# Credit Risk Contagion and Systemic Risk on Networks

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## Abstract

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## 1. Introduction

## 2. Credit Risk

#### 2.1. Probability Density Function of Credit Losses

#### 2.2. Systemic and Idiosyncratic Risk in the Vasicek Loan Portfolio Value Model

#### 2.3. Dynamics of Credit Risk Contagion

## 3. Credit Risk Contagion on Networks

#### Spreading of Systemic Risk Using the SIIS (Susceptible-Infected1-Infected2-Susceptible) Model

## 4. Numerical Experiments

## 5. Conclusions and Research Perspectives

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Unconditional cumulative probabilities of default for a default probability of $p=0.02$ and different values of asset correlation $\rho $.

**Figure 3.**Transfer diagram for the $SIIS$ (Susceptible-Infected1-Infected2-Susceptible) model. Boxes represent compartments, and arrows represent the transition from one compartment to another.

**Figure 4.**Low level of connectivity:Erdös–Rényi graph with 100 nodes and a connection probability of $p=0.01$ (

**left**). Time evolution of the mean value of non-performing loans on the graph (

**right**).

**Figure 5.**Core-periphery graph with five core nodes and 95 periphery nodes (

**left**). Time evolution of the mean value of non-performing loans on the graph (

**right**).

**Figure 6.**Small-world (Watts-Strogatz) network with 100 nodes and a rewiring probability of $\beta =0$ (ring) (

**left**). Time evolution of the mean value of non-performing loans on the graph (

**right**).

**Figure 7.**Medium level of connectivity:Erdös–Rényi graph with 100 nodes and a connection probability of $p\sim 0.02$ (

**left**). Time evolution of the mean value of non-performing loans on the graph (

**right**).

**Figure 8.**Core-periphery graph with 10 core nodes and 90 periphery nodes (

**left**). Time evolution of the mean value of non-performing loans on the graph (

**right**).

**Figure 9.**Small-world (Watts-Strogatz) network with 100 nodes and a rewiring probability of $\beta =0.1$ (

**left**). Time evolution of the mean value of non-performing loans on the graph (

**right**).

**Figure 10.**High level of connectivity: Erdös–Rényi graph with 100 nodes and a connection probability of $p=0.04$ (

**left**). Time evolution of the mean value of non-performing loans on the graph (

**right**).

**Figure 11.**Core-periphery graph with 20 core nodes and 80 periphery nodes (

**left**). Time evolution of the mean value of non-performing loans on the graph (

**right**).

**Figure 12.**Small-world (Watts–Strogatz) network with 100 nodes and a rewiring probability of $\beta =0.2$ (

**left**). Time evolution of the mean value of non-performing loans on the graph (

**right**).

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**MDPI and ACS Style**

Dolfin, M.; Knopoff, D.; Limosani, M.; Xibilia, M.G.
Credit Risk Contagion and Systemic Risk on Networks. *Mathematics* **2019**, *7*, 713.
https://doi.org/10.3390/math7080713

**AMA Style**

Dolfin M, Knopoff D, Limosani M, Xibilia MG.
Credit Risk Contagion and Systemic Risk on Networks. *Mathematics*. 2019; 7(8):713.
https://doi.org/10.3390/math7080713

**Chicago/Turabian Style**

Dolfin, Marina, Damian Knopoff, Michele Limosani, and Maria Gabriella Xibilia.
2019. "Credit Risk Contagion and Systemic Risk on Networks" *Mathematics* 7, no. 8: 713.
https://doi.org/10.3390/math7080713