Degree distribution tries to capture the disparity between nodes that have a very high degree of connections and those that have a very low degree of connections. As such, it can be interpreted as a measurement of inequality; looking at how equal or unequal the distribution of connections is within the whole network. This parameter is important because it is a key contributor to one of the major topological transformations within a network; a transformation that takes us from distributed networks that have a small degree distribution to centralized networks that have a very large degree distribution that is characterized by a power law. One way to understand this degree distribution is that connections are rarely made at random. As with many other types of networks, agents within an economic or financial network will typically make connections based upon transaction cost; they will connect to other nodes that are easier to connect to and provide them with a high marginal return for the cost of making the interaction.
We will start by looking at a network with a low degree distribution, what we call a distributed network where the connections are distributed out evenly across all the nodes in the graph. Everyone has more or less the same amount of connections. Peer-to-peer markets are a good example of distributed networks. In finance, peer-to-peer finance matches many small lenders with many small borrowers. Car sharing services and peer-to-peer renting are other examples of peer markets that connect people directly, meaning everyone has a similar degree of connectivity although it may vary somewhat. Without any form of centralized hubs, distributed networks are typically user-generated. Nodes often have a high degree of autonomy and there is an overall low level of specialization. Everyone is doing largely the same thing, meaning it does not matter too much who you connect to. This low level of specialization also often makes distributed networks robust to attack. With low overhead cost and the capacity to couple or decouple from any other node in the network, they can be highly dynamic and flexible. However, without centralized coordination, they may not be able to operate as a coherent whole, putting them at a disadvantage to more centralized systems.
Most economic networks are not distributed like this though. Networks typically contain some nodes that are significantly more connected than others. One type of this kind of network is called a decentralized network, as we now have the emergence of local hubs. This means that many more nodes have, for some reason, decided to connect to one of these regional hubs. There are many reasons why this might be, but within economics one of them is economies of scale. By a number of agents or organizations combining their resources, they can obtain diminishing marginal cost through performing batch processing. Also related to this is specialization. By leveraging centralization, components can specialize in particular functions that can only be performed given a large enough demand. For example, we need to have a city of a large enough size before we are going to get a shop selling pianos.
Another benefit to these decentralized networks is what is called the small world phenomena. A small-world network is a type of graph in which most nodes are not neighbors of one another, but most nodes can be reached from any other by a small number of connections. For example, if one wants to transfer money to someone on the other side of the planet, one might go to their local Western Union center and they would send it to one of their other centers near to the person who is receiving the money. In a distributed network, this would have taken many links. If this was a large network it might have had to traverse thousands of links, but due to these local hubs, we get the small world phenomenon where we can easily reach any other node.
Although decentralized networks are ubiquitous in our world, even more centralized networks are just as common. Highly centralized networks represent a radically unequal level of connectivity to the nodes. Many nodes have very few connections and some have very many. These centralized networks are also called scale-free networks as their degree distribution follows a power law, meaning, unlike our distributed and random networks that had a normal distribution with most nodes tending towards the average degree, in scale-free networks, there are very many nodes with very few connections, very few with very many. Thus, the vast majority link into just one or few centralized mega hubs. This power law relation is a more exact description of the Pareto principle, which has been identified in many areas, from the distribution of land ownership to that of wealth; where the richest 20% of the world’s population control approximately 80% of the world’s income. Or for example, in a financial context, power laws are also seen within stock market pricing and the interbank network, where the fat tail indicates that there exist few banks interacting with many others, giving us banks that are too big to fail.
One plausible explanation to the dynamics causing the formation of these highly centralized networks is that of preferential attachment. A preferential attachment process is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have so that those who are already wealthy receive more than those who are not.8 For example, these major hubs in scale-free networks can leverage significant economies of scale to reduce marginal cost of interaction and work to make them a default attractor for the formation of new connections.
These different network structures that we have outlined – distributed, decentralized and centralized networks – will all fundamentally alter the flow of resources through the network. In a distributed network, anything entering the system will be relatively slow to flow through it, given the high number of linkages needed to be traversed in order to get across it. Whereas in a centralized hub and spoke system, where every node can be reached in just one or two links, something can flow through it much quicker and more efficiently, but it will be dependent upon these major centralized nodes. With everything being routed through them this will create systemic risks; a few nodes that are capable of affecting the whole network.
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