# Inferring Urban Social Networks from Publicly Available Data
## 4 Characterization of the Urban Social Network
### Geography of the Graph  
Since our model includes a factor purposely designed to penalize long friendship edges, it is especially worth assessing the correlation between topological properties and population density. To this end, we first show in Figure 7d the distribution of the edges' physical length. We immediately see that, as expected, the fitness function and the average degree appear to have a negligible impact on the distribution, contrary to the distance function. Our graph model indeed guarantees that both \(\mu\) and \(f_{u}\) only impact on the number of friends of \(u\), whereas, regardless of \(\mu\) and \(f_{u}\), \(D(u,v)\) is what ultimately determines the ratio of friends \(v\) that \(u\) will have at any given geographical distance. Of course, this works as long as the fitness is distributed independently of the location, which is true by design on a probability base. For what concerns \(D\), with respect to \(\beta=2\), setting \(\beta=0.5\) significantly favors the creation of long edges at the expenses of the very short ones.  
We now look into the obtained communities to see if they are geographically concentrated and/or bounded. To this end, Figure 7e shows the mean and max intra-cluster distance for the first 50 clusters of the graph. By comparing Figure 7e with Figure 7b, we realize that, albeit long intra-cluster edges tend to disappear as the cluster size gets smaller, only clusters that are at least 3 orders of magnitude smaller than the whole graph are geographically bounded--a phenomenon that is especially visible when \(\mu=10\), that is, when the cluster size distribution is steeper. This is partially in line with a previous work showing a sudden rise in the geographical extension of the communities of empirical networks [21]. All sufficiently large clusters behave very similarly to each other and to the whole graph: most intra-cluster edges connect nearby vertices, yet very long edges do exist in each cluster. This plot underlines that the parameter \(\beta\) has a paramount role even in the formation of clusters, with both the mean and max intra-cluster distance being consistently greater when \(\beta=0.5\) compared to the case \(\beta=2\). In particular, the mean distance when \(\beta=2\) is often less than the tile side \(l\) (set to 1 km as per Section 2.1), meaning that most adjacent vertices are at one tile of distance or less; the mean distance is instead between \(2l\) and \(3l\) when \(\beta=0.5\).  
Finally, in Figure 7f, we plotted the average geographical distance of a vertex \(u\) from all others against the degree of the vertex \(u\). Since we introduced a penalization for long edges, it is reasonable to expect that vertices that occupy a favorable position, closer, on average, to the other vertices, will generally have a greater degree. However, setting \(D=d^{-0.5}\)  
Figure 8: **(Florence)** Modularity of the obtained clustering structure—distribution for 10 independent runs per configuration. The quality of the obtained partition increases for small \(\mu\) and large \(\beta\), whereas it is not influenced by \(f_{u}\).  
seems to be enough to significantly dissipate this effect, which is instead clearly visible when \(D=d^{-2}\). Moreover, we also see that all large hubs have a very small average distance from all other vertices, regardless of the specific configuration. In particular, when \(f_{u}\sim 1+\mathcal{LN}(\ln(2),0.25)\), the greatest hubs correspond always to vertices having both a large \(f_{u}\) and a favorable position in the territory. Albeit the introduction of a social fitness in the model allows to have medium-large hubs in sparsely populated regions, the greater prevalence, on average, of hubs in densely populated areas may exacerbate the tendency of other vertices to establish links with individuals in those areas.