# Towards Optimal Robustness of Network Controllability by Nested-Edge Rectification

^{*}

## Abstract

**:**

## 1. Introduction

- Due to the impossibility of theoretical analysis and exhaustive searching for large-sized networks, an exhaustive search was executed on feasible small-sized networks, and NRS was obtained. NRS satisfies ENC and has great controllability robustness.
- NER is proposed to improve the robustness of the network controllability against random attacks by constructing NRS in the network. Meanwhile, NER can be applied to networks with different scales. In addition, NER constructs a backbone ring through maximum matching, which rapidly improves the initial controllability of networks.
- The controllability robustness can be improved on six synthetic networks and real-world networks by NER, and NER is better than other methods of edge rectification. For networks with poor controllability robustness, such as the scale-free network, NER improves controllability robustness more obviously.

## 2. Network Controllability and Its Robustness

## 3. Nested Ring Structure and Optimization Strategy

#### 3.1. Nested Ring Structure

#### 3.2. Nested Edge Rectification

Algorithm 1 Nested ring rectification strategy |

input:The adjacency matrix of the network A; the number of network nodes N; thenumber of network edges M; Number of reconnected edges TIMES |

Output: Adjacency matrix of the optimized network A |

t ← 0; |

if backbone does not exists on the network then |

$STARTNODE\leftarrow $ nodes that are only the started node of an edge in a maximum matching |

$ENDNODE\leftarrow $ nodes that are only the ended node of an edge in a maximum matching |

for
$i\leftarrow 1$ to
$\left|STARTNODE\right|$ do |

$i\leftarrow $ node with the largest out-degree |

$j\leftarrow $ node with the largest in-degree among the successors of node i |

delete edge ${A}_{(i,j)}$ |

add edge ${A}_{\left(STARTNODE\right(i),STARTNODE(mod(i+1,N)\left)\right)}$ |

$t\leftarrow t+1$ |

end for |

Number nodes as $1,2,\dots N$ through backbone; |

end if |

for
$r\leftarrow 2$
to
$M/N$
do |

for
$m\leftarrow 1$
to
N
do |

if do not exist edge ${A}_{(m,m+r)}$ then |

$i\leftarrow $ node with the largest out-degree |

$j\leftarrow $ The node with the largest outdegree among the successors of node i is except $i+1,i+2,\dots ,i+r-1$ |

delete edge ${A}_{(i,j)}$ |

add edge ${A}_{(j,mod(j+r,N)}$ |

$t\leftarrow t+1$ |

if
$t==TIMES$ then |

return A |

end if |

end if |

end for |

end for |

return A |

#### 3.3. Computational Complexity

## 4. Simulation Results

#### 4.1. Results on Synthetic Networks

#### 4.2. Results of Real-World Networks

#### 4.3. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Robustness of the structural controllability of the six original networks with $N=500$ by rewiring 1000 times. ${P}_{N}$ represents the proportion of nodes to be removed. ORI represents the original network without edge rectification. UCR, RER and NER represent the ways of edge rectification.

**Figure A2.**Robustness of structural controllability of the six original networks with $N=1500$ by rewiring 3000 times. ${P}_{N}$ represents the proportion of nodes to be removed. ORI represents original network without edge rectification. UCR, RER and NER represent the ways of edge rectification.

**Table A1.**Robustness of network controllability with different numbers of edge rectification operations for networks with $N=500$.

Number of Edge Rectification | Strategy | ER | SF | RT | RR | SW | QS |
---|---|---|---|---|---|---|---|

0 | 0.2957 | 0.6186 | 0.3135 | 0.2792 | 0.2593 | 0.3927 | |

UCR | 0.2890 | 0.4891 | 0.3022 | 0.2801 | 0.2643 | 0.3783 | |

500 | RER | 0.2512 | 0.4162 | 0.2560 | 0.2477 | 0.2445 | 0.3190 |

NER | 0.2442 | 0.3366 | 0.2478 | 0.2418 | 0.2368 | 0.2811 | |

UCR | 0.2837 | 0.4490 | 0.2998 | 0.2799 | 0.2669 | 0.3621 | |

750 | RER | 0.2448 | 0.3542 | 0.2473 | 0.2435 | 0.2422 | 0.2913 |

NER | 0.2352 | 0.2815 | 0.2368 | 0.2346 | 0.2295 | 0.2538 | |

UCR | 0.2831 | 0.4176 | 0.2977 | 0.2798 | 0.2672 | 0.3487 | |

1000 | RER | 0.2417 | 0.3050 | 0.2436 | 0.2416 | 0.2406 | 0.2710 |

NER | 0.2241 | 0.2549 | 0.2256 | 0.2243 | 0.2236 | 0.2303 |

**Table A2.**Robustness of network controllability with different numbers of edge rectification operations for networks with $N=1500$.

