# Inferring Urban Social Networks from Publicly Available Data
## 4 Characterization of the Urban Social Network
### Network Adjacency Matrices and Degree Density Plot  
In Figure 9, for Florence (and in Figures 11 and 11 for the other cities), we report the scatter plot of the adjacency matrices relative to the obtained social graph. In the plots, nodes are ordered by their age and, within each age-group, by tile.  
In all matrices, the prevalence of intra-groups edges (assortativity by age) over inter-groups edges is clearly visible and in qualitative agreement with previous works on social mixing patterns [49; 50; 51; 52; 57]. The white stripes, clearly visible for \(\beta=2\) (Figure 9b), are a consequence of the sorting by tile (and of the use of tile-to-tile distances) and show the impact of distance on connection patterns. In References [49; 57], the authors also identified sub-diagonals which account for parent-children contacts. These sub-structures cannot be seen in our matrices because we used only four age-groups, while the aforementioned studies relied on a stratification of the population into 5-year segments. In contrast, we see that the adult group, which is the largest group in the population, dominates the inter-groups contacts.  
To further investigate the role of population density in shaping the graph, in Figure 10 (and Figures 11 and 11), we show the number of individuals living in each tile and their average degree, for two selected configurations both with \(\mu=10\), so that the average degree of the whole graph is \(K\approx 12\). It is apparent that the average degree per tile is strongly influenced by the choice of the distance function. While, in Figure 10b, the situation is mostly homogeneous with few tiles above average, in Figure 10c, most tiles are far below average, while the tiles surrounded by a densely populated area have a high average degree. The increase of \(\beta\) has the effect of decreasing the probability of long contacts, a condition that favors the concentration of hubs in high density areas.  
Figure 9: (**Florence**) Adjacency matrix of the social graph with nodes (people) ordered by age-group. In both cases, \(f_{u}\sim 1+\mathcal{LN}(\ln(2),0.25)\), \(\mu=10\), and \(D(u,v)=d(u,v)^{-\beta}\), but \(\beta\) varies between the two figures. The assortativity by age is clearly visible in both cases.  
Finally, to assess the impact of \(f_{u}\), we consider two different configurations having both \(\mu=10\) and \(\beta=2\), but one with \(f_{u}\sim 1+\mathcal{LN}(\ln(2),0.25)\) and the other with \(f_{u}\equiv 1\). Figure 10(a) shows the difference of the mean degree per tile between these two configurations, which results being negligible across the whole city. Figure 10(b) instead shows the difference of the maximum degree per tile, which is huge for central and densely populated areas. In essence, for fixed \(\mu\), the distance function \(D\) governs the average degree of each tile, whereas the fitness scores \(f_{u}\) has a major impact on its variance, as already emerged from Figure 7(c) for the whole graph. If \(f_{u}\equiv 1\), the degree distribution within a single tile is almost flat, except for the age-based variance induced by the social mixing matrix \(S\). With a long-tailed distribution for \(f_{u}\), we instead obtain individuals with different sociability inside each tile.