Rajasekhar Anguluri

Anirudh Rayas, Rajasekhar Anguluri, Gautam Dasarathy

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.

Nafiseh Ghoroghchian, Rajasekhar Anguluri, Gautam Dasarathy et al.

We consider the controllability of large-scale linear networked dynamical systems when complete knowledge of network structure is unavailable and knowledge is limited to coarse summaries. We provide conditions under which average controllability of the fine-scale system can be well approximated by average controllability of the (synthesized, reduced-order) coarse-scale system. To this end, we require knowledge of some inherent parametric structure of the fine-scale network that makes this type of approximation possible. Therefore, we assume that the underlying fine-scale network is generated by the stochastic block model (SBM) -- often studied in community detection. We then provide an algorithm that directly estimates the average controllability of the fine-scale system using a coarse summary of SBM. Our analysis indicates the necessity of underlying structure (e.g., in-built communities) to be able to quantify accurately the controllability from coarsely characterized networked dynamics. We also compare our method to that of the reduced-order method and highlight the regimes where both can outperform each other. Finally, we provide simulations to confirm our theoretical results for different scalings of network size and density, and the parameter that captures how much community-structure is retained in the coarse summary.

Abrar Zahin, Rajasekhar Anguluri, Lalitha Sankar et al.

In Gaussian graphical model selection, noise-corrupted samples present significant challenges. It is known that even minimal amounts of noise can obscure the underlying structure, leading to fundamental identifiability issues. A recent line of work addressing this "robust model selection" problem narrows its focus to tree-structured graphical models. Even within this specific class of models, exact structure recovery is shown to be impossible. However, several algorithms have been developed that are known to provably recover the underlying tree-structure up to an (unavoidable) equivalence class. In this paper, we extend these results beyond tree-structured graphs. We first characterize the equivalence class up to which general graphs can be recovered in the presence of noise. Despite the inherent ambiguity (which we prove is unavoidable), the structure that can be recovered reveals local clustering information and global connectivity patterns in the underlying model. Such information is useful in a range of real-world problems, including power grids, social networks, protein-protein interactions, and neural structures. We then propose an algorithm which provably recovers the underlying graph up to the identified ambiguity. We further provide finite sample guarantees in the high-dimensional regime for our algorithm and validate our results through numerical simulations.

Antos Cheeramban Varghese, Rajasekhar Anguluri, Anamitra Pal

The process of calibrating instrument transformers (ITs) has been greatly simplified by using phasor measurement unit (PMU) data since this process eliminates the need for (a) additional hardware, and (b) taking ITs offline. However, such simplification comes at the cost of knowing the line parameters, whose estimation using PMU data in turn requires calibrated ITs. To solve this interdependency problem, we propose a novel framework that incorporates power system domain knowledge as constraints to perform simultaneous line parameter estimation and IT calibration. We demonstrate the effectiveness of our approach with simulated and real PMU data as well as for a power system application that uses both PMU data and line parameter information.

Anirudh Rayas, Jiajun Cheng, Rajasekhar Anguluri et al.

Complex networked systems driven by latent inputs are common in fields like neuroscience, finance, and engineering. A key inference problem here is to learn edge connectivity from node outputs (potentials). We focus on systems governed by steady-state linear conservation laws: $X_t = {L^{\ast}}Y_{t}$, where $X_t, Y_t \in \mathbb{R}^p$ denote inputs and potentials, respectively, and the sparsity pattern of the $p \times p$ Laplacian $L^{\ast}$ encodes the edge structure. Assuming $X_t$ to be a wide-sense stationary stochastic process with a known spectral density matrix, we learn the support of $L^{\ast}$ from temporally correlated samples of $Y_t$ via an $\ell_1$-regularized Whittle's maximum likelihood estimator (MLE). The regularization is particularly useful for learning large-scale networks in the high-dimensional setting where the network size $p$ significantly exceeds the number of samples $n$. We show that the MLE problem is strictly convex, admitting a unique solution. Under a novel mutual incoherence condition and certain sufficient conditions on $(n, p, d)$, we show that the ML estimate recovers the sparsity pattern of $L^\ast$ with high probability, where $d$ is the maximum degree of the graph underlying $L^{\ast}$. We provide recovery guarantees for $L^\ast$ in element-wise maximum, Frobenius, and operator norms. Finally, we complement our theoretical results with several simulation studies on synthetic and benchmark datasets, including engineered systems (power and water networks), and real-world datasets from neural systems (such as the human brain).

Rajasekhar Anguluri, Abed AlRahman Al Makdah, Vaibhav Katewa et al.

This paper proposes a new framework and several results to quantify the performance of data-driven state-feedback controllers for linear systems against targeted perturbations of the training data. We focus on the case where subsets of the training data are randomly corrupted by an adversary, and derive lower and upper bounds for the stability of the closed-loop system with compromised controller as a function of the perturbation statistics, size of the training data, sensitivity of the data-driven algorithm to perturbation of the training data, and properties of the nominal closed-loop system. Our stability and convergence bounds are probabilistic in nature, and rely on a first-order approximation of the data-driven procedure that designs the state-feedback controller, which can be computed directly using the training data. We illustrate our findings via multiple numerical studies.