Number of Edge Rectification | Strategy | ER | SF | RT | RR | SW | QS |
---|---|---|---|---|---|---|---|

0 | 0.2955 | 0.6709 | 0.3130 | 0.2796 | 0.2591 | 0.4284 | |

UCR | 0.2895 | 0.5242 | 0.3028 | 0.2809 | 0.2640 | 0.4166 | |

1500 | RER | 0.2512 | 0.4524 | 0.2562 | 0.2476 | 0.2439 | 0.3566 |

NER | 0.2437 | 0.3606 | 0.2482 | 0.2417 | 0.2354 | 0.2941 | |

UCR | 0.2885 | 0.4798 | 0.3004 | 0.2796 | 0.2669 | 0.3989 | |

2250 | RER | 0.2439 | 0.3856 | 0.2465 | 0.2433 | 0.2419 | 0.3235 |

NER | 0.2355 | 0.2926 | 0.2368 | 0.2339 | 0.2291 | 0.2618 | |

UCR | 0.2871 | 0.4428 | 0.2971 | 0.2800 | 0.2687 | 0.3811 | |

3000 | RER | 0.2423 | 0.3301 | 0.2425 | 0.2412 | 0.2403 | 0.2951 |

NER | 0.2238 | 0.2632 | 0.2249 | 0.2238 | 0.2231 | 0.2332 |

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**Figure 1.**All cases of adding an edge for original network. (

**a**) original network structure; (

**b**–

**g**) all cases after adding one edge to the original network. The red edge is the added edge.

**Figure 2.**Adding edge results for exhaustive search. (

**a**) Original network; (

**b**–

**f**) the result of incrementally adding edges.

**Figure 3.**The process of a random attack. (

**a**) Original network; (

**b**–

**e**) The network under random attack processes.

**Figure 4.**Construct backbone ring. (

**a**) Original network; (

**b**) a kind of maximum matching of the network; (

**c**,

**d**) the backbone ring is constructed by maximum matching; (

**e**) the final network; (

**f**) equivalent structure of the final network. The red line is maximum matching, the blue edge is a deleted edge and the green edge is an added edge.

**Figure 5.**Robustness of the structural controllability of the six original networks with $N=1000$ by rewiring 2000 times. ${P}_{N}$ represents the proportion of nodes to be removed. ORI represents the original network without edge rectification. UCR, RER and NER represent the ways of edge rectification.

**Figure 6.**Ratio of nodes to be removed when the number of driver nodes changes from 1 to 2 on six synthetic networks after $2N$ NER operations. The blue represents networks after NER, and the red represents networks after RER. ‘network’ represents the types of original networks. ${P}_{N}$ represents the ratio of nodes to be removed.

**Figure 7.**Changes in HO and HI of six networks after NER for different lengths of time under random attacks.

**Figure 8.**Robustness of structural controllability after edge rectification $2N$ times on real-world networks.

**Figure 9.**The changes in out-degree distribution under different numbers of iterations for NER and RER. ${k}^{out}$ represents the out-degree of nodes. ${P}_{{k}^{out}}$ represents the proportion of nodes with ${k}^{out}$ out-degree of all nodes.

**Figure 10.**Relationship between NRS and the structure satisfying ENC. (

**a**) NRS. (

**b**) General structure satisfying ENC. (

**c**) Equivalent structure of (

**b**). (

**d**) Relationship between NRS and ENC.

**Figure 11.**The number of operations required for NER and RER to make the number of driver nodes one. The times axis represents the number of NER and RER.

