# Inferring Urban Social Networks from Publicly Available Data
## 4 Characterization of the Urban Social Network
### Connectivity, Communities, Degree  
Hereafter, we focus on the graph we obtained for the city of Florence, for which we provide an overview of the main characteristics in Figure 7. Similar results hold for the other two cities and are reported in Appendices C and D.  
First of all, by looking at Figure 7a, we notice that the fragmentation of the graph into connected components is very stable across different simulations and different configurations. For all eight considered combinations of parameters, a single giant component exists, covering at least 97.3% of the whole graph, with a few additional components having tenths to hundreds of nodes and a constellation of isolated vertices. We then focus on the modularity-based clustering obtained by applying to our network the well-known Louvain algorithm. From Figure 7b, we see that some parameters do have an impact on the organization of the graph into densely connected communities. In particular, a steeper cluster size distribution is associated with a greater average degree and with a stronger dependence of friendships on distance. These two conditions, in fact, favor the emergence of just a few giant clusters and of a multitude of clusters of variable size. When \(\mu=10\) and \(\beta=0.5\), we notice a sudden drop of the cluster size, with no clusters of size \(\approx 100\). Quite surprisingly, taking \(f_{u}\sim 1+\mathcal{LN}(\ln(2),0.25)\) seems to have a minor impact from this point of view, albeit this choice is supposed to guarantee the existence of large hubs. For the sake of completeness, in Figure 8, we compare the modularity of the obtained clustering structure for different configurations. We see that the modularity ranges from less than 0.3, when \(\mu=10\) and \(\beta=0.5\), to \(\approx\)0.5 when \(\mu=5\) and \(\beta=2\). Not surprisingly and in line with previous findings, the density of the network has a major impact on the quality of the obtained communities. Especially with denser networks, the penalty applied to long edges also has a visible effect, contrarily to the fitness score. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline  & & & & \multicolumn{4}{c}{\(\mu=5\)} \\ \(f_{u}\) & \(\beta\) & \(K\) & \(\langle\)**dist**\(\rangle\) & \(C\) & \(\mathbf{C_{loc}}\) & \(\rho\) & **\# Comp.** & **Giant \%** \\ \hline \(1\) & 0.5 & 6.82 & 6.72 & 0.053 & 0.074 & 0.261 & 4071.1 & 98.8\% \\ \(1\) & 2 & 6.81 & 6.72 & 0.050 & 0.089 & 0.334 & 6824.8 & 97.7\% \\ \(1+\mathcal{LN}(\ln(2),0.25)\) & 0.5 & 6.82 & 6.57 & 0.049 & 0.085 & 0.203 & 5357.4 & 98.4\% \\ \(1+\mathcal{LN}(\ln(2),0.25)\) & 2 & 6.81 & 6.58 & 0.046 & 0.100 & 0.258 & 8109.3 & 97.3\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: (**Florence**) Main features of the social graph as \(f_{u}\), \(\beta\) and \(\mu\) vary, for data-driven population and \(S\). \(K=\mu+\
u\) is the average degree, \(\langle\)dist\(\rangle\) is the average path length, \(C\) and \(C_{\text{loc}}\) are the global and average local clustering coefficients, \(\rho\) is the degree assortativity, “# comp.” denotes the number of connected components, and “giant %” denotes the percentage of nodes in the giant component. Our data-driven social network has the desired clustering and assortativity. The parameters \(f_{u}\) and \(\beta\) have a minor, albeit visible, impact. In Figure 7c, we finally show the degree distribution of the graph (we remind that household edges are included) for all configurations, in a log-log scale. At least when \(f_{u}\sim 1+\mathcal{LN}(\ln(2),0.25)\), we expect the right tail of the degree distribution of the graph to be heavy but not fat (i.e., subexponential but not power-law). To verify this insight, we included in the plot the result of a Lognormal fit of the portion of the distribution corre  
Figure 7: **(Florence)** Overview of the urban social graph. Each plot shows the average with confidence interval for 10 independent runs with the same configuration. The plots are discussed in Sections 4.2 and 4.3. **(a)** Size of the largest 50 connected components.