Inspired by the Microscopic Markov Chain Approach (MMCA) designed for epidemic spreading, first we propose an adaptation of the framework for modeling the default cascades observed in the transactions between different companies in real financial networks. Then we introduce some measures to dynamically analyze the default contagion process and its functional relations with any sectoral financial network.

### 3.1 MMCA model for default contagion

The original MMCA model was designed to cope with the propagation of epidemics [15], where the states of the agents (nodes) forming the network of contacts where binary, namely, susceptible or infected. In well-mixed populations, the differential equations governing the number of susceptible (S) and infected (I) individuals are

$$ \begin{aligned} &\frac{dS}{dt} = - \tilde{\beta } S \frac{I}{N} + \tilde{\mu } I , \\ &\frac{dI}{dt} = \tilde{\beta } S \frac{I}{N} - \tilde{\mu } I, \end{aligned} $$

(1)

where \(N = S(t) + I(t)\) is the (constant) size of the population. The term \(I/N\) accounts for the probability of contacting an infected individual in a well-mixed population of size *N*, *β̃* is the infectivity rate (probability per unit time) for each contact, and *μ̃* is the rate at which one infected individual recovers. Their corresponding differential equations are

$$ \begin{aligned} &S(t+\Delta t) = S(t) \biggl( 1- \tilde{\beta } \Delta t \frac{I(t)}{N} \biggr) + \tilde{\mu } \Delta t I(t) , \\ &I(t+\Delta t) = I(t) \biggl( 1- \tilde{\mu } \Delta t + \tilde{\beta } \Delta t \frac{I(t)}{N} S(t) \biggr) , \end{aligned} $$

(2)

or equivalently

$$ I(t+\Delta t) = I(t) - \tilde{\mu } \Delta t I(t) + \tilde{\beta } \Delta t \frac{I(t)}{N} \bigl[N - I(t)\bigr] . $$

(3)

Defining \(\rho (t)=I(t)/N\) as the fraction of infected individuals in the population, Eq. (3) is written as

$$ \rho (t+\Delta t) = \rho (t) - \tilde{\mu } \Delta t \rho (t) + \tilde{\beta } \Delta t \rho (t) \bigl[1 - \rho (t)\bigr] . $$

(4)

Note that this discrete equation can be mapped to networks, and in a microscopic approximation, the density of infected individuals will correspond to the individual probability of being infected.

In the case of our economic networks, the state of the different agents corresponds to their liquidity in time according to their initial state and economical activity. To keep things as simple and realistic as possible, we assume a discrete-time version of the SIS model in networks, and adopt its terminology, defining \(\mu = \tilde{\mu } \Delta t\) as the probability of default recovery, and \(\beta = \tilde{\beta } \Delta t\) as the probability of default contagion. In this setting, *μ* reflects the average recovery time a company needs to overcome a default process, whereas *β* reveals the average number of interactions with an infected company required to be defaulted. Finally, the network of contacts, in this case, is represented as a weighted matrix, with matrix elements \(w{ij}\) accounting for the total sum of money transfers from node *i* to node *j* in 12 months (from a fixed date). The equation governing the default system is then described as:

$$ p_{i}(t + 1) = \bigl(1-q_{i}(t) \bigr) \bigl(1 - p_{i}(t) \bigr) + (1 - \mu ) p_{i}(t) + \mu \bigl(1 - q_{i}(t) \bigr) p _{i}(t) , $$

(5)

where \(p_{i}(t)\) is the probability of a node to be in default; \(q_{i} = \prod_{j=1}^{N} (1 - \beta w_{ji} p_{j}(t) )\) the probability that a given node *i* is not infected by any of its neighbors. The right hand side of Eq. (5) is explained as follows: \((1 - q_{i}(t) ) (1 - p_{i}(t) )\) is the probability that a given node *i* is susceptible of entering into default \((1 - p_{i}(t) )\) and it is infected, \(( 1 - q_{i}(t) )\), by at least one of its neighbors in default. The term \((1 - \mu )p_{i}(t)\) is the probability that a node *i* in default does not recover \((1 - \mu )\). Finally, the term \(\mu (1 - q_{i}(t) )p_{i}(t)\) corresponds to the probability that a given node *i* recovers from default but is reinfected by at least one of its neighbors already in default \(( 1 - q_{i}(t) )\).

