Networked systems that occur in various domains, such as the power grid, the brain, and opinion networks, are known to obey conservation laws. For instance, electric networks obey Kirchoff's laws, and social networks display opinion consensus. Such conservation laws are often modeled as balance equations that relate appropriate injected flows and potentials at the nodes of the networks. A recent line of work considers the problem of estimating the unknown structure of such networked systems from observations of node potentials (and only the knowledge of the statistics of injected flows). Given the dynamic nature of the systems under consideration, an equally important task is estimating the change in the structure of the network from data -- the so called differential network analysis problem. That is, given two sets of node potential observations, the goal is to estimate the structural differences between the underlying networks. We formulate this novel differential network analysis problem for systems obeying conservation laws and devise a convex estimator to learn the edge changes directly from node potentials. We derive conditions under which the estimate is unique in the high-dimensional regime and devise an efficient ADMM-based approach to perform the estimation. Finally, we demonstrate the performance of our approach on synthetic and benchmark power network data.

翻译：存在于电网、大脑及意见网络等不同领域的网络化系统，已知遵循守恒定律。例如，电网遵守基尔霍夫定律，社交网络呈现意见共识现象。此类守恒定律通常被建模为描述节点注入流量与势能之间关系的平衡方程。近期一系列研究关注从节点势能观测值（及仅已知注入流量的统计特性）中估计此类网络化系统未知结构的问题。鉴于所研究系统的动态特性，评估网络结构随数据的变化——即所谓的差分网络分析问题——同等重要。具体而言，给定两组节点势能观测值，目标是估计潜在网络间的结构差异。我们针对遵循守恒定律的系统提出这一新颖的差分网络分析问题，并设计了一种可直接从节点势能中学习边变化的凸估计器。我们推导出在高维条件下估计量唯一性的条件，并开发了基于ADMM的高效求解方法。最后，我们在合成数据与基准电网数据上验证了该方法的性能。