# Reconstructing Damaged Complex Networks Based on Neural Networks

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## Abstract

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## 1. Introduction

## 2. Model

#### 2.1. Small-World Network

#### 2.2. Scale-Free Network

_{0}nodes. Next, a new node is introduced to the network and attaches to m existing nodes with high degree k. The process consisting of new node introduction and preferential attachment is repeated until a network with N = t + m

_{0}nodes has been constructed. Figure 2 shows the detailed algorithm for scale-free network construction in pseudocode format.

#### 2.3. Network Damage Model

_{RE}, that is defined as follows:

_{L}is the total number of existing links in the complex network that were damaged due to random attack and N

_{L,diff}is the total number of links in the reconstructed network that are different from the links in the original network before any node removal. For example, let us assume that the number of nodes N = 4 and the node pair set in the original network is equal to E

_{o}= {(1, 2), (1, 4), (2, 3), (3, 4)}. If the reconstructed network has node pair set as E

_{r}= {(1, 2), (1, 3), (1, 4), (2, 3)}, then N

_{L,diff}= 2 and N

_{L}= 4, giving us P

_{RE}= 0.5. Note that the estimated links in the reconstructed network that were not in the original network, in additions to links that were not reproduced, are all counted as errors.

## 3. Reconstruction Method

#### 3.1. Neural Network Model

#### 3.2. Neural Network Based Method

## 4. Performance Evaluations

#### 4.1. Simulation Environment

_{RE}described in Section 2. For the network damage model, we assume random attack process, where nodes are randomly removed with attached links. The MLPNN used in our method has two hidden layers with 64 neurons in the first layer and four neurons in the second layer. The nonlinear activation function in the hidden layer is chosen to be triangular activation function. The number of inputs to the MLPNN depends on the number of nodes in the network. To train and test the MLPNN, using complex networks with N = 10, N = 30, and N = 50, the number of inputs are set equal to the possible number of node pair combinations, which are 45, 435, and 1225, respectively. As for the number of outputs, eight are chosen to represent maximum number of 256 complex networks. The training input and output data patterns are randomly chosen from LL of M damaged complex networks with different percentage f of failed nodes out of total N nodes and corresponding indices of the complex networks.

#### 4.2. Small-World Network

_{RE}is less than 0.35 and for f = 0.5, P

_{RE}is less than 0.5. In another words, the proposed method can reconstruct almost close to 70% of the network topology for 10% node failures and more than 60% of the network topology for 50% node failures. Note that lower reconstruction error probability is observed for larger number of nodes. The reason for this results is due to the higher dimension of input data to the MLPNN, e.g., 1225 for N = 50. Furthermore, from the figure, we observe that P

_{RE}is less than what one might expect for the case where most of the nodes are destroyed, e.g., f = 0.7. This phenomenon is due to the large number of overlap in the node connections in LL among the M damaged networks due to small rewiring probability p. To study how the rewiring probability affects the reconstruction accuracy, simulations are performed with p = 0.3, p = 0.5, and p = 0.7, as shown in Figure 7 and Table 2. From the figure, we can see that there is a significant deterioration in performance in reconstruction accuracy with increase in rewiring probability p. This is because the small-world network topology becomes increasingly disordered with increase in rewiring probability and results in decrease in ability of the proposed method to reproduce the original network topology.

_{RE}remains less than 0.3.

#### 4.3. Scale-Free Network

_{0}was set to two and the node degree K = 2 for the preferential attachment process. Figure 9 shows that the reconstruction accuracy in scale-free network model is significantly lower compared to the small-world network. The reason for the poor performance is that the network topologies of M scale-free networks are more complex compared to the small-world network models. Furthermore, the links in LL between the M damaged networks do not overlap as much as in the small-world network models. In Figure 10 and Table 5, the reconstruction error probability performance with N = 30 and m

_{0}= 2, for different number of networks M = 10, M = 30, and M = 50, is shown. Compared to the small-world network environment, the reconstruction error probability values are quite high even in low percentage of node failures for M = 30 and M = 50. Due to the complex topology of the scale-free network model, increase in M affects the link estimation ability of the MLPNN. Finally, Figure 11 and Table 6 shows the reconstruction accuracy performance with different initial number of nodes in constructing scale-free network model. One can observe that there is a small difference in reconstruction performance for high percentage of node failures, regardless of the initial number of nodes. This is because the link estimation difficulty is almost equal to the MLPNN, even if they have different degree distributions, when the number of hubs remains the same in scale-free networks.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 6.**Probability of reconstruction error for small-world networks with N = 10, N = 30, N = 50, M = 10, and p = 0.15.

**Figure 7.**Probability of reconstruction error for small-world networks with p = 0.3, p = 0.5, p = 0.7, M = 10, and N = 50.

**Figure 8.**Probability of reconstruction error for small-world networks with M = 10, M = 30, M = 50, N = 50, and p = 0.15.

