  This two-part paper discusses robustification methodologies for
linear-iterative distributed algorithms for consensus and coordination problems
in multicomponent systems, in which unreliable communication links may drop
packets. We consider a setup where communication links between components can
be asymmetric (i.e., component j might be able to send information to component
i, but not necessarily vice-versa), so that the information exchange between
components in the system is in general described by a directed graph that is
assumed to be strongly connected. In the absence of communication link
failures, each component i maintains two auxiliary variables and updates each
of their values to be a linear combination of their corresponding previous
values and the corresponding previous values of neighboring components (i.e.,
components that send information to node i). By appropriately initializing
these two (decoupled) iterations, the system components can asymptotically
calculate variables of interest in a distributed fashion; in particular, the
average of the initial conditions can be calculated as a function that involves
the ratio of these two auxiliary variables. The focus of this paper to
robustify this double-iteration algorithm against communication link failures.
We achieve this by modifying the double-iteration algorithm (by introducing
some additional auxiliary variables) and prove that the modified
double-iteration converges almost surely to average consensus. In the first
part of the paper, we study the first and second moments of the two iterations,
and use them to establish convergence, and illustrate the performance of the
algorithm with several numerical examples. In the second part, in order to
establish the convergence of the algorithm, we use coefficients of ergodicity
commonly used in analyzing inhomogeneous Markov chains.