**Table 1.**Robustness of network controllability with different numbers of edge rectification operations.

Number of Edge Rectification | Strategy | ER | SF | RT | RR | SW | QS |
---|---|---|---|---|---|---|---|

0 | 0.2963 | 0.6557 | 0.3124 | 0.2798 | 0.2605 | 0.4168 | |

UCR | 0.2884 | 0.5147 | 0.2995 | 0.2796 | 0.2634 | 0.4012 | |

1000 | RER | 0.2510 | 0.4414 | 0.2564 | 0.2478 | 0.2437 | 0.3451 |

NER | 0.2437 | 0.3527 | 0.2482 | 0.2419 | 0.2362 | 0.2900 | |

UCR | 0.2852 | 0.4696 | 0.2962 | 0.2787 | 0.2657 | 0.3852 | |

1500 | RER | 0.2446 | 0.3745 | 0.2472 | 0.2433 | 0.2420 | 0.3121 |

NER | 0.2353 | 0.2891 | 0.2368 | 0.2338 | 0.2301 | 0.2591 | |

UCR | 0.2820 | 0.4348 | 0.2927 | 0.2786 | 0.2641 | 0.3682 | |

2000 | RER | 0.2419 | 0.3221 | 0.2435 | 0.2412 | 0.2402 | 0.2856 |

NER | 0.2242 | 0.2603 | 0.2253 | 0.2233 | 0.2228 | 0.2322 |

**Table 2.**Changes in the basic features of the original network and the network for which the edge rectification operation was performed 2000 times. Average path length (APL), average (node), betweenness centrality (ABC), clustering coefficient (CC), heterogeneity of out-degree (HO), and heterogeneity of in-degree (HI).

Strategy | ER | SF | RT | RR | SW | QS | |
---|---|---|---|---|---|---|---|

ORI | INF | INF | 4.8536 | 4.7131 | 5.1819 | 206.6615 | |

UCR | INF | INF | INF | INF | INF | INF | |

APL | RER | 4.8772 | 4.3943 | 4.9356 | 4.8762 | 4.9145 | 5.0733 |

NER | 6.0785 | 5.8503 | 6.3936 | 6.0961 | 7.0076 | 65.9633 | |

ORI | 3875 | 1351.8 | 4185.7 | 3709.4 | 3849.8 | 205460 | |

UCR | 4002.3 | 2543.7 | 3867.0 | 3881.2 | 3912.7 | 4304.2 | |

ABC | RER | 3873.4 | 3390.9 | 3931.7 | 3972.3 | 3910.6 | 4069.0 |

NER | 5073.4 | 4845.4 | 5388.2 | 5091.0 | 6001.6 | 64898 | |

ORI | 0.0024 | 0.0526 | 0.0014 | 0.0027 | 0.0020 | 0.0002 | |

UCR | 0.0033 | 0.0185 | 0.003 | 0.0032 | 0.0019 | 0.0021 | |

CC | RER | 0.0019 | 0.0072 | 0.0020 | 0.0020 | 0.0019 | 0.0016 |

NER | 0.1540 | 0.1572 | 0.1616 | 0.1307 | 0.3329 | 0.4094 | |

ORI | 1.2521 | 9.3584 | 1.3645 | 1.1989 | 1.1215 | 4.0503 | |

UCR | 1.1291 | 4.0055 | 1.1868 | 1.1929 | 1.1211 | 2.2373 | |

HO | RER | 1.0166 | 2.0102 | 1.0217 | 1.0144 | 1.0009 | 1.5408 |

NER | 1.0115 | 1.1357 | 1.0366 | 1.0070 | 1.0001 | 1.2281 | |

ORI | 1.2486 | 9.2805 | 1.4021 | 1.2171 | 1.1311 | 1.2288 | |

UCR | 1.2401 | 4.2642 | 1.3044 | 1.2238 | 1.1298 | 1.2392 | |

HI | RER | 1.0163 | 2.0038 | 1.0203 | 1.0137 | 1.0106 | 1.0132 |

NER | 1.0280 | 3.3571 | 1.0371 | 1.0248 | 1.0233 | 1.0173 |

Network | N | M |
---|---|---|

Roget | 1022 | 5075 |

ia-email-univ | 1133 | 5451 |

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**MDPI and ACS Style**

Yu, Z.; Nie, J.; Li, J.
Towards Optimal Robustness of Network Controllability by Nested-Edge Rectification. *Axioms* **2022**, *11*, 639.
https://doi.org/10.3390/axioms11110639

**AMA Style**

Yu Z, Nie J, Li J.
Towards Optimal Robustness of Network Controllability by Nested-Edge Rectification. *Axioms*. 2022; 11(11):639.
https://doi.org/10.3390/axioms11110639

**Chicago/Turabian Style**

Yu, Zhuoran, Junfeng Nie, and Junli Li.
2022. "Towards Optimal Robustness of Network Controllability by Nested-Edge Rectification" *Axioms* 11, no. 11: 639.
https://doi.org/10.3390/axioms11110639