Centralized & Power Law Networks
Centralized networks represent networks with a very high degree distribution, meaning in this type of network structure there will be very many nodes with a very low level of connectivity and a 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. A good example of a centralized network might be global banking activity with nodes representing the absolute size of assets booked in the respective jurisdiction and the edges between them the exchange of financial assets. A very few core nodes dominate the global financial network, with approximately 200 countries in the world, but the 19 largest jurisdictions in terms of capital together are responsible for over 90% of the assets.1 This type of centralized structure to a network is surprisingly prevalent in our world and there are many other examples of it, such as social networks where a very few people may have millions of people connected to them and the vast majority very few.
These highly centralized networks are more formally called scale-free or power law networks, that describe a power or exponential relationship between the degree of connectivity a node has and the frequency of its occurrence. These power law networks are really defined by the mathematics that is behind them. The number of nodes with degree x is proportional to 1 over x squared. For example, 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 the nodes. The number of nodes with degree 10 is proportional to one hundredth. Power law distribution like this creates what is called a long tail distribution. The long tail means 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 our 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 on the web that have very few links into them.2 Since then it has been discovered in many types of very different networks,3 such as in metabolic networks where the essential molecules of ATP and ADP that provide the energy to fuel cells play a central role interacting with very many different molecules, whereas most of the molecules interact with very few others, thus making these two molecules hubs in the metabolic networks fueling the cells in our bodies.
This power law has also been documented in the frequency of citations between academic papers and within the social network between of Hollywood actors. This 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 among a number of nodes according to how much they already have so that those who already have a lot receive more than those who have little. In more familiar terms this is called “the rich get richer.”4
Within this model, if you are say, building a website and choosing which other websites 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 that a bit better, the probability that you will make a link to a site is proportional to the size of the site. If a network was created under these rules then we should get a power law distribution, but in reality, this is quite a simplified mode. 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 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, not just its size.
With respect to their robustness, centralized networks can be very robust or very fragile depending on if we remove nodes randomly or strategically. If we were removing nodes randomly, they will be very robust to failure because the vast majority of the 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 are doing, then these centralized networks are very susceptible to failure of this kind because we just have to remove one of the giant hubs that are critical in their role, connecting many smaller hubs and the system will be affected greatly by this.