# Study of Cascading Failure in Multisubnet Composite Complex Networks

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Composite Network Model

#### 2.2. Coupling Network Cascading Failure Model

#### 2.3. Cascading Failure Model of Composite Networks

## 3. Results

- (a)
- Attack node
**v**in subnet A to invalidate it, take the value of the overload function for that node as “0”, and find neighbor nodes**v**that are connected to it. - (b)
- Redistributes the load of node v and its neighbors, and if any node is overloaded, mark its overload function as “0”.
- (c)
- Find the failure node in the subnet A.
- (d)
- Find all the neighbor nodes of one failed node and redistribute the load of the failed node and its neighbors, where the node with an overload function value of “0” does not accept any external load, and if any node is overloaded, mark its overload function as “0”.
- (e)
- Repeat steps (c)–(d) until no nodes fail.
- (f)
- Calculate the number of failed nodes in the entire network as
**F**._{A} - (g)
- Identify the node in subnet B that is coupled to subnet A. If all of its coupled nodes in subnet A fail, then the overload function of this node is marked as “0”.
- (h)
- Identify the failed nodes in subnet B.
- (i)
- Redistribute the load of a failed node and its neighbor nodes in subnet B. A node with an overload function value of “0” will not accept any external load, and when the node fails, its overload function is marked as “0”.
- (j)
- Repeat steps (h)–(i) until no nodes fail.
- (k)
- Calculate the number of failed nodes in the entire network as
**F**._{B} - (l)
- Repeat steps (a)–(h) until each node is attacked once in subnet A, and calculate the failure size
**S**of the composite network.

#### 3.1. Influence of Intranetwork Relationship Strength on Composite Networks under Different Conditions of Relationship Topology within the Network

#### 3.2. Influence of Coupling Relationship Strength on Composite Networks under Different Conditions of Relationship Topology within the Network

#### 3.3. The Average Degree of the Relationship within the Network and Effect of Relationship Strength on Composite Networks under Different Conditions

#### 3.4. The Effect of Relationship Strength on Composite Networks under Different Conditions of Different Average Degree of the Coupling Relationship

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Influence of intranetwork relationship strength on composite networks. (

**a**) WS-WS (Two subnets are WS network); (

**b**) BA-BA (Two subnets are BA network); (

**c**) WS-BA (The first subnet is WS network and the second subnet is BA network); (

**d**) BA-WS (The first subnet is BA network and the second subnet is WS network).

**Figure 5.**Influence of coupling relationship strength on composite networks. (

**a**) WS-WS (Two subnets are WS network); (

**b**) BA-BA (Two subnets are BA network); (

**c**) WS-BA (The first subnet is WS network and the second subnet is BA network); (

**d**) BA-WS (The first subnet is BA network and the second subnet is WS network).

**Figure 6.**The average degree of the relationship on composite networks. (

**a**) The average degree of intranetwork relationships is 2 and 4; (

**b**) the average degree of intranetwork relationships is 2 and 6; (

**c**) the average degree of intranetwork relationships is 4 and 2; (

**d**) the average degree of intranetwork relationships is 6 and 2.

**Figure 7.**The effect of relationship strength on composite networks. (

**a**) The average degree of coupling relationship is 2; (

**b**) the average degree of coupling relationship is 4; (

**c**) the average degree of coupling relationship is 6; (

**d**) the average degree of coupling relationship is 8.

Symbols | Explanation |
---|---|

${\mathit{F}}_{{\mathit{v}}_{\mathit{h}}{\mathit{v}}_{\mathit{l}}}^{{\mathit{r}}_{\mathit{i}}}$ | The relationship ${\mathit{r}}_{\mathit{i}}$ on edge ${\mathit{v}}_{\mathit{h}}{\mathit{v}}_{\mathit{l}}$ |

$\widehat{{\mathit{k}}_{{\mathit{v}}_{\mathit{h}}}^{{\mathit{r}}^{\mathit{\prime}}}}$ | The degree of node ${\mathit{v}}_{\mathit{h}}$ about relationship ${\mathit{r}}^{\mathit{\prime}}$ |

${\mathit{L}}_{{\mathit{v}}_{\mathit{h}}}\left(\mathbf{0}\right)$ | The initial load of node ${\mathit{v}}_{\mathit{h}}$ |

${\mathit{R}}_{\mathit{A}}$ | The relationship set between the failed node and its neighbor nodes |

${\Gamma}_{{\mathit{v}}_{\mathit{h}}}^{{\mathit{r}}_{\mathit{i}}}$ | The set of neighbor nodes of node ${\mathit{v}}_{\mathit{h}}$ about relationship ${\mathit{r}}_{\mathit{i}}$ |

$\mathit{s}{\mathit{f}}_{\mathit{i}}$ | The relationship strength of relationship ${\mathit{r}}_{\mathit{i}}$ |

${\mathit{p}}_{{\mathit{v}}_{\mathit{h}}{\mathit{v}}_{\mathit{l}}}^{{\mathit{r}}_{\mathit{i}}}$ | The number of triangles constituted by edges ${\mathit{v}}_{\mathit{h}}{\mathit{v}}_{\mathit{l}}$ according to the relationship ${\mathit{r}}_{\mathit{i}}$ |

${\mathit{W}}_{{\mathit{v}}_{\mathit{l}}}^{{\mathit{r}}_{\mathit{i}}}$ | The importance of node ${\mathit{v}}_{\mathit{l}}$ about relationship ${\mathit{r}}_{\mathit{i}}$ |

$\Delta {\mathit{L}}_{{\mathit{v}}_{\mathit{h}}{\mathit{v}}_{\mathit{l}}}^{{\mathit{r}}_{\mathit{i}}}$ | The load of the relationship ${\mathit{r}}_{\mathit{i}}$ from node ${\mathit{v}}_{\mathit{h}}$ to node ${\mathit{v}}_{\mathit{l}}$ |

$\mathit{C}{\mathit{F}}_{\mathit{v}}$ | The number of failed nodes caused by node $\mathit{v}$ |

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**MDPI and ACS Style**

Sun, G.; Chen, C.-C.; Bin, S.
Study of Cascading Failure in Multisubnet Composite Complex Networks. *Symmetry* **2021**, *13*, 523.
https://doi.org/10.3390/sym13030523

**AMA Style**

Sun G, Chen C-C, Bin S.
Study of Cascading Failure in Multisubnet Composite Complex Networks. *Symmetry*. 2021; 13(3):523.
https://doi.org/10.3390/sym13030523

**Chicago/Turabian Style**

Sun, Gengxin, Chih-Cheng Chen, and Sheng Bin.
2021. "Study of Cascading Failure in Multisubnet Composite Complex Networks" *Symmetry* 13, no. 3: 523.
https://doi.org/10.3390/sym13030523