Motivated by the increasing risk of data leaks in distributed networks, we consider the privacy-preserving problem in a consensus network in the presence of an eavesdropper who is able to intercept the data transmitted on the network. First, we introduce a consensus protocol with privacy-preserving function, and analyze its convergence and its privacy-preserving effect. Second, we propose a criterion to measure the degree of network privacy leaks in the existence of the eavesdropper. Particularly, we consider the networks with ring topology and small-world topology, where we find a suboptimal eavesdropping strategy that maximizes the probability of privacy leaks. Finally, we verify all the derived results by numerical examples.Researches using cryptography are often involved with quantification and cypher keys [19][20][21][22]. Kishida in [19] proposed a real-time signal encryption method by combining a quantizer with the Paillier cryptosystem, which ensures the privacy of the recursion of the consensus protocol with a length-reduced key. The reduce of the length of the key is due to the changing sensitivity of the quantizer which maps real-valued states into integers. Guo et al. in [20] encoded the information with authority traceability using elliptic curve based chameleon hashing. Yang et al. in [21] introduced homomorphic encryption into a distributed projected gradient-based algorithm to reach privacy preserving distributed optimization. Different from the cryptography methods, in the studies which used intensionally generated noises, properly conceived noises were added into the transmitted data to confuse the eavesdropper from directly reading the data [23][24][25][26]. Huang et al. proposed a privacy-preserving protocol by adding independent and exponentially decaying Laplacian noise in the process of consensus update [23]. Manitara et al. in [24] also proposed a noise-adding protocal to achieve privacy preserving average consensus. However, different from the independent true random Laplacian noise in [23], the noise added in [24] is a pseudo-random sequence with finite steps, within which the sum the the sequence is fixed to zero to ensure unbiased average consensus. Similar works has been done by Mo et al. [25] and He et al. [26]. In [25], the added noise is conceived as a linear combination of a standard normal distributed random sequence with time-varying coefficients, which also ensures exact average consensus. In [26], the distribution of the added noise is not fixed, but a criterion is proposed to measure the privacy-preserving performance. Under the criterion, the optimal noise distribution is obtained. In [27,28], distributed privacy preserving consensus is used in cooperative driving and vehicular networks, which improves driving security, enhances infrastructure utilization, reduces fuel consumption and at the same time protect the vehicle privacy in platooning applications.In most of the above mentioned privacy preserving protocols, a fact is that, if the neighboring set of the nth node c...