Yesterday morning started way better than I expected it to considering the events of the night before. I even managed to wake up earlier then I usually do and finish of the two assignments for the most stubborn lecturer I have ever met! He even had problems with the way I printed my programs. On the plus side he did let me correct it. Anyway, before I rant on about him again lets move onto better things.
As planned, I wanted to go through at least one interesting journal article/magazine article/piece of research a week. In a bid to focus my attention onto my dissertation, I thought I’ll start of with the paper that got me interested in the topic. It’s a relatively short paper, so a good way to ease into things, but it is still packed with information. The paper I am looking at is by Steven Strogatz and Duncan Watts and was published in Nature in 1998. It is titled”Coupled Dynamics of a ‘Small World’ Networks” (link).
Small World Networks finds it’s origin in the similarly named Small World Phenomenon, or as most people probably know it, “Six Degrees Of Separation”. It’s essentially the same thing and having equated it to Six Degrees of Separation made it much easier to understand. The basic small world network can be described as both highly clustered as well as having a small path distance. What this means is, the same as what it means in 6 Degrees of Separation, there is high clustering/grouping (like in a normal network) as well as there not being too many steps between any two nodes.
Going along with the social network aspect of things, the paper discusses the collaboration of film actors using data from IMDB as well as looking at (more interestingly to me) the neural network of the worm Caenorhabditis elegans and the power grid of the western united states.
The paper can seem very dense on the mathematics (considering it’s been written by a Mathematician that’s not exactly surprising), especially in discussing the formation of the small world networks. However, as I understand it, it is not too difficult to grasp what is being discussed. The paper begins by attempting to prove what a small world network is. To do this, they randomly create a load of networks. These range from fully connected “Large world” networks, to completely random networks. The expectation is that normal networks are highly clustered but have high path lengths and random networks have low clustering but have low path lengths. The expectation is that the spectrum between the two will go from low to high path length and low to high cluster size, however there is a point at which this is not true. In comes the small world network. The small world network has a high cluster and a low path length. The important aspect of the small world networks are just a few long-range connections. Those few connections change the whole dynamic of the network.
As mentioned, there are a few real world examples of these small world networks that the paper goes through (worms, power grids and actors…a potent combination). All three of these things present small world properties, the implications of which extend the small world phenomenon past just social network and into nature in general (including, and most fascinatingly, the brain).
Finally to see the functional relationship this has with dynamical systems, the author carries out an experiment where an “infectious disease” is propagated through the networks. They propagate this across a range of networks including small world networks, and the final results indicate that the infectious disease spreads most quickly in a small world network. This fits with the hypothesis however the low number of long-range connections required to make this true was surprising. The paper finishes by saying that this is essentially the beginning and that more research is required. Since the paper was published in 1998, there has been a lot of research into the area of small world networks, most interesting of which is looking at small world brain networks.
There has been a lot of research into this area, and as the worm example in this paper suggests, the findings do show that the brain has small world properties in its neural connections. Most interestingly is the incorporation of small world theory into various potential brain architectures, for example Murray Shannahans’ work on global workspace theory. The small world networks incorporated here are of a different variety to the ones shown in the Watts and Strogatz paper. For example the ones incorporated in Shannahans’ architecture are much more modular and have “hub nodes” (which are when all information to a cluster comes through one specific node in that cluster). Overall, understanding the role of small world networks in the brain could help give us a much better understanding of the dynamics of the brain.
And that’s that! Hopefully that is an informative review. Normally I would do papers that are more recent, however as I said, in a bid to better understand what my dissertation is going to focus on, for the moment at least, I’ll most likely be focused on this topic in these weekly “reviews”.
thepicklesarefickle 