ABSTRACT
Historically the two ways of describing the changing behavior of complex phenomena are through the Langevin equation for the dynamic variables, or equivalently through the phase space equation for the PDF. The dynamic equations are constructed in the physical sciences from general principles, such as the variation of energy in conservative systems. Recall from Chapter 2 that the complete system is often partitioned into the network of interest and the rest of the universe (the environment). In its simplest form the environment contributes to the network dynamics a random force and a linear dissipation of the network’s energy, that constitutes the Langevin equation description of the environment’s dynamic complexity. Also in Chapter 2 the argument for how to replace the system’s trajectory with an ensemble of trajectories characterized by the PDF in phase space was presented. The dynamic equations were replaced with the phase space equation of evolution for the PDF and the environmental complexity entailed a statistical description of the system’s response.
