Theory of Computing
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Title : Distributed Corruption Detection in Networks
Authors : Noga Alon, Elchanan Mossel, and Robin Pemantle
Volume : 16
Number : 1
Pages : 1-23
URL : https://theoryofcomputing.org/articles/v016a001
Abstract
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We consider the problem of distributed corruption detection in
networks. In this model each node of a directed graph is either
truthful or corrupt. Each node reports the type (truthful or corrupt)
of each of its outneighbors. If it is truthful, it reports the truth,
whereas if it is corrupt, it reports adversarially. This model, first
considered by Preparata, Metze and Chien in 1967, motivated by the
desire to identify the faulty components of a digital system by having
the other components checking them, became known as the PMC model. The
main known results for this model characterize networks in which _all_
corrupt (that is, faulty) nodes can be identified, when there is a
known upper bound on their number. We are interested in networks in
which a _large fraction_ of the nodes can be classified. It is known
that in the PMC model, in order to identify all corrupt nodes when
their number is $t$, all in-degrees have to be at least $t$. In
contrast, we show that in $d$ regular-graphs with strong expansion
properties, a $1-O(1/d)$ fraction of the corrupt nodes, and a
$1-O(1/d)$ fraction of the truthful nodes can be identified, whenever
there is a majority of truthful nodes. We also observe that if the
graph is very far from being a good expander, namely, if the deletion
of a small set of nodes splits the graph into small components, then
no corruption detection is possible even if most of the nodes are
truthful. Finally we discuss the algorithmic aspects and the
computational hardness of the problem.