  The asymptotic (non)equivalence of canonical and microcanonical ensembles,
describing systems with soft and hard constraints respectively, is a central
concept in statistical physics. Traditionally, ensemble equivalence (EE) has
been associated with the vanishing of the relative canonical fluctuations of
the constraints in the thermodynamic limit. Recently, it has been reformulated
in terms of a vanishing relative entropy density between microcanonical and
canonical probabilities. The earliest observations of the breakdown of EE
required phase transitions or long-range interactions. More recent research on
binary networks found that an extensive number of local constraints can also
break EE, even in absence of phase transitions. In this work we study for the
first time ensemble nonequivalence in weighted networks with local constraints.
Unlike their binary counterpart, these networks can undergo a form of
Bose-Einstein condensation (BEC) producing a core-periphery structure where a
finite fraction of the link weights concentrates in the core. This phenomenon
creates a unique setting where local constraints coexist with a phase
transition. We find that all the relative fluctuations vanish in the non-BEC
phase while some of them do not vanish in the BEC phase, showing that
fluctuations are sensitive to the phase transition. By contrast, for all
positive temperatures the relative entropy density never vanishes, showing that
EE is broken by the extensivity of the number of constraints, irrespective of
BEC. Only at zero temperature, where the effective number of constraints
becomes finite, EE is broken by BEC in a subtle way. Therefore, in presence of
local constraints, the vanishing of relative fluctuations no longer guarantees
EE.
