论文标题
学习大型网络系统遵守保护法的结构
Learning the Structure of Large Networked Systems Obeying Conservation Laws
论文作者
论文摘要
众所周知，许多网络系统，例如电网，大脑和舆论动态社交网络，以遵守保护法。这种现象的例子包括电网中的基尔乔夫法律和社交网络中的意见共识。 Conservation laws in networked systems may be modeled as balance equations of the form $X = B^{*} Y$, where the sparsity pattern of $B^{*}$ captures the connectivity of the network, and $Y, X \in \mathbb{R}^p$ are vectors of "potentials" and "injected flows" at the nodes respectively.节点电位$ y $会导致边缘流动，并且在节点上注入的流量$ x $是网络动力学的无关紧要的。在几个实用的系统中，网络结构通常是未知的，需要从数据中估算。为此，可以访问节点电位$ y $的样本，但只有节点注射$ x $的统计信息。在这个重要问题的激励下，我们研究了$ n $ y $ y $ $ y $样本矩阵$ b^{*} $的稀疏结构的估计，假设节点注射$ x $跟随高斯分布，并具有已知的共价$σ_x$。我们在高维度中提出了一个新的$ \ ell_ {1} $ - 该问题的正则最大似然估计器，其中网络$ p $的大小大于样本量$ n $。我们表明，此优化问题是目标中的凸，并接受了独特的解决方案。在新的相互不一致的条件下，我们在三重$（n，p，d）$上建立了足够的条件，对于$ b^{*} $的精确稀疏恢复是可能的； $ D $是图的程度。我们还建立了在元素最大，Frobenius和运营商规范中回收$ b^{*} $的保证。最后，我们通过对拟议估计量在合成和现实世界数据的性能进行实验验证来补充这些理论结果。
Many networked systems such as electric networks, the brain, and social networks of opinion dynamics are known to obey conservation laws. Examples of this phenomenon include the Kirchoff laws in electric networks and opinion consensus in social networks. Conservation laws in networked systems may be modeled as balance equations of the form $X = B^{*} Y$, where the sparsity pattern of $B^{*}$ captures the connectivity of the network, and $Y, X \in \mathbb{R}^p$ are vectors of "potentials" and "injected flows" at the nodes respectively. The node potentials $Y$ cause flows across edges and the flows $X$ injected at the nodes are extraneous to the network dynamics. In several practical systems, the network structure is often unknown and needs to be estimated from data. Towards this, one has access to samples of the node potentials $Y$, but only the statistics of the node injections $X$. Motivated by this important problem, we study the estimation of the sparsity structure of the matrix $B^{*}$ from $n$ samples of $Y$ under the assumption that the node injections $X$ follow a Gaussian distribution with a known covariance $Σ_X$. We propose a new $\ell_{1}$-regularized maximum likelihood estimator for this problem in the high-dimensional regime where the size of the network $p$ is larger than sample size $n$. We show that this optimization problem is convex in the objective and admits a unique solution. Under a new mutual incoherence condition, we establish sufficient conditions on the triple $(n,p,d)$ for which exact sparsity recovery of $B^{*}$ is possible with high probability; $d$ is the degree of the graph. We also establish guarantees for the recovery of $B^{*}$ in the element-wise maximum, Frobenius, and operator norms. Finally, we complement these theoretical results with experimental validation of the performance of the proposed estimator on synthetic and real-world data.