According to the European Central Bank definition for risk classification [43], the susceptible state would correspond to a company which is in step 3 (default). In this step, the credit quality of the company is considered equivalent to a probability of default of between 0.10% and 0.40% over a one-year horizon. Therefore, after a given period of time (12–18 months), which depends on its revenue, it can go through the step 2 (cure) and finally come back to step 1 (normal) if it proves to have a good payment behavior.

By using this model, we computationally analyze the behavior of default contagion processes in a real topology created by the interactions among different companies. Additionally, we want to understand the main properties of default propagation and the emergent clusters containing defaulted companies in the full system. Moreover, having a well-defined sector distribution and their annual revenue, see Fig. 1, we can elucidate if default cascades depend on sectors and their economical states. Our objective here is not to build an accurate description of the default process, since for this purpose our model should be more complex and realistic, but to explore the role that the network structure plays on the default propagation process with varying economic scenarios.

### 3.2 Dynamical analysis of default contagion

To analyze the potential for default propagation in a financial network using the defined contagion model, we define a default probability density *ρ* as

$$ \rho = \frac{\sum_{i = 1}^{N} p_{i}^{\mathrm{ss}}}{N} , $$

(6)

where *N* is the number of companies and \(p_{i}^{\mathrm{ss}}\) stands for the default probability of company *i* at the model’s stationary state. Note that since we are considering a SIS modeling framework, by construction, the dynamics will always reach a steady state. As shown in Sect. 3.1, the default probability density depends on the default infection rate *β* and the recovery rate *μ*. For obtaining *ρ*, we monitor \(p_{i}(t)\) as a function of time in a discrete manner, until the contagion model reaches the steady state. Note that time represents the iterations of the MMCA recursive model. To do so, we set \(p_{i}(0)\) to the real company default label. For understanding the system dynamics, without loss of generality, we can apply the classic MMCA framework where the parameter, \(\beta _{i}\), the contagion infectivity per node, is constant and equal for all companies in the network.

However, to make this model more realistic we introduce a variation to this setting using default recovery probability *μ* dependent on the relative in-degree of each company *i*. A particular \(\mu _{i}\) for a company *i* is therefore defined as

$$ \mu _{i} = \frac{k_{i}}{\max_{j=1}^{N}(k_{j})} , $$

(7)

where \(\max_{j=1}^{N}(k_{j})\) is the maximum in-degree in the network. The intuition behind this heterogeneity in the recovery parameter \(\mu _{i}\) is that companies having large number of customers will recover faster than those whose market risk is concentrated in a few customers. Note that this is just one of the possible variations of our default contagion framework. For instance, the generalization of the heterogeneous *μ* to other company characteristics/features as balance sheet information or any other individual attribute is straightforward. Moreover, not only the recovery rate but also the infection probability *β* can be defined as company dependent, and/or even depend on group of companies such as economical sector (*C*). So, Eq. (5) can be generalized as

$$ \textstyle\begin{cases} p_{i}(t + 1) = (1-q_{i}(t) ) (1 - p_{i}(t) ) + (1 - \mu _{i}) p_{i}(t) + \mu _{i} (1 - q_{i}(t) )p _{i}(t) , \\ q_{i} = \prod_{j=1}^{N} (1 - \beta ) w_{ji} p_{j}(t) ) . \end{cases} $$

(8)

Accordingly, the proposed modification, Eq. (7), to obtain a in-degree dependent *μ* is one of the simplest approaches, since it only varies a parameter using a topological characteristic such as the relative in-degree.

