Centralized & Scale Free Networks

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In this module we looked at networks that have the highest degree distribution making their topology very heterogeneous in terms of the distribution of connectivity, these networks may have one or a few nodes with a very high degree of connectivity forming global hubs within the network and very many with a much smaller degree of connectivity



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let's start by taking a few examples of these centralized systems, if we look at his network of 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, with data taken from the IMF. We can then see clearly how a very few core nodes dominate this network, there are approximately 200 countries in the world but these 19 largest jurisdictions in terms of capital together are responsible for over 90% of the assets. This type of centralized structure to a network is surprisingly reverent in our world and we could cite 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 define by the mathematics that is behind them so lets just take a quick look at that it. 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 the nodes
The number of nodes with degree 10 is proportional to one hundredth

If we have a network with a thousand nodes of degree one then we would have 250 nodes with degree two, and proximally 31 with degree ten. If we notice then when we go from degree one to degree two we had a very big drop, but then going from degree two to teen the drop is a lot more gradual and this decline continues to get more gradual giving the graph what is called a long tail. The point to take away from this is that this 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 discover 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 a 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 interacting with a very many different molecules, where as most of the molecule interaction very few others, thus making these two molecule hubs in the metabolic networks fueling the cells in our bodies.