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Ensuring privacy of sensitive data is essential in many contexts, such as healthcare data, banks, e-commerce, wireless sensor networks, and social networks. It is common that different entities coordinate or want to rely on a third party to solve a specific problem. At the same time, no entity wants to publish its problem data during the solution procedure unless there is a privacy guarantee. Unlike cryptography and differential privacy based approaches, the methods based on optimization lack a quantification of the privacy they can provide. The main contribution of this paper is to provide a mechanism to quantify the privacy of a broad class of optimization approaches. In particular, we formally define a one-to-many relation, which relates a given adversarial observed message to an uncertainty set of the problem data. This relation quantifies the potential ambiguity on problem data due to the employed optimization approaches. The privacy definitions are then formalized based on the uncertainty sets. The properties of the proposed privacy measure is analyzed. The key ideas are illustrated with examples, including localization, average consensus, among others.

An epidemic spreading in a network calls for a decision on the part of the network members: They should decide whether to protect themselves or not. Their decision depends on the trade-off between their perceived risk of being infected and the cost of being protected. The network members can make decisions repeatedly, based on information that they receive about the changing infection level in the network. We study the equilibrium states reached by a network whose members increase (resp. decrease) their security deployment when learning that the network infection is widespread (resp. limited). Our main finding is that the equilibrium level of infection increases as the learning rate of the members increases. We confirm this result in three scenarios for the behavior of the members: strictly rational cost minimizers, not strictly rational, and strictly rational but split into two response classes. In the first two cases, we completely characterize the stability and the domains of attraction of the equilibrium points, even though the first case leads to a differential inclusion. We validate our conclusions with simulations on human mobility traces., Comment: 19 pages, 5 figures, submitted to the IEEE Transactions on Automatic Control

When a sensor has continuous measurements but sends limited messages over a data network to a supervisor which estimates the state, the available packet rate fixes the achievable quality of state estimation. When such rate limits turn stringent, the sensor's messaging policy should be designed anew. What are the good causal messaging policies ? What should message packets contain ? What is the lowest possible distortion in a causal estimate at the supervisor ? Is Delta sampling better than periodic sampling ? We answer these questions under an idealized model of the network and the assumption of perfect measurements at the sensor. For a scalar, linear diffusion process, we study the problem of choosing the causal sampling times that will give the lowest aggregate squared error distortion. We stick to finite-horizons and impose a hard upper bound on the number of allowed samples. We cast the design as a problem of choosing an optimal sequence of stopping times. We reduce this to a nested sequence of problems each asking for a single optimal stopping time. Under an unproven but natural assumption about the least-square estimate at the supervisor, each of these single stopping problems are of standard form. The optimal stopping times are random times when the estimation error exceeds designed envelopes. For the case where the state is a Brownian motion, we give analytically: the shape of the optimal sampling envelopes, the shape of the envelopes under optimal Delta sampling, and their performances. Surprisingly, we find that Delta sampling performs badly. Hence, when the rate constraint is a hard limit on the number of samples over a finite horizon, we should should not use Delta sampling., Comment: Submitted to the SIAM journal on control and optimization. 32 pages, 7 figures