In this section, we discuss our methods for identifying systemically important countries, which can cause a significant fraction of the countries to become financially distressed. As an initial approach, we use MultiRank (Rahmede et al. 2018), a centrality measure for multiplex networks. We then study a multiplex threshold model, which allows a contagion to spread between different layers.

### MultiRank

Centrality diagnostics measure the relative importances of nodes (or edges or other structures) in a network (Newman 2018), and it is thus natural to use centrality measures to try to identify systemically important nodes. To identify important countries in a monolayer network, we calculate PageRank centrality, a measure of a node’s importance that is based on a random walk with teleportation (Newman 2018; Gleich 2015). In our calculations, we use the teleportation-parameter value *κ*=0.15, where 1−*κ* is the probability that a random walker moves to an adjacent node and *κ*=0.15 is the probability that a random walker teleports to any node (which we choose uniformly at random) in the network (Gleich 2015). The choice of *κ*=0.15 is the most common choice in PageRank, although it is an arbitrary value.

To examine important countries and important asset types in our multiplex global financial network, we calculate MultiRank (Rahmede et al. 2018), a centrality measure that was inspired by PageRank, to rank both nodes and layers. One assigns a MultiRank centrality to each node *i* as a whole, rather than to each node-layer (*i*,*α*).

Several multilayer centrality measures have been developed (Kivelä et al. 2014; Bravo and Óskarsdóttir 2020). We use MultiRank both because it is easy to implement and because it allows us to rank both nodes and layers.

We briefly explain how to compute node *i*’s MultiRank *x*_{i} and layer *α*’s MultiRank *z*^{[α]}. MultiRank assigns a large centrality to a node when central nodes point to it in layers with high influence (Rahmede et al. 2018). One can determine the MultiRank (i.e., “influence”) of a layer in one of two ways, depending on a parameter *δ*∈{−1,1}. If *δ*=1, layers tend to be more influential if very central nodes in it are adjacent to each other. By contrast, if *δ*=−1, layers tend to be more influential if there are few nodes with nonzero in-degree and if the nodes that do have in-edges have large centralities. In the present paper, we use *δ*=1, because we consider a type of asset to be more important when important countries have an exposure to it.

For multiplex networks in which each node has an out-degree of at least 1 in at least one layer, we compute both node MultiRank and layer MultiRank by solving the following set of equations:

$$\begin{array}{*{20}l} z^{[\alpha]} &= \frac{1}{N}(W^{[\alpha]})^{\phi} \left(\sum_{i=1}^{n} B_{\alpha i}^{\text{in}}(x_{i})^{\delta\gamma}\right)^{\delta}, \notag \\ x_{i} &= \kappa \sum_{j=1}^{n}\frac{G_{ji}}{k_{j}}x_{j} + \frac{1 - \kappa}{n}, \end{array} $$

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where *z*^{[α]} is layer *α*’s MultiRank, *N* is a normalizing factor, \(W^{[\alpha ]} = \sum _{i,j} A_{ij}^{[\alpha ]}\), the adjacency matrix is \(G = \sum _{\alpha =1}^{L} z^{[\alpha ]} A_{ij}^{[\alpha ]}\), there are *n* nodes, the out-degree of node *j* is \(k_{j} = \sum _{i} G_{ji}\), and the normalized in-strength \(B^{\text {in}}_{\alpha i}\) of node *i* in layer *α* is

$$ B^{\text{in}}_{\alpha i} = \frac{\sum_{j} A_{ji}^{[\alpha]}}{W^{[\alpha]}}\,. $$

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As in the PageRank algorithm, *κ* denotes the probability that a random walker moves to an adjacent node and 1−*κ*=0.15 is the teleportation probability, but now we use the adjacency matrix *G*_{ij}. The parameter *γ*>0 influences the contributions of small-centrality nodes by either enhancing (*γ*>1) or suppressing (*γ*<1) them, and *ϕ*∈{0,1} determines whether one considers (if *ϕ*=1) or does not consider (if *ϕ*=0) the total edge weight in layer *α* (Rahmede et al. 2018). In the present paper, we suppose that edge weights are important (because they encode financial exposures), so we use *ϕ*=1. For simplicity, we choose *γ*=1, which implies that nodes contribute to layer influence (i.e., layer MultiRank) in a linear fashion. For details about how to derive this form of the MultiRank algorithm, see the “MultiRank” section of the appendix.

