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## Discrete data based local-to-global network reconstruction algorithm

Xu Xiang, Zhu Cheng, Zhu Xian-Qiang
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• #### 摘要

网络的结构和功能彼此相互影响, 网络上的功能往往体现为网络上的动力学过程, 网络上的动力学过程通过网络中的行为表象数据进行体现. 因此, 根据网络上可观测的相关数据对网络结构进行重构将成为可能. 本文拟解决如何根据网络上可观测的离散数据还原网络拓扑结构的问题, 提出了在网络局部利用每一条离散数据对应节点的相似程度来推测节点间发生连边的可能性, 通过多条离散数据重构网络各个局部拓扑并将由多条数据得到的局部拓扑进行叠加, 最终重构出整个网络的全局拓扑结构的算法. 为了验证算法的可行性与准确性, 在小世界、无标度和随机网络中进行了网络重构实验, 通过在三种不同类型及不同规模的网络中进行网络重构实验可以看出, 网络重构算法在不同类型网络中的表现也不同, 且网络的平均度值也会影响网络重构算法对数据的要求. 为了验证算法的适用性, 对三个实际网络进行了网络重构实验, 结果显示算法能够适用实际较大规模网络的重构. 该算法具有很好的适用性和准确度, 适合不同类型网络的拓扑结构重构场景.

#### Abstract

The structure and the function of network interact with each other. The function of network is often reflected as the dynamic process on the network. The dynamic process on the network is reflected by the behavior data in the network. Therefore, it is possible to reconstruct the network structure according to the observed data. This paper aims to solve the problem of how to restore the network topology according to the observable discrete data on the network. In this paper, an algorithm to infer the possibility of edge connection between nodes is proposed by using the similarity degree of each node corresponding to each discrete datum, and by reconstructing each local topology of the network through multiple discrete data, and by superposing the local topology obtained from multiple data, the global topology of the whole network is reconstructed finally. The data in the network are generated by SIR (Susceptible Infective Removed) model with infection probability of 0.2 and recovery probability of 1. Each time, a single node is selected as the infected node, and the final infection state of the network is counted as a network datum. In order to verify the feasibility and accuracy of the algorithm, the network reconfiguration experiments are carried out in small world, scale-free and random networks. Through the network reconstruction experiments in the networks of three different types and different scales, we can see that the performances of network reconstruction algorithm in different types of networks are different, and the average degree of network will affect the requirements for data of the network reconstruction algorithm. In order to verify the applicability of the algorithm, network reconstruction experiments are carried out on three practical networks. The results show that the algorithm can be applied to the reconstruction of large-scale networks. In order to show the accuracy of the algorithm more intuitively, we analyze the network reconstruction error after each network reconstruction experiment. The experiment shows that with the gradual increase of network data, the network reconstruction error gradually decreases and finally approaches to 0. In a nutshell, the algorithm we proposed in this work has good applicability and accuracy, and is suitable for different types of network topology reconstructions.

#### 作者及机构信息

###### 通信作者: 朱承, zhucheng@nudt.edu.cn
• 基金项目: 国家自然科学基金(批准号: 71571186, 61703416)和湖南省研究生创新基金(批准号: CX20190041)资助的课题

#### Authors and contacts

###### Corresponding author: Zhu Cheng, zhucheng@nudt.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 71571186, 61703416) and the Postgraduate Innovation Project of Hunan Province, China (Grant No. CX20190041)

#### 施引文献

• 图 1  网络初始二值数据矩阵

Fig. 1.  Initial binary data matrix of the network.

图 2  节点1和节点2的相同感染源数量

Fig. 2.  The same number of infection sources in node 1 and node 2.

图 3  子图重构过程

Fig. 3.  Subgraph reconstruction process.

图 4  子图叠加过程

Fig. 4.  Subgraph superposition process.

图 5  不同规模的WS小世界网络重构实验效果

Fig. 5.  Experimental results of WS small world network reconstruction with different scales.

图 6  不同规模的WS小世界网络重构误差分析

Fig. 6.  Error analysis of WS small world network reconstruction with different scales.

图 7  WS小世界网络不同平均度值对网络重构实验效果的影响

Fig. 7.  Influence of different average degrees of WS small world network on network reconstruction experiment.

图 8  不同规模的BA无标度网络重构实验效果

Fig. 8.  Experimental results of BA scale-free network reconstruction with different scales.

图 9  不同规模的BA小世界网络重构误差分析

Fig. 9.  Error analysis of BA scale-free network reconstruction with different scales.

图 10  BA无标度网络不同平均度值对网络重构实验效果的影响

Fig. 10.  The influence of different average degree values of BA scale-free network on network reconstruction experiment.

图 11  不同规模的ER随机网络重构实验效果

Fig. 11.  Experimental results of ER random network reconstruction with different scales.

图 12  不同规模的ER小世界网络重构误差分析

Fig. 12.  Error analysis of ER random network reconstruction with different scales.

图 13  ER随机网络不同平均度值对网络重构实验效果的影响

Fig. 13.  Influence of different average degree of ER random network on network reconstruction experiment.

图 14  三种不同网络在相同数据下的重构效果对比

Fig. 14.  Comparison of reconstruction effects of three different networks under the same data.

图 15  三个实际网络重构实验效果

Fig. 15.  Experimental results of three practical network reconstruction.

图 16  三个实际网络重构误差分析

Fig. 16.  Error analysis of three practical network reconstruction.