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Privacy-preserving average consensus aims to guarantee the privacy of initialstates and asymptotic consensus on the exact average of the initial value. Inexisting work, it is achieved by adding and subtracting variance decaying andzero-sum random noises to the consensus process. However, there is lack oftheoretical analysis to quantify the degree of the privacy protection. In thispaper, we introduce the maximum disclosure probability that the other nodes caninfer one node's initial state within a given small interval to quantify theprivacy. We develop a novel privacy definition, named $(\epsilon,\delta)$-data-privacy, to depict the relationship between maximum disclosureprobability and estimation accuracy. Then, we prove that the generalprivacy-preserving average consensus (GPAC) provides $(\epsilon,\delta)$-data-privacy, and provide the closed-form expression of therelationship between $\epsilon$ and $\delta$. Meanwhile, it is shown that theadded noise with uniform distribution is optimal in terms of achieving thehighest $(\epsilon, \delta)$-data-privacy. We also prove that when allinformation used in the consensus process is available, the privacy will becompromised. Finally, an optimal privacy-preserving average consensus (OPAC)algorithm is proposed to achieve the highest $(\epsilon, \delta)$-data-privacyand avoid the privacy compromission. Simulations are conducted to verify theresults.

Privacy-preserving Average Consensus: Privacy Analysis and Optimal Algorithm Design