 Continuing on with our theme of degree distribution to networks, centralized networks represent networks with a very high degree distribution, meaning that in this type of network structure there will be very many nodes with a very low level of connectivity, and very few or maybe just one node with an exceptionally high degree of connectivity. Thus they are very heterogeneous and unequal in terms of how connected and influential the different nodes in the network are. So let's start by taking a few examples of these centralized networks. If we look at this network of global banking activity with nodes representing the absolute size of assets booked in a respective jurisdiction, and the edges between them the exchange of financial capital with data taken from the IMF, we can then see clearly how a very few core nodes dominate the network. There are approximately 200 countries in the world, but these 19 jurisdictions in terms of capital together are responsible for over 90% of the assets. This type of centralized structure to a network is surprisingly prevalent in our world, and we could cite many other examples of it, such as social networks where very few people may have millions of people connected to them, with the vast majority only a very few connections. These highly centralized networks are more formally called scale free or power law networks that describes a power or exponential relationship between the degree of connectivity a node has and the frequency of its occurrence. Power law networks are really defined by the mathematics that is behind them. In these networks the number of nodes with degree x is proportional to 1 over x squared. So the number of nodes with degree 2 is one fourth of all the nodes. The number of nodes with degree 3 is one ninth of all the nodes, and the number of nodes with degree 10 is proportional to 1 over 100. The point to take away from this is that this long tail means that there can be nodes with a very high degree, but there will also be very many with a very low degree of connectivity, giving us a highly centralized network. This type of power law graph was first discovered within the degree distribution of websites on the internet, with some websites like Google and Yahoo having very many links into them, but there also being very many sites out there on the web that have very few links into them. Since then it has been discovered in many types of very different networks, such as in metabolic networks where the essential molecules of ATP and ADP that provide the energy to fuel cells play a central role in interacting with very many different other molecules, whereas most of the molecules interact with very few others, thus making these two molecules hubs in the metabolic networks that fuel the cells in our body. This power law distribution has also been documented in the frequency of citations between academic papers and within the social networks behind Hollywood actors. The scale-free property to networks is then interesting because it appears regularly and across all forms of networks from the internet to social groups to biological systems. The power law distribution to a network like the World Wide Web is often explained with reference to what is called preferential attachment. Preferential attachment describes how a resource is distributed amongst a number of nodes according to how much they already have, so that those who already have a lot receive much more than those who have little. In more familiar terms, this is called the rich get richer. Within this model, if you are say building a website and choosing which other website to link to, then you will be twice as likely to link to a website that has twice as many links as another. So to formalize this a bit better, the probability that you will make a link to a site is proportional to the size of the site. If the network was created under these rules, then we should get a power law distribution. But in reality, this is quite a simplified model. It should just give you an idea for some of the mechanics behind these power law networks. Why we have these very large centralized nodes in the financial system we saw earlier is of course much more complex than this involving a number of different parameters. Most notable among these is the actual quality of the service that the node is providing and not just its size. One last thing we'll note about these centralized networks in respect to their robustness is that they can be very robust or very fragile, depending if we are removing nodes randomly or strategically. If we were removing nodes randomly, then they will be very robust to failure because the vast majority of nodes have a very low degree of connectivity and thus we would likely be removing one of these insignificant nodes with little effect on the overall network. But inversely, if we were to remove a node strategically, that is to say purposefully choosing the node we remove in order to maximize the damage we cause, then these centralized networks are very susceptible to failure of this kind. We just have to remove one of the giant hubs that are critical in their role connecting many small hubs and the system will be affected greatly.