# Inferring Urban Social Networks from Publicly Available Data
## 3 Impact of System Parameters
### Safeguarding Connectivity  
There are two main factors that impact on the overall connectivity of the graph: the average number of friends \(\mu\) and the reconstructed households structure. In Section 2.3, we found that our efforts to introduce heterogeneity in the model while preserving certain structural properties inferred from contact data yield an upper bound for \(\mu\). Albeit there is no analogous strict lower bound, \(\mu\) must be chosen large enough to guarantee that the graph has the expected level of connectivity. To provide insights into the choice of \(\mu\), we compare the graphs obtained with \(\mu=1\) and \(\mu=5\), measuring the percentage of nodes belonging to the giant component for different configurations. We remind that the household layer \(G_{H}\) is composed of a set of cliques and has average degree \(\
u\approx 2\) for all cities--the average degree of the whole graph being instead \(K=\
u+\mu\). In the following, we will focus on the friendship layer \(G_{F}\) alone and on the entire urban social network \(G\). To prevent the results from being influenced by the composition of the population, we: (i) set the matrix \(S\) to a constant, to force homogeneous mixing between the different age groups (see Appendix B); (ii) ignore the data-driven population density, considering equally frequent age groups and distributing the individuals uniformly at random in the given territory (Actually, we map individuals only to tiles that are non-empty according to real data. This is meant to preserve the realism of the benchmark model by preventing that people are found in uninhabitable areas, such as parks, lakes, sea, etc.). For what  
Figure 1: A graphical representation of our territory and population, through a random sample of 1000 people on the territory of Florence and Sabaudia (some points may fall outside of the territory if the center of the tile to which they belong falls inside). \begin{table}
\begin{tabular}{l l c c c c c c} \hline \hline
**City** & **Boundary** & \(N\) & **C\%** & **Y\%** & **A\%** & **E\%** & \(\
u\) & \((T_{\text{lat}},T_{\text{lon}})\)** \\ \hline
**Florence** & City & 363,060 & 15.1\% & 16.9\% & 43.1\% & 24.9\% & 2.08 & \((15,12)\) \\
**Viterbo** & Municipality & 66,598 & 16.0\% & 17.8\% & 44.2\% & 22.0\% & 2.08 & \((16,20)\) \\
**Sabaudia** & Municipality & 21,274 & 16.8\% & 18.1\% & 45.6\% & 19.5\% & 2.07 & \((26,32)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Reconstructed parameters for the three cities. The initials C, Y, A, E stay for children, young people, adults, and elderly people, respectively. \(\
u\) is the average degree of the reconstructed household network \(G_{H}\), and \(T_{\text{lat}}\) and \(T_{\text{lon}}\) are the number of tiles along the two axes of the grid used to cover the city territory. concerns the parameters \(f_{u}\) and \(D\), we focus on the special configurations highlighted in Section 2.3--i.e., \(f_{u}\sim 1+\mathcal{LN}(\ln(2),0.25)\) and \(D(u,v)=d(u,v)^{-\beta}\) with \(\beta\in\{0.5,2\}\)--and we additionally compare them with the choice of a constant fitness \(f_{u}\equiv 1\) for all \(u\). Ideally, we would like a graph where (almost) all nodes belong to a single giant component, in accordance with empirical findings from the literature [11]. In Table 4, we present the case of Florence; similar results hold for the other cities and are listed in Appendices C and D. Let us first focus on the friendship graph \(G_{F}\) alone (i.e., no households). The case \(\mu=1\) clearly shows that this graph is very poorly connected if the average degree is very low. When the fitness is Lognormally distributed, thanks to the increased degree variability, the giant component includes \(\sim\)20% of the network. When the fitness is constant, the giant component is instead significantly smaller, reaching a critical 1.9% in the configuration that maximizes the homogeneity of the network. When \(\mu=5\), the giant component on the other hand, includes almost the entirety of the network.