by R.S. MacKay, S. Johnson, and B. Sansom
If networks be the food of growth, then on
Supply chain networks something should depend
Exploring them showed that we must extend
Trophic coherence: this we have achieved
With a more general notion that allows
The concept to enlarge to any net
Including ecosystems and genomics
So now we’ll use this on our economics.
Many economic relationships have a direction: the flow of goods and services from suppliersto buyers in the production process; the flows of international trade, labour and investment between countries; the claims of creditors on borrowers; the transfers of risk from sellers to buyers; and the flows of liquidity and collateral chains in financial markets are just some examples.
The direction of these relationships is often crucial in defining for example the direction of risk: e.g. the default of a firm may create problems most immediately for its creditors. While individual links tell us about the bilateral direction of exposure, or “flow of risk” at this very local level, when the overall flow structure set up by the wider web of these relationships is considered, some directed networks are much more strongly directed than others. The question of how directed a directed network is, may be important as this may influence e.g. whether and how instabilities spread or amplify, as well as outcomes for specific nodes (agents, institutions, sectors).
With our recent paper ‘How directed is a directed network?’ we introduce simple but powerful new tools for revealing the direction of flow among nodes and measuring the overall “directedness” of economic, financial and other complex networks.
Figure 1: Different perspectives on the same corporate control network: left standard force directed layout; right trophic levels reveal flow hierarchy (y-axis); [Coherence=0.98]
The above chart (Figure 1) shows two visualizations of the same network of corporate control among firms in the telecoms industry: on the left using a standard force-directed method of the sort commonly used for the visualisation of complex networks; on the right using information on the “flow structure”of the network revealed using our new method. Taking explicit account of flow structure dramatically exposes the highly directed and multi-layered character of this extremely hierarchical network.
Figure 2: Illustrating hierarchical (left), circular (right) and mixed (centre) structures
Broadly the flow structure of a network may be characterised in terms of its vertical/hierarchical vs. cyclic character:
Hierarchical flows describe ‘sources’, intermediating chains and ‘sinks’ in an unbalanced system - e.g. where there are net exporters/importers in a trade system; or net long/short positions in a market. For example: ultimate sellersand buyers of risk may be linked by complex intermediating chains in financial markets; primary materials (including embedded carbon and labour) flow through multiple stages of value-adding intermediate production potentially crossing multiple borders before final goods are delivered for consumption; trade flows originate somewhere, and may flow through multiple ports on their way to final destinations.
Cyclic flows describe two-way relationships or communication between different parts of a system. At an extreme it describes balanced relationships e.g. balanced trade or matched-books. As well as describing balanced flows, cyclic structure may also be a source of destabilising feedback; self-reinforcing growth; or opportunities for arbitrage. For example work in the financial networks literature has identified a trade-off between the benefits from local risk diversification vs. contagion risk from a more densely connected system - a new link on the one hand helps an individual agent diversify its risk; on the other hand it may contribute to the formation of destabilizing feedback loops in the system.
Figure 2 above provides simple illustrations of both a perfectly hierarchical network (left) and a perfectly balanced/circular network (right) as well as a mixed structure (centre) with both some circulating flow and some directionality. In practice of course most real world systems tend to have both some cyclic flow and some overall directionality. The new measure of “trophic-coherence” we introduce conveniently quantifies how “directed” a network is overall on a scale between 1, for a fully coherent network (such as Figure 2 left), and 0 for a completely balanced network (such as the example in Figure 2 right). Where there is any overall directionality to the network the associated measure of “trophic levels” we introduce also provides a hierarchical ordering of the nodes, identifying the direction of flow between them. In our paper we demonstrate how these new methods can identify node function in examples not only from economics, such as supply networks, but also other complex networks. (This work improves on existing notions of trophic-leveland trophic coherence in ecology, fixing a number of issues that limit their use in economic and financial network applications).
We expect these methods will provide a valuable tool for analysing and understanding economic and financial network data. We show that our improved measure of trophic-coherence- like the existing measure in ecology - is theoretically related to network stability and spreading processes. We are particularly interested in whether, empirically, there is a link between trophic-structure and dynamical phenomena on economic networks.
Figure 3: A visual illustration of research questions/strategies
Industrial organization and economic development literatures have long taken an interest in the role of “vertical linkages” in shock transmission and growth processes and the position of individual sectors or countries in these chains (although we find changes in the trophic-coherencedriven by cyclic flows may have been confused in economics with vertical ‘depth’/fragmentation of supply chains). Could the overall directionality of national or global production systems also be important for e.g. sectoral or aggregate fluctuations or growth rates?
Looking at some data we find trophic-coherence of production networks has changed over time and varies between countries and firms: for example, Figure 4shows the evolution of the trophic-coherence of the US sectoral input-output network since 1947. Despite the strongly hierarchical mental model we have of the value chain, we see the directedness/coherence of this network has always been relatively low, but has increased steadily over time.
Figure 4: Trophic coherence of US IO network over time
Figure 5 shows some moderate and significant correlation between the average coherence of national IO-network structure and average growth rates of gross fixed capital formation in a sample of 47 differed national economies (including OECE and G20).
Figure 5: Trophic coherence vs. GFC growth rate for 47 national economies (including G20 and OECD) shows moderate correlation (Pearson 0.48 p=0.001)
Figure 6 shows variation in the trophic coherence of the supply networks of S&P500 firms where we find trophic coherence is rather high overall and varies systematically between industrial sectors.
Figure 6: Firm level nets and coherence
You can find our paper here: How directed is a directed network?. In order to facilitate easy application of these methods by other researchers we are providing an open toolbox – which will be available here and updated and expanded over time [Link: https://github.com/BazilSansom/Trophic-Analysis-Toolbox]. We would welcome hearing from you and happy to engage if you have interest, ideas or questions on the meaning and application of these methods. Do get in touch!