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Resilient Infrastructure Network: Sparse Edge Change Identification via L1-Regularized Least Squares

Rajasekhar Anguluri

TL;DR

This work presents an $\ell_{1}$-norm regularized least-squares framework to identify multiple but sparse edge changes using noisy data, and systematically leverages the inherent structure within the Laplacian matrix, effectively avoiding overparameterization.

Abstract

Adversarial actions and a rapid climate change are disrupting operations of infrastructure networks (e.g., energy, water, and transportation systems). Unaddressed disruptions lead to system-wide shutdowns, emphasizing the need for quick and robust identification methods. One significant disruption arises from edge changes (addition or deletion) in networks. We present an $\ell_1$-norm regularized least-squares framework to identify multiple but sparse edge changes using noisy data. We focus only on networks that obey equilibrium equations, as commonly observed in the above sectors. The presence or lack of edges in these networks is captured by the sparsity pattern of the weighted, symmetric Laplacian matrix, while noisy data are node injections and potentials. Our proposed framework systematically leverages the inherent structure within the Laplacian matrix, effectively avoiding overparameterization. We demonstrate the robustness and efficacy of the proposed approach through a series of representative examples, with a primary emphasis on power networks.

Resilient Infrastructure Network: Sparse Edge Change Identification via L1-Regularized Least Squares

TL;DR

This work presents an -norm regularized least-squares framework to identify multiple but sparse edge changes using noisy data, and systematically leverages the inherent structure within the Laplacian matrix, effectively avoiding overparameterization.

Abstract

Adversarial actions and a rapid climate change are disrupting operations of infrastructure networks (e.g., energy, water, and transportation systems). Unaddressed disruptions lead to system-wide shutdowns, emphasizing the need for quick and robust identification methods. One significant disruption arises from edge changes (addition or deletion) in networks. We present an -norm regularized least-squares framework to identify multiple but sparse edge changes using noisy data. We focus only on networks that obey equilibrium equations, as commonly observed in the above sectors. The presence or lack of edges in these networks is captured by the sparsity pattern of the weighted, symmetric Laplacian matrix, while noisy data are node injections and potentials. Our proposed framework systematically leverages the inherent structure within the Laplacian matrix, effectively avoiding overparameterization. We demonstrate the robustness and efficacy of the proposed approach through a series of representative examples, with a primary emphasis on power networks.
Paper Structure (12 sections, 18 equations, 5 figures, 2 tables)

This paper contains 12 sections, 18 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model and Definitions
  3. Steady-state Model for Infrastructure Networks
  4. Edge Changes Identification Problem
  5. A Sparse Linear Model Formulation for Edge Changes Identification
  6. Vectorization of the EIV model
  7. Solution Strategies
  8. $\ell_1$-regularized least squares: LASSO estimator
  9. Simulations
  10. Synthetic network
  11. Power system networks
  12. Conclusions

Figures (5)

  • Figure 1: problem statement (illustrative): The graph underlying the Laplacian ${L}_0$ before the change with the network system is in (a). The Laplacian after edge changes is ${L}_1$, and is in (b). We want to directly estimate the graph underlying $\mathbf{\Delta}_{\mathbb{L}}$ in (c) (equivalently its sparsity pattern) using node potentials and injected flows.
  • Figure 2: Visualizing synthetic network ($8$ nodes and $12$ edges): (a) The edges shown in dashed lines are removed after the change. (b) White spaces correspond to nodes with no edges; making red check boxes white (including a copy on the upper triangular portion) gives the sparsity pattern for $L_1$. Sparsity patterns in (c) are representative only and should not be confused with the no. of boxes in the vectors and the dimensions of $\beta$'s or $\operatorname{Vec}(L_0)$ denoting these vectors; see second paragraph in \ref{['sec: synthetic network']} for additional details.
  • Figure 3: LASSO performance on the synthetic network in Fig. \ref{['fig: synthetic sparsity']}. The optimal $\lambda$ is for which acc, TP, and TN are closer to one and FP and FN are closer to zero. Thus, the optimal range is $0.15\leq \lambda \leq 0.6$.
  • Figure 4: Sparsity patterns of power networks' Laplacian matrices $(L_0)$ before change: The dimensions of matrices are $(57\times 57)$, $(118\times 118)$, and $(145\times 145)$, from left to right. The dark-colored boxes denote non-zeros, while the light-shaded boxes denote zeros.
  • Figure 5: LASSO performance on power system networks in Fig. \ref{['fig: power-sparsity']}. The optimal range of $\lambda$ are the ones for which acc, TP, and TN are closer to one, and FP and FN are closer to zero. Thus, the optimal range for all three systems is $0.1\leq \lambda \leq 1$.

Theorems & Definitions (1)

  • proof