Over the past decades, our financial system has evolved into a global network of connections between institutions, markets, assets, and liabilities, leading many to the insight that this system may well be best modeled using the science of network theory. Network science is becoming a topic of major interest in financial research, particularly due to its implications for financial crises. Some current applications of network modeling in finance include looking at the spreading of financial contagion and systemic risk, the formation of interbank markets or stock correlation networks, among many others.
Network theory, also called graph theory, is one of the very few major modeling frameworks within complexity theory. It is an abstract formal language which deals with the idea of connectivity. This world of connectivity is very different from the one we are used to. It is all about access, where you are in the network, what is the overall structure of that network and what is flowing through it. With networks, it is the structure of the connections within the system that determines the outcomes and not so much the component parts.
A financial network is a system of financial entities that are linked through a set of connections in some way. A node in the network can be any organization with a balance sheet or any asset or liability. Connections between them can be exchanges of various forms, such as that of ownership, e.g. between shareholders and a company or credit and debt between borrower and lender. A financial network thus forms an interlinked system of interdependence between a group of financial entities, their assets and liabilities.
Network theory is a modeling framework that can help us to understand financial systems by looking at the structure of the connections between nodes. The first thing we really need to know about a financial network is how connected it is. Complexity theory has in many ways taught us that connectivity is a fundamental parameter to a system. At a low level of connectivity, events do not travel far and there are not enough connections for events to return to their source. The more interconnected a network the greater the capacity for feedback loops, nonlinearity, and cascading effects.
Going from a low level of connectivity to a high one is a paradigm shift in that it is a systemic change that affects how the whole system operates. A key parameter here is that of transaction cost. Generally, as transaction costs go down exchanges go up, connectivity goes up, interdependence goes up. As interdependence is the fundamental source of nonlinearity the consequence is that the system starts to behave in a nonlinear fashion. The transaction cost within a financial system can take many forms, such as with telecommunication technology, i.e. how easy is it to send information around or it could be regulatory, how easily can capital be moved into or out of a jurisdiction. For example, we have seen how much the global financial system has changed over the past decades and a lot of this is due to a reduction in transaction cost increasing interconnectivity and interdependence – the advancement of communication and information technology and the deregulation of capital markets. One way of quantifying this concept of the overall connectivity to a network is with reference to its density. The density of a network is defined as a ratio of the number of connections to the number of possible connections, and this will also correlate to the average degree of connectivity to the nodes in the network.
How important a node is within a financial network is a function of both how much of the network’s resources are flowing through it and how critical that node is to the system. A node’s real significance within a network – what is called its centrality – is not a trivial feature to analyze. As an example of centrality analysis, we might think about government bailouts during a financial crisis. As the government is interested in maintaining the functionality of the entire network, it needs to ask a number of questions about its connections including: how many links does this bank node have and what volume are those links? Does the node play some critical role within the financial network that no other institution could perform? How closely connected is it to all the other nodes and how important are the other nodes that it is directly connected to? By answering all these questions, we would be able to get some understanding of its importance in maintaining the entire network.
The way in which a network is connected plays a large part in how networks are analyzed and interpreted, this overall structure to a financial network we would call its topology. The topology of the network defines how things flow through it. Information and resources flow very differently in a centralized star structured network as opposed to a distributed network, likewise spreading will happen very differently on a network that is homogeneous versus heterogeneous in terms of its clustering. Due to some common set of properties shared by a subset of the system, we often get subsystems forming within networks. These subsystems are called clusters and often have a significant effect on the makeup of the network. For example, we might think here about the clustering in different commodity markets or different geographic areas, such as that of Anglo-America or the Chinese market, that create discontinuity and disparities between them resulting in resistance to something flowing evenly across the whole network.
The degree distribution to a network is a key factor in its overall structure and dynamics. Degree distribution answers the question how evenly distributed are the connections in the system, do some people or institutions have a lot of connectivity while others have little, thus creating a very unequal system, or do all have roughly the same degree of connectivity creating a relatively equal system. As in many cases, connectivity equates to the flow of resources and opportunities this metric can tell us much about the level of equality within the system and how centralized or distributed it is – analyzing degree distribution may tell us whether some institutions in the financial system dominate over all others or if power and influence are more evenly distributed in the system.
Centralized – Decentralized
Degree distribution helps to capture a critical aspect to a network of any kind, how centralized or distributed it is. The degree of centralization to the overall network is a major determinant of many factors, such as its robustness and criticality, how resources flow across the network and how one might go about intervening in the network. With a relatively equal distribution of connections across the nodes, we get a distributed system, e.g. everyone has more or less the same amount of connections. One example of this in a financial network might be peer-to-peer lending which works to match many small lenders with many small borrowers. Without any form of centralized hubs, distributed networks are typically user-generated. However, distributed systems are often not what we see when we look around us and typically not the case for financial systems; often we see networks that are more concentrated creating centralized hubs.
Highly centralized networks represent a radically unequal level of connectivity to the nodes. Many nodes have very few connections while some have very many. These centralized networks are also called scale-free networks as their degree distribution follows a power law, meaning, unlike a distributed or random network – that have 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 a 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.
Highly centralized systems are often the product 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. For example, these major hubs in scale-free networks can leverage significant economies of scale to reduce the marginal cost of interaction, working to make them a default attractor for the formation of new connections e.g. large markets that have high liquidity reducing exchange costs.
The process under which a network was created and matured will play a large role in how something will spread across it and ultimately how resilient it is to failure. There are a few key parameters that will greatly affect this process of spreading. Firstly, how contagious is the phenomenon that is spreading? An important consideration here is whether this is being driven by some positive feedback loop, as is typically the case within financial markets where loss of confidence begets more loss of confidence, or inversely, increase in confidence creates even more confidence. Secondly, how resistant are the nodes in the network to this phenomenon? So for a financial institution facing a mass of defaults, this resistance might represent how much capital they are holding. Thirdly, we need to consider the overall structure of the network; is it centralized or decentralized? Centralized networks are more susceptible to certain kinds of attack. This is one of the great benefits of distributed ledger technology. It reduces the current cybersecurity vulnerability of having large amounts of financial data within centralized repositories. Lastly, we need to also take into account whether this failure is being spread strategically or at random, as different network topologies exhibit different vulnerability characteristics depending on how random the failure is.
Resilience and robustness have become a hot topic since the financial crisis and many researchers have since tried to apply network models to financial systems in order to make an assessment of their resilience to shocks and spreading. On a very general level connectivity can both add and reduce to the system’s robustness. It works both ways; greater connectivity can provide channels for spreading risk and supporting each other but it also works as channels for failures and shocks to spread.
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