Inferring network topologies from observed graph-structured data (also known as graph signals) is a crucial task in many applications of network science. Existing papers on network topology inference typically assume that the observations at all nodes are available. However, there are many situations where only partial observations can be collected due to application-specific constraints. To handle the missing data problem, we propose a framework that relies on heterogeneous incomplete data from a collection of related networks to identify multiple network topologies simultaneously. This work advocates the Gaussian graphical model (GGM) and casts the topology inference problem in terms of estimating the precision matrix that has a form of graph Laplacian. Firstly, an unbiased estimator for the covariance matrix of incomplete data is established and then algorithms based on the alternating direction method of multipliers (ADMM) are developed to jointly estimate graph topologies, rather than estimate each graph topology separately. So that we can borrow information across the multiple related networks to eliminate the impact of missing data. Moreover, non-asymptotic statistical analysis is provided, which proves the consistency of the graph estimator and enables us to investigate the effect of several key factors on the graph estimation error bound. Furthermore, based on the consistent graph estimator, an adaptive algorithm that utilizes the reweighting scheme is proposed to improve the estimation accuracy of the graph-edge structure. Finally, we evaluate our method on both real and synthetic datasets, and the experimental results demonstrate the advantage of our method in comparison with benchmarking algorithms.

中文翻译：

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异构不完全图信号的网络拓扑推断

从观察到的图结构数据（也称为图信号）推断网络拓扑是网络科学许多应用中的一项关键任务。现有的关于网络拓扑推理的论文通常假设所有节点的观察都是可用的。但是，由于特定于应用程序的限制，在很多情况下只能收集部分观察结果。为了处理缺失数据问题，我们提出了一个框架，该框架依赖于来自相关网络集合的异构不完整数据来同时识别多个网络拓扑。这项工作提倡高斯图模型（GGM），并根据估计具有拉普拉斯图形式的精度矩阵来解决拓扑推理问题。首先，建立了不完整数据协方差矩阵的无偏估计器，然后开发了基于乘法器交替方向法（ADMM）的算法来联合估计图拓扑，而不是单独估计每个图拓扑。这样我们就可以跨多个相关网络借用信息，以消除丢失数据的影响。此外，提供了非渐近统计分析，证明了图估计器的一致性，并使我们能够研究几个关键因素对图估计误差界限的影响。此外，基于一致性图估计器，提出了一种利用重加权方案的自适应算法来提高图边结构的估计精度。最后，我们在真实和合成数据集上评估我们的方法，