Privacy-preserving weighted average consensus and optimal attacking strategy for multi-agent networks

Abstract

This paper proposes a privacy-preserving consensus algorithm which enables all the agents in the directed network to eventually reach the weighted average of initial states, and while preserving the privacy of the initial state of each agent. A novel privacy-preserving scheme is proposed in our consensus algorithm where initial states are hidden in random values. We also develop detailed analysis based on our algorithm, including its convergence property and the topology condition of privacy leakages for each agent. It can be observed that final consensus point is independent of their initial values that can be arbitrary random values. Besides, when an eavesdropper exists and can intercept the data transmitted on the edges, we introduce an index to measure the privacy leakage degree of agents, and then analyze the degree of privacy leakage for each agent. Similarly, the degree for network privacy leakage is derived. Subsequently, we establish an optimization problem to find the optimal attacking strategy, and present a heuristic optimization algorithm based on the Sequential Least Squares Programming (SLSQP) to solve the proposed optimization problem. Finally, numerical experiments are designed to demonstrate the effectiveness of our algorithm.

Introduction

In the past few decades, considerable interests have been focused on the average consensus studies of multi-agent networks [1], [2], [3], [4], [5] due to its extensive real applications, such as dynamic load balancing [6], coordination of groups of mobile autonomous agents [7] and cooperative control of vehicle formations [8]. A survey of theory and applications of consensus problems in networked systems is presented in [9]. One of the fundamental problems in average consensus studies is how to design a consensus algorithm which can enable the states of all agents reach the average initial value of all agents. In traditional average consensus algorithms, if one agent knows the update rules of all other agents, then under some observability conditions, it can infer the initial states of all other agents [22] such that the privacy of initial states is leaked. From the privacy-preserving perspective, the participating agents may not want to expose their initial values in the process of achieving the average consensus. For example, in social networks, a group of individuals can execute a prescribed procedure to obtain the common opinion on a subject [10]. However, they may not want to reveal their exact personal opinion on the subject. Another example is the multi-agent rendezvous problem [11], where all agents are committed to eventually rendezvousing at a certain location, but they want to keep their initial location secret to the others. Hence, preserving the privacy of initial states becomes an important research topic.

This paper develops a privacy-reserving average consensus algorithm for the directed connected network in the presence of an eavesdropper. Specially, we propose a novel privacy-preserving scheme to hide the initial states in random noises. Subsequently, we introduce an index to measure the degree of agent privacy leakage under attacks. An optimization problem to find the optimal attacking strategy is established, and a heuristic optimization algorithm based on the Sequential Least Squares Programming (SLSQP) for solving the optimization problem is presented. The major contributions of this paper are summarized as the following four aspects:

• A privacy-reserving average consensus algorithm for the directed connected network is developed, which not only preserves the privacy of the initial states, but also achieves the weighted average consensus. Unlike the existing privacy-preserving method, the initial states of agents are hidden in random values, that is, the initial state is equal to the specific random values generated. In this case, the consensus point is the linear combination of random values, and depends on the random values.

• The theoretical analysis results are developed, including consensus analysis and topology condition analysis of privacy leakages for each agent. These results are also verified by numerical examples.

• We propose a measure index for the degree of privacy leakage, based on which, the degree of privacy leakage under attacks is derived for agents and network, respectively.

• A heuristic optimization algorithm based on the SLSQP to search for the optimal attacking strategy is presented.

The remainder of this paper is structured as follows. In Section 2, we provide the related work. In Section 3, we introduce the notations, and present the graph theory for communication network. Also, we give a novel privacy-preserving scheme. The main results are established in Section 4, including four parts. First, the consensus analysis based on the proposed algorithm is presented. Second, the privacy preserving condition analysis is provided. Third, the degree of agent privacy leakage under attacks is derived based on the proposed measure index. Fourth, a heuristic optimization algorithm based on the SLSQP to find the optimal attacking strategy is established. Section 5 provides some numerical examples to testify the effectiveness of our results. Section 6 contains some main conclusions and further research.