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This article is a study on the use of additive obfuscation signals to keep the reference values of the agents in the continuous-time Laplacian average consensus algorithm private from eavesdroppers. Obfuscation signals are perturbations that agents add to their local dynamics and their transmitted-out messages to conceal their private reference values. An eavesdropper is an agent inside or outside the network that has access to some subset of the interagent communication messages, and its knowledge set also includes the network topology. Rather than focusing on using a zero-sum and vanishing additive signal, our work determines the necessary and sufficient conditions that define the set of admissible obfuscation signals that do not perturb the convergence point of the algorithm from the average of the reference values of the agents. Of theoretical interest, our results show that this class includes nonvanishing signals as well. Given this broader class of admissible obfuscation signals, we define a deterministic notion of privacy preservation. In this definition, privacy preservation for an agent means that neither the private reference value nor a finite set of values to which the private reference value of the agent belongs to can be obtained. Then, we evaluate the agents’ privacy against eavesdroppers with different knowledge sets. 

This paper studies distributed submodular optimization subject to partition matroid. We work in the value oracle model where the only access of the agents to the utility function is through a black box that returns the utility function value. The agents are communicating over a connected undirected graph and have access only to their own strategy set. As known in the literature, submodular maximization subject to matroid constraints is NP-hard. Hence, our objective is to propose a polynomial-time distributed algorithm to obtain a suboptimal solution with guarantees on the optimality bound. Our proposed algorithm is based on a distributed stochastic gradient ascent scheme built on the multilinear-extension of the submodular set function. We use a maximum consensus protocol to minimize the inconsistency of the shared information over the network caused by delay in the flow of information while solving for the fractional solution of the multilinear extension model. Furthermore, we propose a distributed framework of finding a set solution using the fractional solution. We show that our distributed algorithm results in a strategy set that when the team objective function is evaluated at worst case the objective function value is in 1−1/e−O(1/T) of the optimal solution in the value oracle model where T is the number of communication instances of the agents. An example demonstrates our results. 

In this paper, we study the problem of privacy preservation of the continuous-time Laplacian static average consensus algorithm using additive perturbation signals. We consider this problem over a strongly connected and weight-balanced digraph. Starting from a local reference value, in static average consensus algorithm each agent constantly communicates with its neighboring agents to update its local state to compute the average of the reference values across the network. Since every agent transmits its local reference value to its in-neighbors, the reference value of the agents are trivially disclosed. In this paper, we investigate the possibility of preserving the privacy of the reference value of the agents by adding admissible perturbation signals to the local dynamics and the transmitted out signals of the agents. Admissible additive perturbation signals are those signals that do not perturb the final convergence point of the algorithm from the average of the reference values of the agents. Our results show that if an adversarial agent has access to the output of another agent and all the input signals transmitted to that agent, the adversary can discover the private reference value of that agent, regardless of the perturbation signals. Otherwise, the privacy of the agent can be preserved. We demonstrate our results through a numerical example.