## 4 Experimental Evaluation
### Results
The variation in the results using this amount of simulations is small enough and straightforward so as to produce a clear curve, as shown in Figure 4b. More simulations are not expected to alter in any significant way the shape of the curve, which is what the aim of the experiment is. Figure 4a provides a screenshot from the simulation framework during an experiment where there are 600 users at the simulated centre of Athens and the missing child. The experiments conducted for the Athens centre highlighted how this simulation framework could be used to examine the necessary adoption rates of such a citizen sensing mobile application per area to be successful. The results for the first indicator, days for the child to be found, are shown in Figure 4b,c. A horizontal dashed line at days equal to 2 has been added to show the baseline value of the experiment. It can be seen in Figure 4b that the median values of the days for the missing child to be found for the different number of users create a curve resembling a hyperbola. Indeed, fitting the data through nonlinear regression to different curves, the best fit, by using nonlinear least-squares minimisation, was found to be:  
\[f(x)=\frac{a}{x},\text{ where }\;\;a=466.18 \tag{1}\]  
The optimal value for parameter \(a\) was determined so that the sum of the squared residuals (SSR) of \(f(x)-y_{data}\) is minimised, namely, the minimum of:  
\[SSR=\sum_{i=1}^{108}(f(x_{i})-y_{i})^{2}, \tag{2}\]  
where \(y_{i}\) is the \(i\)th value of the sample, \(f(x_{i})\) is the predicted value of \(y_{i}\), and \(x_{i}\) is the \(i\)th value of the independent variable (i.e., _nb_people_). The fitted curve over the original data is also depicted in Figure 4b. The resulting fitted curve can then be used in decision-making as it can be considered a utility function that quantifies the marginal utility of the users for the citizen sensing application in question. In other words, a diminishing marginal utility [94]--which is a concept also used lately in a broader sense than its original economics scope [95,96]--can be observed in Figure 4b as its first users yield more utility than the subsequent ones, with a continuous decline for more users. Therefore, the decision-makers--the missing children organisations in that case--will be able to determine the expected benefit from the acquisition of new users of the mobile application to adapt their marketing strategies and compare this solution over their other approaches for engaging the public. According to the data, the median value for the 'days to be found' falls below the baseline for the first time at _nb_people_ = 210. After 500 users, a horizontalisation of the curve appears, approaching a horizontal asymptote. Therefore, while the critical mass cannot be an exact number, it can be claimed that around 500 users of the mobile application are needed in the examined geographical area to outperform the current situation and have significantly better results. Less than 500 users mean that the area is not yet sufficiently covered, and, thus, each new user offers significance to the investigation. More than 500 users would increase the effectiveness, but with diminishing returns per new user. An ANOVA test also concurred that until they reach 500, more users would make a significant difference. In particular, the differences between the results for the time needed to find the child with _nb_people_ = 50 and with _nb_people_ = 550 are statistically significant (_p_-value = 0.0016), while for _nb_people_ = 550 and _nb_people_ = 1050, they are not (_p_-value = 0.394). Figure 4: _Cont._  
The simulated area is the centre of the city of Athens, Greece and is around 2.5 km\({}^{2}\). Overall, the average population density for the entire city is 7381 people/km\({}^{2}\)[97], while for its centre, it reaches more than 25,000 people/km\({}^{2}\). This is also close to the estimation from the mobile operator's data (\(\mu=\) 27,340 people/km\({}^{2}\) considering a 24.5% market share for the operator [98]).