Internal observation systems and a theory of reaction-diffusion equation on a graph

Some kinds of spatiotemporal pattern generator systems are expressed by evolution equations. Such evolution equations are composed of many dynamic units which behave from only its local information. This is one example to coordinate many subsystems which observe the system from inside and to generate a global order. This paper shows how to treat these evolution equations and how to apply it to network systems. First, a continuous media system which is expressed by a gradient system in function space is considered. One of the simplest potential functionals derives a reaction diffusion equation which decreases the value of potential functional monotonously. The similar result is found in graph space. That is, a reaction diffusion equation on a graph decreases the value of a potential functional monotonously. This means that a network of dynamic units which observe only their connecting units' states can generate a global order in the same way of continuous media. This theory can treat some internal dynamic network system which should coordinate without some kinds of central controllers.