### 3.3 Dynamical properties of the network: onset slope and sensitivity

Naturally, the observed dynamical behavior is the result of an interplay between the MMCA framework and the network topology. To understand the reasons behind the different sectors’ response to default propagation, we will characterize the dynamical behavior of each sector by the onset threshold \(\mathcal{R}_{0} \) and the *sensitivity*
*S* to the initial set of defaulted companies. Both metrics are descriptors of the expected behavior in the steady state regime. The onset slope is measured by estimating numerically the critical \(\beta _{c}(s)\) at which the first default cases start to appear in sector *C*. Practically, fixing *μ*, the parameter *β* is increased until the number of default cases in the sector in the stationary state goes over 1% of the real data defaults found in the sector, marking \(\beta _{c}(s)\). All these calculations are done in the stationary state of the system. When \(\beta _{c}(s) \) is plotted versus *μ*, one finds a noisy linear increase and, therefore, we define \(\mathcal{R}_{0}\) as the slope of the linear fit.

This property reveals the spreading capacity of the infectious process in each sector. Larger values imply that when the life times of defaults in the companies of the sector become shorter, one needs higher infectivity to overcome the threshold. Sectors with larger \(\mathcal{R}_{0}\) should be more resilient to general default. From the moment they start to show significant default, other sectors with lower \(\mathcal{R}_{0}\) may be in very bad shape already. Furthermore, given a certain set of parameters, an isolated default event in one of the sectors with larger \(\mathcal{R}_{0}\) can trigger an avalanche of default on weaker sectors, for which the conditions are favorable for contagion. In this sense, the \(\mathcal{R}_{0}\) value of a sector is also related to the capacity of the sector to spread default.

Regarding the sector sensitivity to default propagation, this dynamical property measures the rate of change of \(\rho (\beta ,\mu )\) at the transition point (which is normally known as *β* cut-off). Computing sensitivity involves fitting a linear regression to the model response and using its standardized regression coefficients as direct measures of sensitivity. Therefore this metric describes how susceptible a sector is to default, quite the opposite to \(\mathcal{R}_{0}\), which characterizes how a sector affects the system. The relationship between these two dynamical descriptors and the network structure will be explored next. As mentioned before, these are defined at the steady state, but it is also important to understand how dynamics evolve in the transient regime. This analysis will be carried out by synthetically concentrating defaulted companies (in specific proportions) in the different sectors and exploring pair-wise sectoral interactions at the initial steps of the simulation.

### 3.4 How do sectoral properties of the nodes affect network dynamics?

Since economic crisis often start in a given sector and later expand to others (see for example the 2000 energy crisis and the 2008 financial crisis), we are also interested in exploring the dependence of default propagation on attributes related to each sector, such as sectoral default probability density and sectoral inter-connectivity. In particular, to study the dependence on the sectoral default probability density we rewrite Eq. (6) for a each sector (*C*) as:

$$ \rho _{C} = \frac{\sum_{i \in C} p_{i}^{\mathrm{ss}}}{N_{C}} . $$

(9)

We also propose a metric of in-sectoral inter-connectivity \(I_{ \mathrm{in}}\) that measures how well, on average, a sector is connected to other sectors by incoming links:

$$ I_{\mathrm{in}} = \frac{\sum_{i,j}(w_{ij}- \frac{s_{i}^{\mathrm{in}}}{17})^{2}}{N_{C}}, $$

(10)

where \(N_{C}\) is the number of companies a certain sector *C* has, \(i \in C\), \(w_{ij}\) are incoming links of *i* (\(j \rightarrow i\)) and \(s_{i}^{\mathrm{in}}\) is the company *i* in-strength. In short, the in-sector inter-connectivity measures the mean square error with respect to a hypothetical equally distributed situation where we have a node with incoming weights from all *C* sectors with equal probability (\(\frac{s _{i}^{\mathrm{in}}}{C}\)). So, the larger the value, the more heterogeneously connected the sector is to other sectors by incoming links.