We will discuss interaction networks that spontaneously form in particulate-based packings. These networks, most commonly known as `force chains' in granular systems, are weighted, dynamic structures, which are known to be of fundamental importance for the purpose of revealing the underlying physical causes of a number of physical phenomena involved in statics and dynamics of particulate-based systems. The presentation will focus on applications of algebraic topology, and in particular of persistent homology (PH) to analysis of such networks found in both simulations and physical experiments. PH allows for a simplified representation of the complex interaction field in both two and three spatial dimensions in terms of persistent diagrams (PDs), which are essentially point clouds. These point clouds could be compared in a meaningful manner, meaning that they allow for the analysis of both static and dynamic properties of the underlying packings. Such representation is robust with respect to small perturbations, which is crucial in particular when applying the method to the analysis of experimental data. In this context, we fill show that PDs allow for extraction of the properties of interaction networks even if contact forces are not well resolved, as it is often the case in physical experiments. In the second part of the talk, we will focus on a case study of evolving granular systems experiencing stick-slip, intermittent type of dynamics.