IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 6, NO. 1, JANUARY 2019 75 Event-Triggered Differentially Private Average Consensus for Multi-agent Network Aijuan Wang, Xiaofeng Liao, Senior Member, IEEE, Haibo He, Fellow, IEEE Abstract—This paper investigates the differentially private problem of the average consensus for a class of discrete-time multi-agent network systems (MANSs). Based on the MANSs, a new distributed differentially private consensus algorithm (DPCA) is developed. To avoid continuous communication be- tween neighboring agents, a kind of intermittent communication strategy depending on an event-triggered function is established in our DPCA. Based on our algorithm, we carry out the detailed analysis including its convergence, its accuracy, its privacy and the trade-off between the accuracy and the privacy level, respectively. It is found that our algorithm preserves the privacy of initial states of all agents in the whole process of consensus computation. The trade-off motivates us to find the best achievable accuracy of our algorithm under the free parameters and the fixed privacy level. Finally, numerical experiment results testify the validity of our theoretical analysis. Index Terms—Average consensus, differentially private, event- triggered communication, multi-agent network systems (MANSs). I. I NTRODUCTION M ULTI-AGENT network systems (MANSs) have been one of the focal points from many researchers due to its extensively real applications in a variety of scopes, such as biological systems, robotic teams, sensor networks, unmanned air vehicles formations, just to name a few [1]−[7]. Particularly, consensus problem, as a basic building block of MANSs’ collective behaviors, has been widely investigated [8]−[14]. Moreover, a survey about theory and applications of consensus problems for MANSs is presented in [15]. The consensus of MANSs is that the states of all agents reach Manuscript received July 12, 2018; revised September 11, 2018; accepted October 7, 2018. This work was supported in part by the National Key Research and Development Program of China (2016YFB0800601). Recom- mended by Associate Editor Derong Liu. (Corresponding author: Xiaofeng Liao.) Citation: A. J. Wang, X. F. Liao, and H. B. He, “Event-triggered differ- entially private average consensus for multi-agent network,” IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, pp. 75-83, Jan. 2019. A. J. Wang is with the College of Computer Science, Chongqing University, Chongqing 400044, the School of Electronics and Information Engineering, Southwest University, Chongqing 400715, China, and also with the Depart- ment of Electrical, Computer and Biomedical Engineering, University of Rhode Island Kingston, RI 02881 USA (e-mail: aijuan321@foxmail.com). X. F. Liao is with the College of Computer Science, Chongqing Uni- versity, Chongqing 400044, and also with Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, China (e-mail: xfliao@cqu.edu.cn). H. B. He is with the Department of Electrical, Computer and Biomedical Engineering, University of Rhode Island Kingston, RI 02881 USA (e-mail: haibohe@uri.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2019.1911327 a common value by exchanging information among agents [16]−[22]. Recently, there is an increasing interest that is to apply the notion of privacy preserving into the consensus behaviors of the dynamical network systems (DNSs) [23]. It means that the network agents may not want to expose their initial states in the process of consensus computation. For example, in social networks (SNs), a group of individuals execute a prescribed procedure to obtain the common opinion on a subject [24] in which individual, however, may not want to divulgate its own true views on the subject. Subsequently, a few literatures have successfully studied the preserving-privacy problem in the context of consensus of MANSs [25]−[27]. It means that the algorithm not only achieves the agents’ consensus, but also preserves the privacy of each agent’ initial state against the other all agents. For example, privacy-preserving average consensus algorithms for MANSs are developed in [25] and [26]. The former just gives the condition that the initial state of one agent can be exactly recognized by the other agents. This motivates the latter to provide a quantitative condition that the initial state can be estimated perfectly. To extend the research of this filed, Duan et al. [27] propose a new privacy-preserving scheme and apply it to the maximum consensus algorithm for MANSs. The probability that the maximum state owner’s identity is recognized by its neighbors is calculated. Later, dif- ferentially private, as an existing preserving-privacy approach, has gained remarkable attention due to its rigorous formulation and proven security properties, including resilience to post- processing and side information, and independence from the model of the adversary. Recently, there are a large number of results concerning the differentially private consensus problem for MANSs [28]−[31]. Huang et al. [28] firstly investigate the differentially private problem in the context of consensus of MANSs. Differentially private maximum consensus algorithm which achieves ǫ-differentially private for initial states of all agents and makes all agents converge to the maximum initial state value is developed [29]. For the common average consen- sus of MANSs, Nozari et al. [30] show that any differentially private algorithm cannot achieve exact average consensus. Following it, the authors in [30] systematically analyze the differentially private average consensus problem of MANSs in expectation [31]. This observation motivates us to develop different types of differentially private consensus algorithms (DPCAs) to preserve the initial states of the MANSs as well as achieve consensus for all agents’ states. It is worth noting that, on the other hand, the privacy- preserving consensus algorithms in [28]−[31] are contin- uous communication between agents and their neighbors.