Temporal network, centrality measures, and ordinary differential equations

Nowadays, centrality indexes are popular measures of the "importance" of nodes in a network. In the analysis of temporal networks, these indexes are extended to sys- tems where interactions evolve over time. In this temporal framework, commonly used centrality metrics typically rely on discrete snapshot approximations, potentially failing to capture the continuous nature of changes in a network. This thesis addresses this limitation by defining and analyzing new centrality metrics rooted in the solution of non- autonomous Ordinary Differential Equations (ODEs) governing the network evolution. Specifically, the solution of such ODEs is known as Time-Ordered Exponential (TOE) and it has a convenient expression in the so-called ⋆-algebra. This expression allows us to define other new metrics, unrelated to the ODE, and can be transformed into a computable matrix representation. Numerical experiments validate the accuracy of the matrix representation approach, achieving sufficient relative errors compared to classical ODE solvers. Furthermore, com- parative analysis against established temporal centrality measures (based on the Katz and Matrix Exponential centralities) reveals that the centralities based on the ⋆-product exhibit heightened "temporal memory" and sensitivity to local dynamics within isolated...