# What mathematical models can analyze and optimize such message passing system?

I look for a mathematical model that can accommodate, analyze and suggest optimizations for a system that can be humanly described as message passing black box programs to which where optimal message directions shall be suggested depending on their reactions to different messages.

## System description:

We have a graph. Its nodes can appear and disappear in time (we know only if node is alive at a discreet moment of time). We can assume that all nodes can be connected one to another via $N$ types of connections (message passing channels and patterns) any node to any node alike ZeroMQ.

Nodes send messages to one to another, messages are different, we know some hash function-based signature for each message and its size. We have f(size,t,c) that describes how fast a message can be passed from one node to another, it uses message size, current time, and c for size of all other messages being sent to this node concurrently.

Each node has a set $M$ of parameters describing their state given via monitoring utilities. We do not know lows of how parameters change, but we know that they change as time goes on and new messages arrive so we can try to approximate them for each node looking at incoming messages and time.

## Task at hand

A node can be called "useful" if it receives and sends as much messages as possible, while its values of set M are as low as possible, or if it sends and receives none messages and values of set M are as low as possible.

We cannot control when a node sends a new message we can only tell to the node having current system time, message hash and size where to send it. Our task is to make as much nodes useful as possible.

I seek for a theoretical studies that encountered/solved such or similar problem. At least mathematical models which allows such problem to be solved in your opinion. So a name of the theory + reference to a paper with a model would be the answer. So what game or any other mathematical model can suggest to which node to send the next message (to make whole system as "useful" as possible)?

• Ignore this if its unrelated: Isn't message passing in junction trees (belief propagation in Bayesian Networks) similar to your system ? – seteropere Mar 6 '15 at 3:34