Maximum Entropy Method and Underdetermined Systems Applied to Computer Network Topology and Routing

The maximum entropy method (MEM) is a relatively new technique for solving underdetermined systems. It has been successfully applied in many different areas. All methods for solving underdetermined systems introduce some additional, artificial constraints. The advantage of the maximum entropy method is that it uses the most natural additional constraint: one that does not introduce any new, arbitrary and unwarranted information. One important property of entropy maximization is that it favors uniform distribution. Network design and analysis almost always involve underdetermined systems, especially when routing policy has to be determined. The number of possible routings grows with the factorial of the number of the nodes in the network and the number of possible topologies is exponential in the number of links. The number of constraints is typically polynomial in the number of nodes in the network. That makes the network design problem a good candidate for the maximum entropy method application. It is intuitively clear that an optimal network should not have overloaded or underutilized links. The hope is that the maximum entropy constraint will give a starting topology and routing with smoothly distributed traffic that would lead to the solution that is closer to the optimal. The problem is computationally feasible and with proper identification and selection of certain parameters the method gives reasonable topology and routing. It is possible to apply MEM if we start our analysis with totally interconnected network of n nodes. Some lines will be dropped later in the process of improving utilization or reducing the cost. To apply the maximum entropy method we have to decide what will be the variables of the system. Some combination of required traffic values can be used for that if,we remember that for MEM application we do not need to start with probabilities, but an arbitrary set of numbers which can be normalized. Additional parameters are introduced which allow the control of optimization process. Philosophical discussions about the real meaning of the maximum entropy method are interesting, but since the method was successfully applied in many areas, for any new area the most important criterion is not how well can we explain the relation between the MEM and that area, but how useful are the results we get by applying the method.