more pronounced risks for the safety of the missing child. In other words, there are two categories of outcomes for missing children: they are either found in good health or not. A larger upwards variation in the time needed to locate each missing child will inevitably lead to fewer children ending up being found in good health. More users aware of the disappearance reduce the uncertainty and make the discovery not only timelier but also more predictable. Figure 4c also confirms our initial assumption that the critical mass of users cannot be a precise number. Nevertheless, for the selected area of the centre of Athens, we found that if the Amber Alert system was not in place, we would need at least 233 moving users to achieve the same result of less than 48 h until a missing runaway is located. In a scenario  
Figure 4: (**a**) Screenshot from the simulation for the Athens centre with the missing child (red dot) and 600 users (blue dots) (**b**), (**c**), (**d**) Results of simulation runs: (**b**) median value for the days needed for the missing child to be found, removing outliers using a z-score with a \(\pm\)3 standard deviations threshold. (**c**) Simulation results using boxplots for _nb_people_ mod 50 = 0. (**d**) Median value for the number of times the missing child is found in a 2-day simulation for different numbers of users. such as this, where multiple unpredictable parameters affect the turn of events, the aim is to reduce the uncertainty as much as possible or needed, to maximise the utility of the application. In the end, the resulting estimation of more than 200 users/km2 for the centre of this city cannot be decoupled from the reduction of the variance achieved in larger numbers of users when considering real cases of missing children. The real-world situation confirms that the variation that the simulation results illustrate aligns with the data from actual cases. As shown in Table 1, presenting data for the resolution of past cases, even cases of the same type are resolved at very different times, even though the same search and rescue mechanisms are activated. In other words, if most children are found within a couple of hours, that one which will be found a couple of days later will face greater risks.