Network Theory is fascinating stuff. The thing I find most interesting about the mathematics and applied mathematics of network theory, like graph theory is the fact that only in the 20th century have we really taken a look at networks.
Network theory is deceptively simple and yet powerfully revealing, quite a bit like a good bikini :) Basically from a purely mathematical construct a network has 2 components Nodes, and connections. That is it, the graph shows some basic networks. The amazing things is the dynamics of networks. In theory any causal event or chain of causal events can be modeled with network theory. Social interactions can be model led with Network theory. Yes you really are only 6 people away from all the people on the planet (six degrees of separation)
Most natural phenomenae exhibit a unique aspect or type of network in that the connections and connectedness of the nodes follows a power law, this type of network is known as a scale free network, now before you think this all theory and math, please realize that the connectedness of the web follows a scale free network model. Most dynamic systems with independent components settle into some form of a scale free network. Thus understanding them is important. Whether modeling the spread of disease in a population of people, the transmission of ideas or the dynamics of some biological systems scale free networks hold the key. During the summer of 2002, I read a few papers and books on the topic, two of the best, I would recommend are Sync by Stephen Strogatz and Linked by Laszo Barbasi. Both of these men are fine mathematicians in their own right and great popularizers of science. They have excellent skills at communicating and sharing some of the amazing worlds and phenomenae they have discovered.
The more interconnected we as a culture and species become the more important it is to understand network theory. I think network theory will eventually become as important as thermodynamics. The thesis being that modeling individuals nodes or components will be useless, but understanding complex systems in aggregate will allow for powerful understanding and control of those systems. Network theory and the emerging metrics and models associated with it will lead to whole new theories of "things" and relationships. Much like thermodynamics allowed us to scale up from Newtonian reductionism and ultimately harness energy, Network theory could very well allow us to model and control some of the most powerful networks of all. The list is long, but could include, neurochemical, social, environmental and communications networks. The implications and impacts will be profound. And it will have started with a Connection and a node. Check out these books if you get a chance, I personally fascinated by social networks, memetic spread and of course financial networks.
One of my friends from Starlab is Keith Still, he works primarily in Crowd simulations and modelling. He is considered one of the experts in this realm. Keith is doing soem really interesting stuff in terms of modelling autonomous agents in an environment to assess their behaviour, anyone interested in crowds, safety or cellular automata may be interested in visiting his page or emailing him. Keith is a great guy and very generous with his time.