69,345 research outputs found
Universal Reliability Bounds for Sparse Networks
Consider a graph with perfect nodes and edges subject to independent random failures with identical probability.The all-terminal reliability (ATR) is the probability that the resulting subgraph is connected. First, we fully characterize uniformly least reliable graphs (ULRG) whose co-rank is not greater than four. Universal reliability bounds are here introduced for those graphs. It is formally proved that ULRG are invariant under bridge-contractions, and maximize the number of bridges among all connected simple graphs with a prescribed number of nodes and edges. A closed-form for the maximum number of bridges is also given, which has an intrinsic interest from a graphtheoretic point of view. Finally, the cost-reliability trade-off is discussed, comparing the number of edges required to reduce the reliability gaps between the least and most reliable graphs. A remarkable conclusion is that the network design is critical under rare event failures, where the reliability-gap between least and most-reliable networks is monotonically increasing with the number of terminalsAgencia Nacional de Investigación e Innovació
Conditional Reliability in Uncertain Graphs
Network reliability is a well-studied problem that requires to measure the
probability that a target node is reachable from a source node in a
probabilistic (or uncertain) graph, i.e., a graph where every edge is assigned
a probability of existence. Many approaches and problem variants have been
considered in the literature, all assuming that edge-existence probabilities
are fixed. Nevertheless, in real-world graphs, edge probabilities typically
depend on external conditions. In metabolic networks a protein can be converted
into another protein with some probability depending on the presence of certain
enzymes. In social influence networks the probability that a tweet of some user
will be re-tweeted by her followers depends on whether the tweet contains
specific hashtags. In transportation networks the probability that a network
segment will work properly or not might depend on external conditions such as
weather or time of the day. In this paper we overcome this limitation and focus
on conditional reliability, that is assessing reliability when edge-existence
probabilities depend on a set of conditions. In particular, we study the
problem of determining the k conditions that maximize the reliability between
two nodes. We deeply characterize our problem and show that, even employing
polynomial-time reliability-estimation methods, it is NP-hard, does not admit
any PTAS, and the underlying objective function is non-submodular. We then
devise a practical method that targets both accuracy and efficiency. We also
study natural generalizations of the problem with multiple source and target
nodes. An extensive empirical evaluation on several large, real-life graphs
demonstrates effectiveness and scalability of the proposed methods.Comment: 14 pages, 13 figure
Smoothed Analysis of Dynamic Networks
We generalize the technique of smoothed analysis to distributed algorithms in
dynamic network models. Whereas standard smoothed analysis studies the impact
of small random perturbations of input values on algorithm performance metrics,
dynamic graph smoothed analysis studies the impact of random perturbations of
the underlying changing network graph topologies. Similar to the original
application of smoothed analysis, our goal is to study whether known strong
lower bounds in dynamic network models are robust or fragile: do they withstand
small (random) perturbations, or do such deviations push the graphs far enough
from a precise pathological instance to enable much better performance? Fragile
lower bounds are likely not relevant for real-world deployment, while robust
lower bounds represent a true difficulty caused by dynamic behavior. We apply
this technique to three standard dynamic network problems with known strong
worst-case lower bounds: random walks, flooding, and aggregation. We prove that
these bounds provide a spectrum of robustness when subjected to
smoothing---some are extremely fragile (random walks), some are moderately
fragile / robust (flooding), and some are extremely robust (aggregation).Comment: 20 page
The impact of partially missing communities~on the reliability of centrality measures
Network data is usually not error-free, and the absence of some nodes is a
very common type of measurement error. Studies have shown that the reliability
of centrality measures is severely affected by missing nodes. This paper
investigates the reliability of centrality measures when missing nodes are
likely to belong to the same community. We study the behavior of five commonly
used centrality measures in uniform and scale-free networks in various error
scenarios. We find that centrality measures are generally more reliable when
missing nodes are likely to belong to the same community than in cases in which
nodes are missing uniformly at random. In scale-free networks, the betweenness
centrality becomes, however, less reliable when missing nodes are more likely
to belong to the same community. Moreover, centrality measures in scale-free
networks are more reliable in networks with stronger community structure. In
contrast, we do not observe this effect for uniform networks. Our observations
suggest that the impact of missing nodes on the reliability of centrality
measures might not be as severe as the literature suggests
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