Complex networks, formed by nodes and links contain a significant amount of crucial information that can reflect the nature of the systems. In this paper, we propose a network-structure-transmission (NST) framework based on $k$ -shell decomposition and compressed sensing for transmitting network structures and spectral properties with limited data. Specifically, we reduce the average degree and degree heterogeneity through the $k$ -shell decomposition, compress the decomposed networks with a linear dimensionality reduction, and transmit the compressed data to different receivers. After receiving the compressed data, the receivers can recover network structures through compressed sensing and assigned keys. The NST framework can achieve various network recovery by assigning different levels of keys to different receivers and controlling the amounts of compressed data transmitted to them. Experimental results on both artificial and empirical networks show that the recovery results can be classified into three phases, i.e., inaccuracy, spectral accuracy, and structural accuracy. The spectral properties of the original networks can be transmitted to the receivers through small amounts of data in the spectral-accuracy phase without recovering accurate network structures. All results suggest that the NST framework is flexible and efficient in transmitting network structures and spectral properties through limited data.

中文翻译：

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通过压缩感知传输有限数据的网络结构

由节点和链接组成的复杂网络包含大量重要信息，这些信息可以反映系统的性质。在本文中，我们提出了一种基于网络结构传输（NST）的框架$ k $ 壳分解和压缩感测，用于以有限的数据传输网络结构和频谱特性。具体而言，我们通过以下方式降低平均度和度异质性：$ k $ -壳分解，以线性降维压缩分解的网络，并将压缩后的数据传输到不同的接收器。接收到压缩数据后，接收器可以通过压缩感测和分配的密钥来恢复网络结构。NST框架可以通过将不同级别的密钥分配给不同的接收器并控制传输给它们的压缩数据量来实现各种网络恢复。人工和经验网络的实验结果表明，恢复结果可分为三个阶段，即不准确性，光谱精度和结构精度。原始网络的频谱特性可以在频谱准确度阶段通过少量数据传输到接收器，而无需恢复准确的网络结构。