## 4 Discussion  
The analyses carried out in this work shed light on several aspects of complex networks synthesised from the origin-destination commuting flows within the Lombardy region. First of all, the correlation analysis between the graph metrics strength and degree shows that although the three networks exhibit the same scale-free behaviour, they are characterised by different weight dynamic growth rates, as revealed by the \(\alpha\) exponent. In particular, the network \"work\" shows the most critical dynamic growth of the weights. The critical role of this network in the diffusion phenomena is highlighted by the simulations of the epidemic spread over the complex networks. Indeed, as shown in Figure 6, the network \"work\" exhibits higher values of the two parameters linked to the number of infected nodes compared to the other two networks, thus proving the greatest contribution to the epidemic spread.  
More sophisticated models have been proposed to analyze the COVID-19 outbreak by using the mobility flows. As an example, in [27] Graph Neural Networks were combined with Recurrent Neural Networks for modeling the spatial and temporal components of the epidemic diffusion among provincial and regional nodes. The authors combined the complex network modeling with deep learning techniques to tune the time-varying parameters of an epidemiological model (e.g., contact rates and recovery rates). In [28], the authors investigated the early COVID-19 global transmission from one country to another by examining the human mobility networks related to inflows of migration and  
Figure 6: Error bars (mean value and standard deviation) for the distributions of (**A**)the peak values of \(I(t)\), (**B**) the temporal location of the peak of \(I(t)\), (**C**) the area under \(I(t)\) computed for different values of \(R_{0}\).  
tourism. The authors tested the impact of incoming migration and inbound tourism upon the initial global spread of COVID-19 showing the significance of complex networks modelling in shaping social interactions and mobility fluxes for epidemic forecast. Freely available data allowed to test COVID-19 diffusion models at different spatial scales such as spreading between countries, regions and provinces within national limits [29; 30; 31; 32].  
In contrast to these studies, our work did not aim to predict the variables of the epidemic phenomenon, but one of the main goals was to provide a multivariate detailed description of the mobility flows within the Italian Lombardy region by means of complex networks. Indeed, here we exploited the information about the daily commuting flows to construct four mobility networks with very high spatial granularity. A limitation of this work is that the OD matrices did not refer to the period of the COVID-19 epidemic spread, not allowing a quantitative comparison with the actual epidemiological conditions. On the other hand, our findings could be exploited to understand both the topological and dynamic structure of the mobility networks derived by classifying the flows of individuals. This static and dynamic information, as well as the epidemic simulation scenarios, can be leveraged to optimize containment actions such as targeted lockdowns that can impact each of the three networks and specific hub nodes within them.