Privacy-preserving bipartite consensus with cooperative–competitive interactions via a node decomposition strategy

Abstract

This paper describes our investigation of the privacy protection problem of multi-agent systems under cooperative–competitive networks. A node decomposition strategy is used to protect the privacy of the initial node values, in which a node vi is split into ni nodes. By designing inter-node weights, the initial value of each node is protected from honest-but-curious nodes and eavesdroppers without relying on external algorithms. The purpose is to design a privacy-preserving consensus algorithm such that the privacy performance is guaranteed by using the node decomposition strategy, while the bipartite consensus is achieved for the cooperative–competitive multi-agent systems. Two numerical simulations are given to validate the effectiveness of the proposed privacy-preserving bipartite consensus algorithm.

1. Introduction

With the development of control theory and its continuous integration with other disciplines, multi-agent systems based on the natural laws of group behaviors have proven to be a powerful tool to complete complex tasks (Mi et al., 2023; Li CY et al., 2024). Accordingly, the coordination control problem of multi-agent systems has attracted widespread attention in various fields (Yang et al., 2008; Wang LC et al., 2022; Huang, 2024; Zheng et al., 2024) due primarily to its low network resource consumption, fast execution speed, strong fault tolerance, and high reliability for tasks with a poor structural performance (Chanfreut et al., 2022; Wang LC et al., 2023, 2024, 2025; Fang et al., 2024). As the basis for cooperation and coordination in multi-agent systems, the consensus problem has been increasingly studied and achievements have been obtained (Talebi et al., 2006; Jiang et al., 2024).

In recent years, the consensus problem of multi-agent systems has been extended to various situations, such as second-order multi-agent systems (Wu XH and Mu, 2022), delayed multi-agent systems (Olfati-Saber and Murray, 2004), and cooperative–competitive networks (Dou and Song, 2023). In the study of multi-agent systems, a purely collaborative relationship between agents is often assumed, thus neglecting adversarial situations with negative connection weights (Zhai and Zheng, 2019). However, competitive relationships are indeed very common in multi-agent systems. For example, in robot soccer competitions, robots from different teams need to compete to kick the ball into the opponent’s goal. In distributed systems, different nodes may compete for limited computational resources or storage space. Altafini (2013) proved that the bipartite consensus problem in a structurally balanced signed network is equivalent to the standard consensus problem in a non-negative network under gauge transformation. To this end, some properties of standard consensus in non-negative networks can also apply to networks with cooperative–competitive relationships.

In networked systems (Yaghoubi et al., 2023; Liang et al., 2024), communication security challenges have become increasingly prominent (Sun LC et al., 2023; Sakthivel et al., 2024). Transmission channels are vulnerable to security threats such as eavesdropping and tampering. Against this backdrop, research on privacy-enhancing technologies has emerged as a critical area of focus. Up to now, homomorphic encryption and differential privacy have been the two most widely used methods to protect sensitive information (Kefayati et al., 2007; Li QX et al., 2019, 2020; Chen XM et al., 2023; Cheng et al., 2024; Wang W et al., 2024). Homomorphic encryption is a special encryption method that allows certain types of calculations to be performed directly on encrypted data without decryption (Gao H et al., 2018; Chen W et al., 2023; Gao PX et al., 2024). After computation, the result remains encrypted, and only those with the decryption key can view the computation result. This encryption method is very useful for protecting data privacy and achieving secure computation. Another widely used method is differential privacy. Differential privacy is a technique used to protect individual privacy, especially during data analysis and the release of statistical information. Because differential privacy is based on probability, all differential privacy methods must incorporate randomness (Mo and Murray, 2017; Wu L et al., 2023). Differential privacy mechanisms effectively ensure the privacy of the initial state values, but a major drawback is that they can only achieve average consensus in the probabilistic sense.