Gaussian graphical model selection is an important paradigm with numerous applications, including biological network modeling, financial network modeling, and social network analysis. Traditional approaches assume access to independent and identically distributed (i.i.d) samples, which is often impractical in real-world scenarios. In this paper, we address Gaussian graphical model selection under observations from a more realistic dependent stochastic process known as Glauber dynamics. Glauber dynamics, also called the Gibbs sampler, is a Markov chain that sequentially updates the variables of the underlying model based on the statistics of the remaining model. Such models, aside from frequently being employed to generate samples from complex multivariate distributions, naturally arise in various settings, such as opinion consensus in social networks and clearing/stock-price dynamics in financial networks. In contrast to the extensive body of existing work, we present the first algorithm for Gaussian graphical model selection when data are sampled according to the Glauber dynamics. We provide theoretical guarantees on the computational and statistical complexity of the proposed algorithm's structure learning performance. Additionally, we provide information-theoretic lower bounds on the statistical complexity of learning GGMs from Glauber dynamics.

Networked systems that occur in various domains, such as electric networks, the brain, and opinion networks, are known to obey conservation laws. For instance, electric networks obey Kirchoff’s laws, and social networks obey opinion consensus. 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 highdimensional 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.

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 are modeled as balance equations of the form $X = B^\ast Y$, where the sparsity pattern of $B^\ast \in \mathbb{R}^{p\times p}$ captures the connectivity of the network on $p$ nodes, 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 which aim to balance out the potential difference, 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 to facilitate modeling, management, and control. To this end, 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 $B^\ast$ from $n$ samples of $Y$ under the assumption that the node injections $X$ follow a Gaussian distribution with a known covariance $\Sigma_X$. We propose a new $\ell_{1}$-regularized maximum likelihood estimator for tackling this problem in the high-dimensional regime where the size of the network may be vastly larger than the number of samples $n$. 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 $B^\ast$ is possible with high probability; $d$ is the degree of the underlying graph. We also establish guarantees for the recovery of $B^\ast$ 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.