  This paper considers the problem of privacy preservation against passive
internal and external malicious agents in the continuous-time Laplacian average
consensus algorithm over strongly connected and weight-balanced digraphs. For
this problem, we evaluate the effectiveness of use of additive perturbation
signals as a privacy preservation measure against malicious agents that know
the graph topology. Our results include (a) identifying the necessary and
sufficient conditions on admissible additive perturbation signals that do not
perturb the convergence point of the algorithm from the average of initial
values of the agents; (b) obtaining the necessary and sufficient condition on
the knowledge set of a malicious agent that enables it to identify the initial
value of another agent; (c) designing observers that internal and external
malicious agents can use to identify the initial conditions of another agent
when their knowledge set on that agent enables them to do so. We demonstrate
our results through a numerical example.
