Graph theory vs. Diff Eq approaches towards studying complex networks

[gravenewworld]

Can anyone comment on the advantages and disadvantages of both graph theory vs. using a system of differential equations to study a complex network? For example, how much computing power and running time would a graph theory approach use compared to say solving a system of 100 differential equations? Memory space? How about the ability to add data or new equations on the fly (thinking of data structures, linked lists)? Example, let's just say I had a large network of pipes and wanted to see if water got from point A to point B (all the way at the end) and at what rate. What would happen if a certain node where pipes meet got knocked out? Would water still get to B? These are some of the types of questions I'm interested in. I'm just wondering if one decided to model such a network, which methods used would be more efficient, still provide powerful analysis and good results.

 

[h6ss]

Example, let's just say I had a large network of pipes and wanted to see if water got from point A to point B (all the way at the end) and at what rate. What would happen if a certain node where pipes meet got knocked out? Would water still get to B? These are some of the types of questions I'm interested in. I'm just wondering if one decided to model such a network, which methods used would be more efficient, still provide powerful analysis and good results.

Although I will be receding from both graph theory and systems of differential equations, I will try to give you a perspective related to my field of studies: the mathematical-artificial neural networks, or ANNs.

Briefly, ANNs are sort of like dynamic/chaotic systems. They're a powerful data modeling tool that are able to capture and represent complex input/output relationships. Applications of ANNs often occur in design of systems in which it is impossible to illustrate their functions in the form of mathematical models, e.g. network of pipes.

ANNs are suitable to perform computational operations on different parameters on given information and date, and thus predict the possible relationships between them, in the near or far future. It's good because you don't need to know the exact relationships at the beginning- basic information on the variables is often more than enough to train your network and make it able to give you relatively accurate predictions.

In my opinion, this is the case in any water/pipe distribution network, system, design or operation, in which input variables could represent pipe lengths, diameters, strength, and maybe even more complex variables like water flow, etc. The output variable could simply represent the reliability of the network. It is probably not a surprise that a strong relationship would probably be given by a set of nonlinear equations such as continuity equations in the case of problems dealing with water flow rates and such. (http://en.wikipedia.org/wiki/Continuity_equation)

In order to determine the parameters that will affect the likelihood of pipe breaks, one can predict the number of breaks for each individual pipe in the water distribution system after a given amount of time and then study the results given by the network.

I guess in your example there are many ways to categorize leaks. It could be anything from pipe diameters, change of pressure in the canalization area, holes, detection of flow loss rate, etc. That's why in my opinion, the optimal way one could design an ANN for such problem would be by using a multilayer perceptron network, a powerful intelligent modular system based on classification with great computational abilities.

For example, one could easily set-up a multilayer perceptron network to study a network of pipes by:

1. Setting up a "pipe-breaking" categorization algorithm
2. Classification set-up of the artificial network
3. Training & learning of the network
4. Testing of the results after the learning cycle(s)
5. Analyzing the results
6. Maybe repeat the cycle using different weights, hidden layers, etc.

The theory behind neural networks, especially complex forms like the multilayer perceptron, can be quite heavy, but the learning curve is reasonable and the programming skills needed are usually accessible.