Centrality Measurement of the Mexican Large Value Payments System from the Perspective of Multiplex Networks

Abstract

With the purpose of going further in the understanding of the payment flows among the participants in the large value payment system in Mexico, SPEI, we elaborate payment networks using historical data for a period of seven years. We conceptualize the SPEI large value payment system as a multiplex network and we study it accordingly. Based on transactions performed on a daily basis, we present three layers built on the following types of payments, i.e. transactions sent from participant to participant, from participant to third party and from third party to third party. We observe that those layers exhibit dissimilar topology: the participant to participant layer reveals the behaviour of banks settling their own obligations, which proved to be sensitive to the failure of Lehmann Brothers; the participant to third party payments layer presented stable properties; and the third party to third party layer resulted in an increasingly dense network since the system has been adopted for the settlement of low-value obligations between accountholders. In order to identify relevant players in those layers, we compare some well-known centrality measures and also a novel centrality measure specifically designed for payment systems, SinkRank. The rankings assigned by SinkRank show a low degree of coincidence across layers.

Appendices

Appendix 1: Topological Measures

In this section we describe the topological or structural measures, which describe the way in which banks are establishing connections in the payment system network. The most simple metrics are the number of banks in the system $n=|N|$ and the number of arcs $m=|A|$, or number of payment relationships among such banks.

1.1 Degree

Degree of a bank $i$ is defined as the number of banks to which bank $i$ has either sent payments to or received payments from in a particular day, let $N(i)$ denote the set of such banks, degree is calculated as follows:

$$\begin{aligned} d_i=\sum _{j \in N(i)}{a_{ij}} \end{aligned}$$

In a similar fashion we define the in-degree, $d^{-}_i$ and the out-degree, $d^{+}_i$ as the number of banks that paid $i$ in one day, and the total number of banks that $i$ has paid to, respectively.

1.2 Clustering Coefficient

Clustering coefficient is a measure of the degree to which banks tend to cluster together in triplets, i.e. it assesses whether if two counterparties of bank $i$ have also sent or received payments among them:

$$\begin{aligned} c_i= \frac{2}{d_i(d_i-1)} \sum _{j,h}{a_{ij}a_{ih}a_{jh}} \end{aligned}$$

1.3 Reciprocity

Reciprocity measures the fraction of arcs in any direction for which there exists an arc in the opposite direction, i.e. it assesses whether payment relationships in the system are reciprocated between banks:

$$\begin{aligned} r= \frac{\sum _{i\in N}\sum _{j\in N(i)}a_{ij} a_{ji}}{\sum _{i\in N} \sum _{j\in N(i)}a_{ij}} \end{aligned}$$

1.4 Completeness Index (Density)

Density of a graph measures how close is the network to be complete. In the context of payment systems a complete network is a situation in which each bank is sending and receiving payments from each other bank in the system:

$$\begin{aligned} \textit{CI}= \frac{ \sum _{i}^{} \sum _{j}^{}a_{ij}}{n(n-1)} \end{aligned}$$

1.5 Network Components

In Dorogovstev et al. (2001) the authors propose to partition the set of nodes (banks) of networks in different components each one characterized by the connectivity properties of the nodes in it. For this study we only divide the banks which belong to the Giant Strongly Connected Component (GSCC) and the banks that do not. The (GSCC) is the largest component in which, for each pair of banks $i$ and $j$, there exists a path from $i$ to $j$ and a path from $j$ to $i$.

Appendix 2: Weigh-Based Measures

The following measures are based on the amount of payments (weights) associated to each of the arcs between banks.

1.1 Strength

The strength of bank $i$ is the total amount of sent payments as well as received payments:

$$\begin{aligned} s_i = \sum _{j \in N(i)}{w_{ij}} \end{aligned}$$

In an analogous manner in-strength, $s_i^{-}$, and out-strength, $s_i^{+}$ are defined as the total amount received and the total amount sent, respectively.

1.2 Flow

The flow between the banks $i$ and $j$ is the net amount paid between bank $i$ and bank $j$: $f_{ij}=w_{ij}^{+}-w_{ij}^{-}$, where $w_{ij}^{+}$ denotes the total amount bank $i$ paid to bank $j$, and $w_{ij}^{-}$ denotes the total amount paid by $j$ to $i$. Then, the total flow of a node is defined as follows: ‘

$$\begin{aligned} f_i=\sum _{j \in N(i)}{f_{ij}} \end{aligned}$$

If $f_i>0$ then the bank $i$ is a net payer or liquidity provider whereas if $f_i<0$ bank $i$ can be considered as a net payee.

