Using information fields to study structure and dynamics of uncertain networks

Network geometry provides different tools to study the effective geometry induced by dynamical processes in networks. These different approaches led to different definitions of latent space, where distances between nodes correspond to distances in this high-dimensional manifold. In this thesis, we use UMAP, a manifold reconstruction and low-dimensional projection technique, to reconstruct the latent space induced both by the structure of the network and the dynamics running over it. To this aim, we use a notion of metric distance found in the literature, the Jacobian distance. This metric captures the spatiotemporal dynamics of perturbations within a network, offering a nuanced view of the latent geometric structure inherent to these systems. Since UMAP has no restriction on embedding dimension, we can project the original latent space of any dimension into a field, where we can use the tools of information field theory (IFT), a mathematical framework originally developed for astrophysics, to make optimal field reconstruction. Since real-world networks often present incomplete information, with unobserved nodes or edges introducing uncertainty, the IFT framework is suitable for their analysis. Accordingly, we employ IFT to infer network properties in scenarios of partial observability. We found that, in this case, the optimal reconstruction of the incomplete problem retains significant information about the full network. This information remains constant with respect to the fraction of observed nodes until a critical threshold where an abrupt transition takes place and above which the crucial information about network structures are lost.