  Privacy-preserving average consensus aims to guarantee the privacy of initial
states and asymptotic consensus on the exact average of the initial value. In
existing work, it is achieved by adding and subtracting variance decaying and
zero-sum random noises to the consensus process. However, there is lack of
theoretical analysis to quantify the degree of the privacy protection. In this
paper, we introduce the maximum disclosure probability that the other nodes can
infer one node's initial state within a given small interval to quantify the
privacy. We develop a novel privacy definition, named $(\epsilon,
\delta)$-data-privacy, to depict the relationship between maximum disclosure
probability and estimation accuracy. Then, we prove that the general
privacy-preserving average consensus (GPAC) provides $(\epsilon,
\delta)$-data-privacy, and provide the closed-form expression of the
relationship between $\epsilon$ and $\delta$. Meanwhile, it is shown that the
added noise with uniform distribution is optimal in terms of achieving the
highest $(\epsilon, \delta)$-data-privacy. We also prove that when all
information used in the consensus process is available, the privacy will be
compromised. Finally, an optimal privacy-preserving average consensus (OPAC)
algorithm is proposed to achieve the highest $(\epsilon, \delta)$-data-privacy
and avoid the privacy compromission. Simulations are conducted to verify the
results.
