Self-organizing system theory by use of reaction-diffusion equation on a graph with boundary

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 a typical structure of self-organizing system, which have many desirable properties for large scale systems, such as flexibility, adaptability, fault tolerability, and so on. We showed that such a system can be modeled by a reaction diffusion equation on a graph in the previous work. The result was that 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. Generally, the pattern which is generated by a self-organizing system is suitable for its environment. This means that such pattern is very affected by its environment. Therefore, the theory which treats such a system must be able to formulate the environment. One of the most suitable theory to treat it is evolution equation on a graph with boundary. In this paper, we derive the Stokes' formula on a graph with boundary. Using this formula, the reaction diffusion equation on a graph with boundary is derived by constructing a gradient system in function space on the graph.