MultiRank has been used to study various multiplex networks (Bonaccorsi et al. 2019; van Lidth de Jeude 2019), and it is useful for exploring important countries in a multiplex financial network. However, to properly study a global financial contagion, it is important to use tools that are designed specifically for that topic.

### Systemic importance of nodes

Several centrality measures have been developed to study the systemic importance of financial entities in networks (Battiston et al. 2012; Bardoscia et al. 2015; Bardoscia et al. 2016; Battiston et al. 2012). Most notably, DebtRank, a scalar quantity that one can compute either for a single node or for a set of nodes, gives an estimate of the monetary loss from the financial distress of that node (or those nodes).

Unfortunately, DebtRank has an issue that makes it inadequate for our problem^{Footnote 4}. Debt-Rank is not additive, in the sense that the DebtRank of the set {*i*,*j*} of two nodes (*i* and *j*) can be smaller than the DebtRank of the individual nodes *i* or *j*. This lack of additivity makes it difficult to generalize DebtRank to multiplex networks. In particular, a straightforward extension of DebtRank to multiplex networks can yield a larger DebtRank for node-layer (*i*,*α*) than for the set \(\left \{(i,\alpha)\right \}_{\alpha =1}^{L}\) of node-layers. In such a generalization, the financial distress of a country in one layer causes a larger monetary loss than the loss from a financial distress of that country in all layers. From an economic perspective, this seems unreasonable. For details about this issue, see the “DebtRank” section of the appendix.

Given the above issue, we propose a different approach: we develop a threshold model of financial contagions for multiplex networks that can be informative about the systemic importance of different countries in the global financial network. We model the propagation of a financial contagion and identify the countries that cause a sufficiently large fraction of the network to become distressed. In particular, we examine the fraction of countries whose assets have a total value that exceeds 90*%* of the total value of all assets in a network.

**A multiplex threshold model of financial contagions.** We develop a discrete-time, deterministic model of threshold dynamics; and we simulate it on multiplex networks that are weighted and directed. Like the nodes in the Watts threshold model (WTM) of a social contagion (Watts 2002) and the Gai–Kapadia model of a financial contagion (Gai and Kapadia 2010), node-layers in our networks have binary states: node-layers (i.e., country-assets) can be either undistressed or distressed^{Footnote 5}. The state of node-layer (*i*,*α*) at time *t* is

$$ s_{i\alpha}(t) = \left\{\begin{array}{ll} 0, &\text{if}\ (i,\alpha)\ \text{is undistressed at time } t\\ 1, &\text{if}\ (i,\alpha)\ \text{is distressed at time } t\,.\end{array}\right. $$

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As in the Gai–Kapadia model (Gai and Kapadia 2010), in which nodes have capital buffers, we suppose that each country has assets in each layer that it holds as capital buffers. The purpose of these buffers is to absorb some amount of loss and avoid distress. Inspired by how national regulators track the CARs of banks based on information about their risky assets, we suppose that the value of the capital buffers that are held by country *i* in layer *α* is proportional to the value *a*_{iα} of the assets that it holds in that layer.

We model the spread of a contagion as follows. If a country loses more assets in a layer than its layer buffer, the country becomes distressed and loses all assets. Additionally, if the sum of the lost assets of a country over all layers exceeds a fraction *c*_{i} of its total assets *a*_{i}, then the country becomes distressed in all layers. This last contagion mechanism is based on risk propagation within countries: when investors perceive that the risk of financial distress of a country increases, the value of that country’s assets decreases.

