# Critical Nodes Identification in Complex Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- We draw inspiration from the theory of structural holes [33] and only consider the node’s local neighborhood information to evaluate the importance of nodes. This makes the algorithm more computationally attractive for large-scale networks;
- (2)
- The proposed algorithm can effectively identify the hubs with numerous structural holes, which play important role in bridging different clusters of the network;
- (3)
- Empirical analyses on real and synthetic networks demonstrate that the proposed method can outperform Deg, k-shell, ME, CI, DDN [34], and random ranking method (Rand).

## 2. Materials and Methods

_{ij}), where a

_{ij}= 1 represents there is a connection linking node i and node j, and a

_{ij}= 0 otherwise. The degree value of node i is represented by k

_{i}.

#### 2.1. Measurement of Node Importance Based on Degree and Structural Hole Count

_{i}represents the neighbor set of node i and Δ

_{ij}represents the number of structural holes formed between node i and j with node i as the intermediary. According to Equation (1), the larger the degree of a node and its neighbors, and the higher the number of structural holes between the node and its neighbors, this means that the stronger the irreplaceability of nodes in the structure, the smaller the value of DSHC will be. According to Figure 1, we calculate the DSHC

_{Ego}

_{_A}between node Ego and node A, there are two structural holes between node Ego and node A,{A-Ego-C, A-Ego-D}. Therefore, we can get

_{Ego}

_{_A}= ((1/4 + 1/2) × (1/(1 + 2)))

^{2}= 1/16

_{Ego}

_{_B}, DSHC

_{Ego}

_{_C}, and DSHC

_{Ego}

_{_D}, and sum them to get DSHC

_{Ego}.

Algorithm 1 the DSHC Method |

Input: Network adjacency matrix A(a_{ij}), degree of network nodes k, the size of network N |

Output: The DSHC value of each node |

1: for i = 1 to N |

2: S_{i} = find (A(a_{ij}) =1) // Find the neighbors S_{i} of node i |

3: len(S_{i}) = length(S_{i}) // Number of neighbors of node i |

4: for j = 1:len(S_{i}) |

5: S_{i}_{_j}= find (A(a_{jk}) =1) // Find the neighbors S_{i}_{_j} of neighbor j of node i |

6: Δ_{ij}= |S_{i}_{_j} - intersect(S_{i}, S_{i}_{_j})| // calculate the number of structural holes formed between node i and j with node i as the intermediary |

7: ${DSHC}_{i\_j}={\left(\left(\frac{1}{{k}_{i}}+\frac{1}{{k}_{j}}\right)\ast \frac{1}{1+{\mathsf{\Delta}}_{ij}}\right)}^{2}$ |

8: end for |

9: DSHC_{i} = sum(DSHC_{i}_{_j}) // according to Equation(1) |

10: end for |

11: Return DSHC value of each node |

#### 2.2. Benchmark Methods

## 3. How to Evaluate the Performance

_{s}denotes the number of components with size equal to s and N represents the size of the network nodes. Obviously, the smaller the value of p

_{c}, the better the ranking algorithm.

_{ij}means the efficiency of node i and j, η

_{ij}= 1/d

_{ij}, and d

_{ij}denotes the shortest path length between node i and j. As network nodes are gradually removed, the average shortest path between nodes becomes larger, which makes the connectivity of the network worse.

_{0}denotes the network efficiency of the original network.

## 4. Data Description

## 5. Results and Analysis

#### 5.1. Experiments in Real Networks

#### 5.2. Experiments in Synthetic Networks

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The network maximum connectivity coefficient (G) as a monotonic function of the number of removed nodes as compared with k-shell, degree, mapping entropy centrality (ME), CI, DDN, and Rand in USAir, Erdos, Facebook, USAirport, Yeast, and Power networks, respectively.

**Figure 4.**The decline rate of network efficiency (η) as a monotonic function of the number of blocked nodes as compared with k-shell, degree, ME, collective influence algorithm (CI), DDN, and Rand in USAir, Erdos, Facebook, USAirport, Yeast, and Power networks, respectively.

**Figure 5.**Comparison of the maximum connectivity coefficient (G) under different attack strategies in synthetic small-world networks, where the size of networks N = 1000, the number of neighbor nodes of each node in the nearest-neighbor coupled network is α = 8, and the randomization reconnection probabilities μ = 0.06, 0.08, and 0.1 for (

**a**), (

**b**), (

**c**), respectively.

**Figure 6.**Comparison of the decline rate of network efficiency (η) under different attack strategies in synthetic small-world networks, where the size of networks N = 1000, the number of neighbor nodes of each node in the nearest-neighbor coupled network is α = 8, and the randomization reconnection probabilities μ = 0.06, 0.08, and 0.1 for (

**a**), (

**b**), (

**c**), respectively.

**Figure 7.**Comparison of the susceptibility (S) under different attack strategies in synthetic small-world networks, where the size of networks N = 1000, the number of neighbor nodes of each node in the nearest-neighbor coupled network is α = 8, and the randomization reconnection probabilities μ = 0.06, 0.08, and 0.1 for (

**a**), (

**b**), (

**c**), respectively.

**Table 1.**The basic statistical features of Facebook, Erdos, USAir, USAirport, Yeast, Power networks where N and M represent the number of nodes and connections, respectively. <k> and C is the average degree and average clustering coefficient of the network, ks

_{max}represents the largest k-shell values, and L denotes the average shortest path length.

Network | N | M | <k> | C | ks_{max} | L |
---|---|---|---|---|---|---|

USAir | 332 | 2126 | 12.807 | 0.625 | 26 | 2.738 |

324 | 2218 | 13.69 | 0.466 | 18 | 3.054 | |

Erdos | 446 | 1417 | 6.33 | 0.296 | 9 | 3.952 |

USAirport | 1574 | 28,236 | 21.901 | 0.505 | 64 | 3.113 |

Yeast | 2375 | 11,693 | 9.847 | 0.306 | 40 | 5.094 |

Power | 4961 | 6964 | 2.669 | 0.080 | 5 | 18.989 |

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**MDPI and ACS Style**

Yang, H.; An, S.
Critical Nodes Identification in Complex Networks. *Symmetry* **2020**, *12*, 123.
https://doi.org/10.3390/sym12010123

**AMA Style**

Yang H, An S.
Critical Nodes Identification in Complex Networks. *Symmetry*. 2020; 12(1):123.
https://doi.org/10.3390/sym12010123

**Chicago/Turabian Style**

Yang, Haihua, and Shi An.
2020. "Critical Nodes Identification in Complex Networks" *Symmetry* 12, no. 1: 123.
https://doi.org/10.3390/sym12010123