Sensor Placement for Learning in Flow Networks

TL;DR

This work studies how to place sensors on a subset of edges in a flow network to accurately infer all edge flows under a flow-conservation constraint. It formalizes the problem, proves NP-hardness, and proposes an efficient greedy heuristic augmented with lazy evaluation and Woodbury-based recursive updates to scale to large networks. Empirical results on real road networks show the approach outperforms topology-only baselines across multiple error metrics, especially when the sensor budget is small, and demonstrates robustness to imperfect ground-truth information and to noisy observations. The practical impact is improved, scalable inference of network flows with few sensors, enabling better monitoring and control for transportation, utilities, and other critical infrastructures.

Abstract

Large infrastructure networks (e.g. for transportation and power distribution) require constant monitoring for failures, congestion, and other adversarial events. However, assigning a sensor to every link in the network is often infeasible due to placement and maintenance costs. Instead, sensors can be placed only on a few key links, and machine learning algorithms can be leveraged for the inference of missing measurements (e.g. traffic counts, power flows) across the network. This paper investigates the sensor placement problem for networks. We first formalize the problem under a flow conservation assumption and show that it is NP-hard to place a fixed set of sensors optimally. Next, we propose an efficient and adaptive greedy heuristic for sensor placement that scales to large networks. Our experiments, using datasets from real-world application domains, show that the proposed approach enables more accurate inference than existing alternatives from the literature. We demonstrate that considering even imperfect or incomplete ground-truth estimates can vastly improve the prediction error, especially when a small number of sensors is available.

Figures (6)

• Figure 1: Reducing SUM to SENSOR, with $X\!=\!\{3,4,6\}$ and $t\!=\!10$, candidate set $C\!=\!X$ (blue), and the target $T\!=\!\{(t_1, t_2)\}$ (red). Choosing the two edges that sum to $t$ for $S$ generates a perfect prediction for $T$.
• Figure 2: Conserved flow prediction results when flows are fully observed for validation purposes. The plots show the correlation between the prediction $\mathbf{\hat{f}}$ and ground truth flow $\mathbf{f}$. Our greedy heuristic (Greedy) outperforms all four baselines in all datasets.
• Figure 3: Predicted vs. ground-truth flows for the Anaheim road network using the flow conservation algorithm (Equation \ref{['eqn:flowleastsquares']}) for various sensor placement algorithms. Values for the selected sensors are shown in red and inferred values in black. The budget $k$ is fixed at 10% of all edges. Both the predicted and actual flows are normalized to [0,1]. Our method (Greedy) chooses a more representative set of edges than the baselines.
• Figure 4: Example of sensor placements (in red) using our heuristic (Greedy) and the Recursive Bisection (RB) baseline for the Anaheim road network. Edge traffic counts are represented by edge thickness. Unlike the baseline, our approach targets a few high-traffic paths.
• Figure 5: Conserved flow prediction results for sensor selection based on a noisy estimate of ground-truth flows for varying noise levels. The number of sensors placed ($k$) is fixed at 10% of the edges. The sensor placement is quite robust to noise target values, outperforming the baselines under noise levels of up to 2$\times$ the standard deviation of the flows.

Theorems & Definitions (2)

• Definition 1: The Sensor Placement Problem
• Theorem 3.1