The main theoretical contribution of this work is a geometric formalism in which to cast dis-
tributed systems. This has numerous advantages and “naturally” parametrizes a wide class of
distributed interaction mechanisms in a uniform way. We make use of this framework to present
a model for distributed optimization, and we introduce the distributed gradient as a general design
tool for synthesizing dynamics for distributed systems. The distributed optimization model is a
useful abstraction in its own right and motivates a definition for a distributed extremum. As one
might expect, the distributed gradient is zero at a distributed extremum, and the dynamics of a
distributed gradient flow must converge to a distributed extremum. This forms the basis for a wide
variety of designs, and we are in fact able to recover a widely studied distributed averaging algorithm
as a very special case.
We also make use of our geometric model to introduce the notion of coordination capacity;
intuitively, this is an upper bound on the “complexity” of coordination that is feasible given a
particular distributed interaction structure. This gives intuitive results for local, distributed, and
global control architectures, and allows formal statements to be made regarding the possibility of
“solving” certain optimization problems under a particular distributed interaction model.
Finally, we present a number of applications to illustrate the theoretical approach presented;
these range from “standard” distributed systems tasks (leader election and clock synchronization)
to more exotic tasks like graph coloring, distributed account balancing, and distributed statistical
computations.
PhD Dissertation
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