# CFM-RFM: A Cascading Failure Model for Inter-Domain Routing Systems with the Recovery Feedback Mechanism

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Optimal Valid Path Prediction Method for Inter-Domain Networks

#### 2.2. Types of Attacks and Costs

## 3. Analysis of the Cascading Failure Principle for the Inter-Domain Routing System

## 4. CFM-RFM

#### 4.1. Notations

#### 4.1.1. The Load and Capacity of Nodes

#### 4.1.2. The Load and Capacity of Links

#### 4.2. Propagation Mechanism of UPDATE Messages

#### 4.2.1. Failure of the Node Causes the Propagation Process of the UPDATE Message

#### 4.2.2. Re-Triggered UPDATE Message Propagation

#### 4.2.3. Pseudocode

Algorithm 1. The propagation algorithm. |

Input:G, LoadNode, InvalidNodeSet, RecoverNodeSet |

Output: The updated state of the node |

1: While InvalidNodeSet $\ne \mathsf{\Phi}$ do |

2: Find all the neighbors (NeighborSet) of the node in the InvalidNode; |

3: NeighborSet sends UPDATE messages to reachable ${v}_{p}$,${v}_{p}\in \mathbb{R}$; |

4: LoadNode [${v}_{p}$] ← LoadNode [${v}_{p}$] + 1; |

5: IF LoadNode [${v}_{p}$] > R, InvalidNodeSet.add (${v}_{p}$), RecoverNodeSet.add (${v}_{p}$); |

6: End while |

7: While RecoverNodeSet $\ne \mathsf{\Phi}$ and $t=\Delta T$ do |

8: Find all the neighbors (NeighborSet) of the node in the RecoverNodeSet; |

9: NeighborSet sends UPDATE messages to reachable ${v}_{q}$,${v}_{q}\in \mathbb{R}$; |

10: LoadNode [${v}_{q}$] ← LoadNode [${v}_{q}$] + 1; |

11: IF LoadNode [${v}_{q}$] > R, InvalidNodeSet.add (${v}_{q}$), RecoverNodeSet.add (${v}_{q}$); |

12: End while |

#### 4.3. Traffic Redistribution

Algorithm 2. Traffic redistribution algorithm. |

Input: G, LoadLink, InvalidLinkSet, OptimalValidPaths,$\alpha $ |

Output: The updated state of the link |

1: Check p in OptimalValidPaths; |

2: While ${e}_{mn}\in p\mathrm{and}{e}_{mn}\in InvalidLinkSet$ do |

3: For ${e}_{ij}\in p$ do |

4: LoadLink [${e}_{ij}$] ← LoadLink [${e}_{ij}$] − $\alpha $; |

5: End for |

6: OptimalValidPaths.delete (p); |

7: End while |

8: Find the new optimal valid path ${p}_{ab}$; |

9: OptimalValidPaths.add (${p}_{ab}$); |

10: For ${e}_{ij}\in {p}_{ab}$ do |

11: LoadLink [${e}_{ij}$] ← LoadLink [${e}_{ij}$] + $\alpha $; |

12: IF LoadLink [${e}_{ij}$] > ${C}_{ij}$, InvalidLinkSet.add (${e}_{ij}$); |

13: End for |

#### 4.4. Modeling the Process of Cascading Failure

## 5. Experiments and Analysis

#### 5.1. Data and Parameter Setup

#### 5.2. Metric of Evaluation for Cascading Failure

#### 5.3. Impact of Recovery Feedback Mechanism

#### 5.4. Analysis of Attack Costs

#### 5.5. Validity of CFM-RFM

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Under the three attack strategies, the top 1% nodes or links of the two networks were removed to observe the change in failure rate and number of UPDATE as step increases by different models. The solid blue line refers to the CFM-DDF, and the solid red line refers to the CFM-RFM. The dotted line indicates the number of UPDATE. The types of attacks and countries have been indicated in the figure.

Notations | Descriptions |
---|---|

$G=(V,E)$ | Topology of inter-domain routing network. |

V | The set of nodes. |

E | The set of links. |

${v}_{i}$ | A node ${v}_{i}\in V$. |

${e}_{ij}$ | A link ${e}_{ij}\in E$. |

${L}_{i}$/${L}_{ij}$ | The load of ${v}_{i}$ or ${e}_{ij}$. |

${R}_{i}$ | The capacity of ${v}_{i}$. |

${C}_{ij}$ | The capacity of ${e}_{ij}$. |

$\beta $ | Tolerance parameter. |

$\Delta {T}_{i}$ | Recovery delay of ${v}_{i}$. |

$\Delta {T}_{ij}$ | Recovery delay of ${e}_{ij}$. |

$\gamma $ | Basic unit flow. |

VIRS | Betweenness. |

EIRS | Betweenness of links. |

${\mathbb{R}}_{i}$ | The set of reachable nodes of ${v}_{i}$. |

Parameter | Value_India | Value_Canada |
---|---|---|

Num_V | 2406 | 1523 |

Num_E | 4052 | 2508 |

Step | 50 | 50 |

$\gamma $ | 1 | 1 |

$\beta $ | 0.3 | 0.3 |

${R}_{i}$ | 250 | 150 |

$\Delta {T}_{i}$ | 5 | 5 |

$\Delta {T}_{ij}$ | 5 | 5 |

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**MDPI and ACS Style**

Zhao, W.; Wang, Y.; Xiong, X.; Li, Y.
CFM-RFM: A Cascading Failure Model for Inter-Domain Routing Systems with the Recovery Feedback Mechanism. *Information* **2021**, *12*, 247.
https://doi.org/10.3390/info12060247

**AMA Style**

Zhao W, Wang Y, Xiong X, Li Y.
CFM-RFM: A Cascading Failure Model for Inter-Domain Routing Systems with the Recovery Feedback Mechanism. *Information*. 2021; 12(6):247.
https://doi.org/10.3390/info12060247

**Chicago/Turabian Style**

Zhao, Wendian, Yongjie Wang, Xinli Xiong, and Yang Li.
2021. "CFM-RFM: A Cascading Failure Model for Inter-Domain Routing Systems with the Recovery Feedback Mechanism" *Information* 12, no. 6: 247.
https://doi.org/10.3390/info12060247