**Figure 9.**Probability of reconstruction error for scale-free networks with N = 10, N = 30, N = 50, M = 10, and m

_{0}= 2.

**Figure 10.**Probability of reconstruction error for scale-free networks with M = 10, M = 30, M = 50, N = 30, and m

_{0}= 2.

**Figure 11.**Probability of reconstruction error for scale-free networks with m

_{0}= 2, m

_{0}= 3, m

_{0}= 4, N = 30, and M = 10.

**Table 1.**Probability of reconstruction error for small-world networks with N = 10, N = 30, N = 50, M = 10, and p = 0.15.

N/f | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 |
---|---|---|---|---|---|---|---|---|

10 | 0.307 | 0.325 | 0.343 | 0.361 | 0.379 | 0.397 | 0.415 | 0.431 |

30 | 0.273 | 0.284 | 0.295 | 0.308 | 0.321 | 0.335 | 0.347 | 0.358 |

50 | 0.186 | 0.196 | 0.205 | 0.216 | 0.225 | 0.234 | 0.241 | 0.246 |

**Table 2.**Probability of reconstruction error for small-world networks with p = 0.3, p = 0.5, p = 0.7, M = 10, and N = 50.

P/f | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 |
---|---|---|---|---|---|---|---|---|

0.3 | 0.315 | 0.319 | 0.334 | 0.356 | 0.373 | 0.399 | 0.431 | 0.477 |

0.5 | 0.430 | 0.438 | 0.457 | 0.477 | 0.494 | 0.524 | 0.566 | 0.616 |

0.7 | 0.534 | 0.529 | 0.534 | 0.556 | 0.590 | 0.653 | 0.705 | 0.770 |

**Table 3.**Probability of reconstruction error for small-world networks with M = 10, M = 30, M = 50, N = 50, and p = 0.15.

M/f | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 |
---|---|---|---|---|---|---|---|---|

10 | 0.186 | 0.196 | 0.205 | 0.216 | 0.225 | 0.234 | 0.241 | 0.246 |

30 | 0.218 | 0.226 | 0.233 | 0.239 | 0.246 | 0.253 | 0.260 | 0.268 |

50 | 0.26 | 0.267 | 0.267 | 0.269 | 0.274 | 0.279 | 0.285 | 0.293 |

**Table 4.**Probability of reconstruction error for scale-free networks with N = 10, N = 30, N = 50, M = 10, and m

_{0}= 2.

N/f | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 |
---|---|---|---|---|---|---|---|---|

10 | 0.411 | 0.423 | 0.437 | 0.455 | 0.484 | 0.518 | 0.551 | 0.588 |

30 | 0.455 | 0.471 | 0.487 | 0.510 | 0.542 | 0.582 | 0.628 | 0.671 |

50 | 0.490 | 0.507 | 0.525 | 0.547 | 0.575 | 0.618 | 0.669 | 0.730 |

**Table 5.**Probability of reconstruction error for scale-free networks with M = 10, M = 30, M = 50, N = 30, and m

_{0}= 2.

M/f | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 |
---|---|---|---|---|---|---|---|---|

10 | 0.455 | 0.471 | 0.487 | 0.510 | 0.542 | 0.582 | 0.628 | 0.671 |

30 | 0.643 | 0.652 | 0.669 | 0.685 | 0.702 | 0.717 | 0.730 | 0.738 |

50 | 0.708 | 0.718 | 0.729 | 0.735 | 0.745 | 0.755 | 0.761 | 0.766 |

**Table 6.**Probability of reconstruction error for scale-free networks with m

_{0}= 2, m

_{0}= 3, m

_{0}= 4, N = 30, and M = 10.

m_{o}/f | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 |
---|---|---|---|---|---|---|---|---|

2 | 0.455 | 0.471 | 0.487 | 0.510 | 0.542 | 0.582 | 0.628 | 0.671 |

3 | 0.468 | 0.483 | 0.503 | 0.528 | 0.559 | 0.597 | 0.636 | 0.679 |

4 | 0.508 | 0.522 | 0.530 | 0.558 | 0.582 | 0.612 | 0.646 | 0.689 |

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Lee, Y.H.; Sohn, I.
Reconstructing Damaged Complex Networks Based on Neural Networks. *Symmetry* **2017**, *9*, 310.
https://doi.org/10.3390/sym9120310

**AMA Style**

Lee YH, Sohn I.
Reconstructing Damaged Complex Networks Based on Neural Networks. *Symmetry*. 2017; 9(12):310.
https://doi.org/10.3390/sym9120310

**Chicago/Turabian Style**

Lee, Ye Hoon, and Insoo Sohn.
2017. "Reconstructing Damaged Complex Networks Based on Neural Networks" *Symmetry* 9, no. 12: 310.
https://doi.org/10.3390/sym9120310