  We study the robustness--the invariance under definition changes--of the
cluster class CL#P [HHKW05]. This class contains each #P function that is
computed by a balanced Turing machine whose accepting paths always form a
cluster with respect to some length-respecting total order with efficient
adjacency checks. The definition of CL#P is heavily influenced by the defining
paper's focus on (global) orders. In contrast, we define a cluster class,
CLU#P, to capture what seems to us a more natural model of cluster computing.
We prove that the naturalness is costless: CL#P = CLU#P. Then we exploit the
more natural, flexible features of CLU#P to prove new robustness results for
CL#P and to expand what is known about the closure properties of CL#P.
  The complexity of recognizing edges--of an ordered collection of computation
paths or of a cluster of accepting computation paths--is central to this study.
Most particularly, our proofs exploit the power of unique discovery of
edges--the ability of nondeterministic functions to, in certain settings,
discover on exactly one (in some cases, on at most one) computation path a
critical piece of information regarding edges of orderings or clusters.
