Risk CDT - Circular Layouts Representation of the Interbank Systems

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This is a project within the multi-disciplinary EPSRC and ESRC Centre for Doctoral Training (CDT) on Quantification and Management of Risk & Uncertainty in Complex Systems & Environments, within the Institute for Risk and Uncertainty. The studentship is granted for 4 years and includes, in the first year, a Master in Decision Making under Risk & Uncertainty. The project includes extensive collaboration with prime industry to build an optimal basis for employability.

Many interbank systems have been shown to follow (approximately) a tiered core-periphery structure (Craig and von Peter, 2014), where the institutions classified as core maintain relationships (payments, trades, exposures, etc.) with all other core institutions while periphery institutions only have relationships with core institutions. In practice such systems rarely follow a perfect core-periphery structure; however, the classification of core and periphery has proved a useful generalization.

Network visualizations, where institutions are represented as nodes and relationships between them as links, are useful tools for understanding the complex structure of interbank systems and are becoming more widely used by regulators and the industry. Deeper insight into the network structure may lead to improved regulation, highlight systemically important institutions, or suggest areas of potential growth. A primary objective in a core-periphery visualization is to show clearly which core institutions interact most with which periphery institutions. Standard graph-drawing theory, which focuses on choosing a node layout to minimize link crossings, may not be applicable for the particular structure of interbank networks and the specific goals associated with visualizing them.

Objectives: A core-periphery visualization has four primary goals, listed below:
•Identify which nodes belong to core and which in periphery;
•Identify core nodes that are connected to many periphery nodes;
•Identify periphery nodes associated with each core node;
•Represent the identified core-periphery nodes in a visually accessible and meaningful form.

Methods: One method for visualizing core-periphery networks, based mainly on standard graph drawing techniques, is to use two concentric circles with core institutions in and around the inner circle and the periphery institutions along the circumference of the outer circle. Placement of nodes within each circle attempts to minimize link crossings, as more crossings can make it more difficult to see which nodes are linked. It is well-known, however, that minimizing the number of crossings is an NP-complete problem (Masuda, Kashiwabara, Nakajima, and Fujisawa (1987)). And although some approximations and heuristics are available, in most real banking systems, the number of nodes and the number of links between core and periphery institutions are large enough that no such optimal layout exists. In a such network, the goal for visualization may then become finding the node layout that minimizes the crossing of links between core and periphery institutions. One solution in current practice is the algorithm of Bachmaier, Buchnerb, Forstera, and Hong (2010), based on bipartite networks. However, the Bachmaier method is approximate and not guaranteed to find the optimal layout. Moreover, minimizing link crossings may not be the optimal criteria for best visualizing relationships between core and periphery institutions.

The research proposed here could take any or all of the following directions.

Extend the Bachmaier et. al. approach:

1. Minimize link crossings even further, using, e.g., simulated annealing or gradient descent methods.

2. Reduce crossings by extending the current algorithm to more than two concentric circles in the layout, using, e.g., the k-core decompositions.

Use proximity of nodes as a further visual cue:

3. Develop algorithms for minimizing link lengths rather than crossings; that is, periphery nodes should be close to the core nodes that they connect to.

4. Take into account link weights, making links with higher weights shorter so that more strongly connected institutions are are placed more closely together.

5. Allow nodes to cluster along the circumference of either circle (i.e., they need not be uniformly spaced).

Combine both approaches:

6. Expand algorithm to minimize crossings or lengths of any existing periphery-periphery links as well.

7. Explore network embeddings into hyperbolic (rather than Euclidian) spaces, e.g., into the Poincaré disk (Papadopoulos, Psomas, and Krioukov (2014)). Given the exponential (rather than poly-nomial) volume growth, the core nodes are likely to lie near the center of the disk, while the periphery nodes are expected to be close to the boundary.