# Inferring Urban Social Networks from Publicly Available Data
## 4 Characterization of the Urban Social Network
### Global Metrics  
We assessed the impact of using the real population density and an inverse-power-law distance penalization in Section 3.3. In this section, we focus again on the main global features of the graph, but considering the fully data-driven model that includes the household layer \(G_{H}\) and the age-based social mixing imposed through the matrix \(S\) (see Appendix B).  
By using the data-driven values of \(S\), we are adding to the model an age-specific homophily and we expect an increase in the assortativity of the resulting graph. Indeed, for the city of Florence, considering the other parameters as in Section 3.3 with \(\beta=2\) and \(\mu=10\), the assortativity grows from 0.2 to 0.48. Moreover, by connecting with higher probability nodes belonging to specific subsets (i.e., those in the same age groups), we report a four-fold increase of the global clustering coefficient from 0.00012 to 0.0004. On the other hand, the layer \(G_{H}\) introduces, by construction, a large amount of cliques--albeit each has a small size--so we can expect a great impact on the clustering coefficient (i.e., the number of triangles). In fact, considering again the city of Florence, with the same parameters of the previous example and adding the households, the global clustering coefficient increases \(\sim\)40 times from 0.0004 to 0.0151.  
An overview of the resulting graph for the city of Florence with the parameter chosen as described at the beginning of Section 4 is reported in Table 6--for Viterbo and Sabaudia see Appendices C and D. For each set of parameters, we repeated the construction of the social graph 10 times. The values reported in the table are the mean computed over each set of runs, the variance is negligible for all the values, thus being omitted. By looking at Table 6, we see that the contribution of the fitness to the clustering coefficient, the assortativity and the connectivity is only marginal compared to \(S\) and to the households (for a direct comparison with the configurations considered in Section 3.3, see Table 5). The value of the global clustering is several orders of magnitude greater than the corresponding ER graphs, it shows a small decrease with the fitness and for higher values of \(\beta\) (and \(\mu\)). Both circumstances favor the creation of hubs that attract edges from the rest of the graph, thus making \"local\" triangles less likely to occur. Besides the presence of hubs, there will be a higher number of nodes at a lower degree and, therefore, a higher probability of having many connected components. The degree heterogeneity also penalizes the assortativity which is consistently lower for the run with the fitness but increases with \(\beta\). It is likely that, on average, the higher value of \(\beta\) reinforces the group-to-group assortativity of \(S\). On the other hand, the average local clustering has the opposite behavior and constantly increases with the fitness, \(\beta\) and \(\mu\). Local connections are favored by a stronger dependence on distance and by the presence of local hubs. Almost regardless of the configuration, the average shortest path length is comparable to \(\frac{\ln(N)}{\ln(K)}\), where \(K=\mu+\
u\) is the average degree of the graph \(G\). This value is typical for small world networks.