1.3 Herfindahl–Hirschman Index

This measure is usually associated with the assessment of competitive conditions in a market; in our context, the Herfindahl-Hirschman Index (HHI) will be used to describe the concentration of amounts of payments sent or received by individual banks in the network. The sending or outer HHI is calculated as follows:

$$\begin{aligned} \textit{HHI}^{+}(i)=\sum _{j\in \mathcal {N}(i)}{\left( \frac{w^{+}_{ij}}{\sum _{j}{w^{+}_{ij}}}\right) ^2} \end{aligned}$$

The receiving or inner HHI can be calculated as follows:

$$\begin{aligned} \textit{HHI}^{-}(i)=\sum _{j\in \mathcal {N}(i)}{\left( \frac{w^{-}_{ij}}{\sum _{j}{w^{-}_{ij}}}\right) ^2} \end{aligned}$$

Appendix 3: Centrality Measures

The concept of centrality has been widely used in social network analysis and it has various interpretations such as relevance, influence or control. In the context of financial network analysis centrality has been related to interconnectedness and the capability of spreading contagion in the system.

In the present study, centrality measures are used to rank the banks in the payment system according to their relevance in the network according to different criteria provided by each centrality measure.

For this study we use the most common centrality measures found in social and financial network analysis literature:

1.1 Strength Centrality

The strength of a bank, $i$, is defined as the sum of the value of payments sent and received by such bank through the system:

$$\begin{aligned} C_S(i)= s_{i} \end{aligned}$$

Analogously, inner strength is defined as the total amount of payments received by an institution in a day of operation, and outer strength is the total amount of payments sent by an institution.

As banks might play different roles in a payment system: a provider of liquidity or an institution that mostly receive payments, it is important to assess the importance of institutions according to their role.

1.2 Degree Centrality

Degree centrality of bank, $i$ is defined as the total number of counterparties such bank has sent or received payments from, through the system in a day of operation:

$$\begin{aligned} C_D(i)= d_{i} \end{aligned}$$

In a similar fashion, centrality metrics in-degree and out-degree are defined as the total number of banks that paid a particular bank in one day, and the total number of banks that such institution has paid to, respectively.

1.3 Betweenness Centrality

Betweenness centrality is associated with being located in the intersection of multiple paths for liquidity, between many banks of the system. If $\sigma _{jk}=\sigma _{kj}$ denotes the number of paths between banks $j$ and $k$, and $\sigma _{jk}(i)$ denotes the number of paths between $j$ and $k$ that pass through bank $i$, betweenness is defined as:

$$\begin{aligned} C_{B}(i)=\sum _{j\ne i \ne k\in N}{\frac{\sigma _{jk}(i)}{\sigma _{jk}}} \end{aligned}$$

1.4 Closeness Centrality

Closeness centrality assesses how far are banks from each other considering the shortest paths to other banks, it is regarded as a measure of how long it will take to spread contagion from one bank to the others:

$$\begin{aligned} C_C(i) = \sum _{j\in N\setminus \{i\}} \frac{1}{d_{G}(i,j)} \end{aligned}$$

(1)

where $d_{G}(i,j)$ denotes the length of the shortest path between banks $i$ and $j$.

1.5 Eigenvector Centrality

The assumption of eigenvector centrality is that the centrality of a bank depends on the centrality of its counterparties. Eigenvector of bank $i$ is the $i$-th entry of the eigenvector $e$ of adjacency matrix $A$ associated to the largest eigenvalue $\lambda$:

$$\begin{aligned} \lambda \mathbf e =A\mathbf e \end{aligned}$$

Some studies have related eigenvector centrality of financial institutions to the capability of spreading contagion, and used this measure as an index for systemic risk: Markose (2012)

1.6 Principal Components Centrality Index

This index was proposed in Martinez-Jaramillo et al. (2014), in order to get a single measure that preserved the information provided by all the centrality measures considered above. It consists in performing the Principal Component Analysis (PCA) over the standardized results of the centrality measures, to get a linear combination (first principal component) of centrality measures with optimal coefficients to maintain the maximum variance of data. For more details regarding this measure, the reader should refer to the paper.

Appendix 4: Overlapping Congruence Measure

Congruence measures attempt to measure the difference between the most central banks between two networks (or layers), in the present study we only used the overlapping congruence measure, defined as follows:

$$\begin{aligned} O_{L,K}(n) = \frac{|\Gamma _L(n) \cap \Gamma _K(n)|}{|\Gamma _L(n) \cup \Gamma _K(n)|}, \end{aligned}$$

where $\Gamma _L(n)$ denotes the set of the $n$ most central banks of layer (or network) $L$. This measure is also known as Jaccard similarity index in various contexts including social network analysis.