## Mathematics of Networks and Autonomic Rules

Monday, March 21st, 2011 by Antonio Manzalini

We know that the behaviour of electrons in an electrical network can be described in terms of random walks; amazingly this local behaviour allows the network, as a whole system, to solve a complex optimization problem, which is minimizing heat dissipation for a given level of current flow. One might be tempted to extend this metaphor to communication networks.

Well, apparently, it’s even better: while the rules governing physical systems are fixed, for future communication networks we may be able, in principle, to engineer microscopic “autonomic rules” so as to achieve the desired macroscopic consequences at the network level. Specifically, as future communication networks will dramatically grow in size, I think that phase-transitions and meta-stability will be a highly interesting issues for the implications that these phenomena might have in network management and control.

As an example, in the project Univerself (http://www.univerself-project.eu/), I’m working on a microscopic description of a communication network as a Markov process with a large number of locally interacting components. Even if the number of possible states of each component is finite, there is the possibility of having meta-stable, i.e., persistent, states on the time scale of practical interest. This possibility is not just theoretical mathematics, but it may have a number of nice practical implications for management and control of network flows. For example, have a look at this paper:

http://www.statslab.cam.ac.uk/~frank/PAPERS/kmbk1.pdf

it shows how the integration of streaming traffic and file transfers can have a stabilizing effect on the variability of the number of flows in a network. Other examples show the impact of flow admission control strategies, or dynamic routing on the instability regions of a network.  A solid mathematical modelling of future networks will allow designing the said “autonomic rules” without compromising network stability.

Share