  Graph theoretical analysis of the community structure of networks attempts to
identify the communities (or modules) to which each node affiliates. However,
this is in most cases an ill-posed problem, as the affiliation of a node to a
single community is often ambiguous. Previous solutions have attempted to
identify all of the communities to which each node affiliates. Instead of
taking this approach, we introduce versatility, $V$, as a novel metric of nodal
affiliation: $V \sim 0$ means that a node is consistently assigned to a
specific community; $V \gg 0$ means it is inconsistently assigned to different
communities. Versatility works in conjunction with existing community detection
algorithms, and it satisfies many theoretically desirable properties in
idealised networks designed to maximise ambiguity of modular decomposition. The
local minima of global mean versatility identified the resolution parameters of
a hierarchical community detection algorithm that least ambiguously decomposed
the community structure of a social (karate club) network and the mouse brain
connectome. Our results suggest that nodal versatility is useful in quantifying
the inherent ambiguity of modular decomposition.
