# Biblio

This paper considers a 2-player strategic game for network routing under link disruptions. Player 1 (defender) routes flow through a network to maximize her value of effective flow while facing transportation costs. Player 2 (attacker) simultaneously disrupts one or more links to maximize her value of lost flow but also faces cost of disrupting links. This game is strategically equivalent to a zero-sum game. Linear programming duality and the max-flow min-cut theorem are applied to obtain properties that are satisfied in any mixed Nash equilibrium. In any equilibrium, both players achieve identical payoffs. While the defender's expected transportation cost decreases in attacker's marginal value of lost flow, the attacker's expected cost of attack increases in defender's marginal value of effective flow. Interestingly, the expected amount of effective flow decreases in both these parameters. These results can be viewed as a generalization of the classical max-flow with minimum transportation cost problem to adversarial environments.

The objective of this work is to develop an efficient and practical sensor placement method for the failure detection and localization in water networks. We formulate the problem as the minimum test cover problem (MTC) with the objective of selecting the minimum number of sensors required to uniquely identify and localize pipe failure events. First, we summarize a single-level sensing model and discuss an efficient fast greedy approach for solving the MTC problem. Simulation results on benchmark test networks demonstrate the efficacy of the fast greedy algorithm. Second, we develop a multi-level sensing model that captures additional physical features of the disturbance event, such as the time lapsed between the occurrence of disturbance and its detection by the sensor. Our sensor placement approach using MTC extends to the multi-level sensing model and an improved identification performance is obtained via reduced number of sensors (in comparison to single-level sensing model). In particular, we investigate the bi-level sensing model to illustrate the efficacy of employing multi-level sensors for the identification of failure events. Finally, we suggest extensions of our approach for the deployment of heterogeneous sensors in water networks by exploring the trade-off between cost and performance (measured in terms of the identification score of pipe/link failures).

In this paper, we study the sensor placement problem in urban water networks that maximizes the localization of pipe failures given that some sensors give incorrect outputs. False output of a sensor might be the result of degradation in sensor's hardware, software fault, or might be due to a cyber attack on the sensor. Incorrect outputs from such sensors can have any possible values which could lead to an inaccurate localization of a failure event. We formulate the optimal sensor placement problem with erroneous sensors as a set multicover problem, which is NP-hard, and then discuss a polynomial time heuristic to obtain efficient solutions. In this direction, we first examine the physical model of the disturbance propagating in the network as a result of a failure event, and outline the multi-level sensing model that captures several event features. Second, using a combinatorial approach, we solve the problem of sensor placement that maximizes the localization of pipe failures by selecting m sensors out of which at most e give incorrect outputs. We propose various localization performance metrics, and numerically evaluate our approach on a benchmark and a real water distribution network. Finally, using computational experiments, we study relationships between design parameters such as the total number of sensors, the number of sensors with errors, and extracted signal features.