  It is generally believed that, in the thermodynamic limit, the microcanonical
description as a function of energy coincides with the canonical description as
a function of temperature. However, various examples of systems for which the
microcanonical and canonical ensembles are not equivalent have been identified.
A complete theory of this intriguing phenomenon is still missing. Here we show
that ensemble nonequivalence can manifest itself also in random graphs with
topological constraints. We find that, while graphs with a given number of
links are ensemble-equivalent, graphs with a given degree sequence are not.
This result holds irrespective of whether the energy is nonadditive (as in
unipartite graphs) or additive (as in bipartite graphs). In contrast with
previous expectations, our results show that: (1) physically, nonequivalence
can be induced by an extensive number of local constraints, and not necessarily
by long-range interactions or nonadditivity; (2) mathematically, nonquivalence
is determined by a different large-deviation behaviour of microcanonical and
canonical probabilities for a single microstate, and not necessarily for almost
all microstates. The latter criterion, which is entirely local, is not
restricted to networks and holds in general.
