  Given a network of agents, each characterized by an initial scalar value, we
study the problem of designing distributed linear algorithms such that the
agents agree in a generalized sense on a vector quantity that belongs to a
k-dimensional subspace. This problem is motivated by applications in
distributed computing and sensing, where agents seek to simultaneously evaluate
k independent functions at a common vector point by running a single
distributed algorithm. We show that linear protocols can agree only on
quantities that are oblique projections of the vector of initial conditions,
and we provide an algebraic characterization of all agreement protocols that
are consistent with a certain communication graph. By leveraging this
characterization, we propose a design procedure for constructing agreement
protocols, and we investigate what are the structural properties of
communication networks that can reach an agreement on arbitrary weights. More
broadly, our results show that agreement algorithms are capable of
simultaneously solving k consensus problems at a fraction of the communication
volume and space complexity of classical algorithms but, in general, require
higher network connectivity. The applicability of the framework is illustrated
via simulations on two problems in robotic formation and distributed
regression.
