  Diffusion processes are instrumental to describe the movement of a continuous
quantity in a generic network of interacting agents. Here, we present a
probabilistic framework for diffusion in networks and propose to classify agent
interactions according to two protocols where the total network quantity is
conserved or variable. For both protocols, our focus is on asymmetric
interactions between agents involving directed graphs. Specifically, we define
how the dynamics of conservative and non-conservative networks relate to the
weighted in-degree Laplacian and the weighted out-degree Laplacian. Our
framework allows the addition and subtraction of the considered quantity to and
from a set of nodes. This enables the modeling of stubborn agents with
time-invariant quantities, and the process of dynamic learning. We highlight
several stability and convergence characteristics of our framework, and define
the conditions under which asymptotic convergence is guaranteed when the
network topology is variable. In addition, we indicate how our framework
accommodates external network control and targeted network design. We show how
network diffusion can be externally manipulated by applying time-varying input
functions at individual nodes. Desirable network structures can also be
constructed by adjusting the dominant diffusion modes. To this purpose, we
propose a Markov decision process that learns these network adjustments through
a reinforcement learning algorithm, suitable for large networks. The presented
network control and design schemes enable flow modifications that allow the
alteration of the dynamic and stationary behavior of the network in
conservative and non-conservative networks.
