  There has been a long-standing and at times fractious debate whether complex
and large systems can be stable. In ecology, the so-called `diversity-stability
debate' arose because mathematical analyses of ecosystem stability were either
specific to a particular model (leading to results that were not general), or
chosen for mathematical convenience, yielding results unlikely to be meaningful
for any interesting realistic system. May's work, and its subsequent
elaborations, relied upon results from random matrix theory, particularly the
circular law and its extensions, which only apply when the strengths of
interactions between entities in the system are assumed to be independent and
identically distributed (i.i.d.). Other studies have optimistically generalised
from the analysis of very specific systems, in a way that does not hold up to
closer scrutiny. We show here that this debate can be put to rest, once these
two contrasting views have been reconciled --- which is possible in the
statistical framework developed here. Here we use a range of illustrative
examples of dynamical systems to demonstrate that (i) stability probability
cannot be summarily deduced from any single property of the system (e.g. its
diversity), and (ii) our assessment of stability depends on adequately
capturing the details of the systems analysed. Failing to condition on the
structure of dynamical systems will skew our analysis and can, even for very
small systems, result in an unnecessarily pessimistic diagnosis of their
stability.
