Multi pathways temporal distance unravels the hidden geometry of network-driven processes

Network-based interactions allow one to model many technological and natural systems, where understanding information flow between nodes is important to predict their functioning. The complex interplay between network connectivity and dynamics can be captured by scaling laws overcoming the paradigm of information spread being solely dependent on network structure. Here, we capitalize on this paradigm to identify the relevant paths for perturbation propagation. We introduce a multi-pathways temporal distance between nodes that overcomes the limitation of focussing only on the shortest path. This metric predicts the latent geometry induced by the dynamics in which the signal propagation resembles the traveling wave solution of reaction-diffusion systems. We validate the framework on a set of synthetic dynamical models, showing that it outperforms existing approaches in predicting arrival times. On a set of empirical contact-based social systems, we show that it can be reliably used also for models of infectious diseases spread - such as the Susceptible-Infected-Susceptible - with remarkable accuracy in predicting the observed timing of infections. Our framework naturally encodes the concerted behavior of the ensemble of paths connecting two nodes in conveying perturbations, with applications ranging from regulatory dynamics within cells to epidemic spreading in social networks.