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This paper develops a framework for privatizing the spectrum of the Laplacian of an undirected graph using differential privacy. We consider two privacy formulations. The first obfuscates the presence of edges in the graph and the second obfuscates the presence of nodes. We compare these two privacy formulations and show that the privacy formulation that considers edges is better suited to most engineering applications. We use the bounded Laplace mechanism to provide (epsilon, delta)-differential privacy to the eigenvalues of a graph Laplacian, and we pay special attention to the algebraic connectivity, which is the Laplacian's the second smallest eigenvalue. Analytical bounds are presented on the accuracy of the mechanisms and on certain graph properties computed with private spectra. A suite of numerical examples confirms the accuracy of private spectra in practice. 

In cooperative multi-agent reinforcement learning (Co-MARL), a team of agents must jointly optimize the team's longterm rewards to learn a designated task. Optimizing rewards as a team often requires inter-agent communication and data sharing, leading to potential privacy implications. We assume privacy considerations prohibit the agents from sharing their environment interaction data. Accordingly, we propose Privacy-Engineered Value Decomposition Networks (PE-VDN), a Co-MARL algorithm that models multi-agent coordination while provably safeguarding the confidentiality of the agents' environment interaction data. We integrate three privacy-engineering techniques to redesign the data flows of the VDN algorithm-an existing Co-MARL algorithm that consolidates the agents' environment interaction data to train a central controller that models multi-agent coordination-and develop PE-VDN. In the first technique, we design a distributed computation scheme that eliminates Vanilla VDN's dependency on sharing environment interaction data. Then, we utilize a privacy-preserving multi-party computation protocol to guar-antee that the data flows of the distributed computation scheme do not pose new privacy risks. Finally, we enforce differential privacy to preempt inference threats against the agents' training data-past environment interactions-when they take actions based on their neural network predictions. We implement PE-VDN in StarCraft Multi-Agent Competition (SMAC) and show that it achieves 80% of Vanilla VDN's win rate while maintaining differential privacy levels that provide meaningful privacy guarantees. The results demonstrate that PE-VDN can safeguard the confidentiality of agents' environment interaction data without sacrificing multi-agent coordination. 

Stochastic matrices are commonly used to analyze Markov chains, but revealing them can leak sensitive information. Therefore, in this paper we introduce a technique to privatize stochastic matrices in a way that (i) conceals the probabilities they contain, and (ii) still allows for accurate analyses of Markov chains. Specifically, we use differential privacy, which is a statistical framework for protecting sensitive data. To implement it, we introduce the Matrix Dirichlet Mechanism, which is a probabilistic mapping that perturbs a stochastic matrix to provide privacy. We prove that this mechanism provides differential privacy, and we quantify the error induced in private stochastic matrices as a function of the strength of privacy being provided. We then bound the distance between the stationary distribution of the underlying, sensitive stochastic matrix and the stationary distribution of its privatized form. Numerical results show that, under typical conditions, privacy introduces error as low as 5.05% in the stationary distribution of a stochastic matrix. 

Privacy-aware multiagent systems must protect agents’ sensitive data while simultaneously ensuring that agents accomplish their shared objectives. Towards this goal, we propose a framework to privatize inter-agent communications in cooperative multiagent decision-making problems. We study sequential decision-making problems formulated as cooperative Markov games with reach-avoid objectives. We apply a differential privacy mechanism to privatize agents’ communicated symbolic state trajectories, and analyze tradeoffs between the strength of privacy and the team’s performance. For a given level of privacy, this tradeoff is shown to depend critically upon the total correlation among agents’ state-action processes. We synthesize policies that are robust to privacy by reducing the value of the total correlation. Numerical experiments demonstrate that the team’s performance under these policies decreases by only 6 percent when comparing private versus non-private implementations of communication. By contrast, the team’s performance decreases by 88 percent when using baseline policies that ignore total correlation and only optimize team performance. 

Graphs are the dominant formalism for modeling multi-agent systems. The algebraic connectivity of a graph is particularly important because it provides the convergence rates of consensus algorithms that underlie many multi-agent control and optimization techniques. However, sharing the value of algebraic connectivity can inadvertently reveal sensitive information about the topology of a graph, such as connections in social networks. Therefore, in this work we present a method to release a graph’s algebraic connectivity under a graph-theoretic form of differential privacy, called edge differential privacy. Edge differential privacy obfuscates differences among graphs’ edge sets and thus conceals the absence or presence of sensitive connections therein. We provide privacy with bounded Laplace noise, which improves accuracy relative to conventional unbounded noise. The private algebraic connectivity values are analytically shown to provide accurate estimates of consensus convergence rates, as well as accurate bounds on the diameter of a graph and the mean distance between its nodes. Simulation results confirm the utility of private algebraic connectivity in these contexts. 

As multi-agent systems proliferate, there is in-creasing demand for coordination protocols that protect agents’ sensitive information while allowing them to collaborate. To help address this need, this paper presents a differentially private formation control framework. Agents’ state trajectories are protected using differential privacy, which is a statistical notion of privacy that protects data by adding noise to it. We provide a private formation control implementation and analyze the impact of privacy upon the system. Specifically, we quantify tradeoffs between privacy level, system performance, and connectedness of the network’s communication topology. These tradeoffs are used to develop guidelines for calibrating privacy in terms of control theoretic quantities, such as steady-state error, without requiring in-depth knowledge of differential privacy. Additional guidelines are also developed for treating privacy levels and network topologies as design parameters to tune the network’s performance. Simulation results illustrate these tradeoffs and show that strict privacy is inherently compatible with strong system performance. 

In decision-making problems, the actions of an agent may reveal sensitive information that drives its decisions. For instance, a corporation’s investment decisions may reveal its sensitive knowledge about market dynamics. To prevent this type of information leakage, we introduce a policy synthesis algorithm that protects the privacy of the transition probabilities in a Markov decision process. We use differential privacy as the mathematical definition of privacy. The algorithm first perturbs the transition probabilities using a mechanism that provides differential privacy. Then, based on the privatized transition probabilities, we synthesize a policy using dynamic programming. Our main contribution is to bound the "cost of privacy," i.e., the difference between the expected total rewards with privacy and the expected total rewards without privacy. We also show that computing the cost of privacy has time complexity that is polynomial in the parameters of the problem. Moreover, we establish that the cost of privacy increases with the strength of differential privacy protections, and we quantify this increase. Finally, numerical experiments on two example environments validate the established relationship between the cost of privacy and the strength of data privacy protections.