Proof-labeling schemes are known mechanisms providing nodes of
networks with

*certificates* that can be

*verified*
locally by distributed algorithms. Given a boolean predicate on
network states, such schemes enable to check whether the
predicate is satisfied by the actual state of the network,
by having nodes interacting with their neighbors only.
Proof-labeling schemes are typically designed for enforcing
fault-tolerance, by making sure that if the current state of the
network is illegal with respect to some given predicate, then at
least one node will detect it. Such a node can raise an alarm,
or launch a recovery procedure enabling the system to return to
a legal state. In this paper, we introduce

*error-sensitive*
proof-labeling schemes. These are proof-labeling schemes which
guarantee that the number of nodes detecting illegal states is
linearly proportional to the edit-distance between the current
state and the set of legal states. By using error-sensitive
proof-labeling schemes, states which are far from satisfying the
predicate will be detected by many nodes, enabling fast return
to legality. We provide a structural characterization of the set
of boolean predicates on network states for which there exist
error-sensitive proof-labeling schemes. This characterization
allows us to show that classical predicates such as, e.g.,
acyclicity, and leader admit error-sensitive proof-labeling
schemes, while others like regular subgraphs don't. We also
focus on

*compact* error-sensitive proof-labeling schemes.
In particular, we show that the known proof-labeling schemes for
spanning tree and minimum spanning tree, using certificates on
$O(\log n)$ bits, and on $O\left(\log^2n\right)$ bits,
respectively, are error-sensitive, as long as the trees are
locally represented by adjacency lists, and not just by parent
pointers.