With these ideas in mind, our spreading rule for a financial contagion is

$$ s_{i\alpha}(t+1) = \left\{\begin{array}{ll} 1, &\text{if } \frac{\text{loss}_{i\alpha}(t)}{a_{i\alpha}} > b_{i\alpha} \\ 1, &\text{if } \frac{\text{loss}_{i}(t)}{a_{i}} > c_{i}\\ 0, &\text{otherwise},\end{array}\right. $$

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where *s*_{iα}(*t*) denotes the state of country *i* in layer *α* at time *t*, the quantity *b*_{iα} is the fraction of assets that are held by country *i* in layer *α* as cash reserves,

$$ \text{loss}_{i\alpha}(t) = \sum_{j} A_{ij}^{[\alpha]} s_{j\alpha}(t) $$

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is the monetary value of the lost assets by the end of time *t* (i.e., including both at and before time *t*) for country *i* in layer *α*, and

$$ \text{loss}_{i}(t) = \sum_{\beta} \text{loss}_{i\beta}(t)\ $$

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is the monetary value of the lost assets of country *i* by the end of time *t*.

We refer to each *b*_{iα} (which encode capital buffers) as a “horizontal threshold” and each *c*_{i} (which encode intra-country risk-propagation buffers) as a “vertical threshold”. To the best of our knowledge, our incorporation of vertical thresholds is a novel feature of our multiplex threshold model. We expect that the idea of having both vertical and horizontal thresholds will also be useful for studying other types of contagions, such as the spread of information and ideas in multichannel communication networks.

For our monolayer financial networks, we run a similar contagion model, but we now neglect the interlayer (i.e., intracountry) contagions. Specifically, we use the update rule

$$ s_{i}(t+1) = \left\{\begin{array}{ll} 1, &\text{if } \frac{\text{loss}_{i}(t)}{a_{i}} > b_{i} \\ 0, &\text{otherwise,}\end{array}\right. $$

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where *s*_{i}(*t*) denotes the state of country *i* at time *t*, the quantity *a*_{i} denotes the value of the assets of country *i*, and *b*_{i} is a threshold that represents the capital buffer of country *i*. The total value of the financial assets that are lost by country *i* in a monolayer network is

$$ \text{loss}_{i}(t) = \sum_{j} A_{ij} s_{i}(t)\,. $$

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**Initial conditions and parameters of the models.** To calibrate the horizontal thresholds (i.e., capital buffers), we use CAR data. A CAR measures the ratio of a bank’s capital to its risk-weighted assets^{Footnote 6}.

The Basel Committee on Banking Supervision (BCBS) recommends minimum CARs that financial institutions must maintain to have a sufficient buffer to provide a cushion against losses due to financial distress (Basel Committee on Banking Supervision 2019).

Although CARs are only for banks, we use them to calibrate the horizontal thresholds in the layers as a proxy for the heterogeneity in the robustness of different countries towards shocks. We assume that *b*_{iα}=*b*_{i}=CAR_{i} for all *α* as a baseline scenario. We then consider several scalar multiples of CAR _{i} as the value of *b*_{i} to examine alternative scenarios. For the vertical thresholds, we suppose that the countries are homogeneous (i.e., that *c*_{i}=*c* for all *i*). As a reference value, we take *c*_{i}=1/*L*, where *L* is the number of layers in our multiplex global financial network. In the “Results” section, we also examine our threshold model using different values of the vertical threshold *c*.

We run our threshold model on our multiplex global financial network and on the monolayer networks (i.e., the aggregate, equity, debt, and banking networks). All of these networks are weighted and directed. We initialize our model with a single country *i*^{∗} that is distressed in all layers, and we measure the subsequent number of countries that become distressed. Our model has a deterministic update rule, so each individual simulation has a deterministic outcome. Once a node-layer is distressed (and, in each monolayer network, once a node is distressed), it remains distressed for the rest of a simulation. A simulation ends when no node-layers change their state anymore, and we use *τ* to denote the time step at which a simulation ends. We consider a country to be distressed when it is distressed in all layers. The extent *R*(*i*^{∗}) of a contagion from a seed country *i*^{∗} is the percent of assets that are collectively held by the distressed countries at the end of the simulation. That is,

$$ R(i^{*}) = 100 \times \frac{\sum_{i} a_{i} \prod_{\alpha} s_{i\alpha}(\tau)}{\sum_{i} a_{i}}\,. $$

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We say that country *i*^{∗} is *systemically important* when *R*(*i*^{∗})>90*%*. Analogously, for the monolayer networks, the extent of a contagion is

$$ R(i^{*}) = 100 \times \frac{\sum_{i} a_{i} s_{i}(\tau)}{\sum_{i} a_{i}}\,. $$

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