This paper studies the feasibility of reaching consensus in an anonymous dynamic network. In our model, n anonymous nodes proceed in synchronous rounds. We adopt a hybrid fault model in which up to f nodes may suffer crash or Byzantine faults, and the dynamic message adversary chooses a communication graph for each round.
We introduce a stability property of the dynamic network – (T,D)-dynaDegree for T ≥ 1 and n−1 ≥ D ≥ 1 – which requires that for every T consecutive rounds, any fault-free node must have incoming directed links from at least D distinct neighbors. These links might occur in different rounds during a T -round interval. (1, n−1)-dynaDegree means that the graph is a complete graph in every round. (1, 1)-dynaDegree means that each node has at least one incoming neighbor in every round, but the set of incoming neighbor(s) at each node may change arbitrarily between rounds.
We show that exact consensus is impossible even with (1, n − 2)-dynaDegree. For an arbitrary T , we show that for crash-tolerant approximate consensus, (T , ⌊n/2⌋)-dynaDegree and n > 2f are together necessary and sufficient, whereas for Byzantine approximate consensus, (T , ⌊(n + 3f )/2⌋)- dynaDegree and n > 5f are together necessary and sufficient.
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Distributed Corruption Detection in Networks
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.
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- NSF-PAR ID:
- 10144273
- Date Published:
- Journal Name:
- Theory of Computing
- Volume:
- 16
- Issue:
- 1
- ISSN:
- 1557-2862
- Page Range / eLocation ID:
- 1 to 23
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4086/toc.2020.v016a001
