Characterization and control of conservative and non-conservative network dynamics

Diffusion processes are instrumental to describe the movement of a continuous quantity in a network of interacting agents. Here, we present a framework for diffusion in networks and study in particular two classes of agent interactions depending on whether the total network quantity follows a conservation law. Focusing on probabilistic, asymmetric interactions between agents, we define how the dynamics of conservative and non-conservative networks relate to the weighted in-degree and out-degree Laplacians. For uncontrolled networks, we compare the convergence behavior of both types of networks as a function of the eigenvectors of the weighted graph Laplacians. For networks with exogenous controls, we also analyze convergence and provide a method to measure the difference between conservative and non-conservative network dynamics based on the comparison of their respective reachable sets. The presented network control framework enables the comparative study of the dynamic and asymptotic network behavior for conservative and non-conservative networks.