29,340 research outputs found
On the Escape Probability Estimation in Large Graphs
We consider the large graphs as the object of study and deal with the problem of escape probability estimation. Generally, the required characteristic cannot be calculated analytically and even numerically due to the complexity and large size of the investigation object. The purpose of this paper is to offer the effective method for estimating the probability that the random walk on graph фЂ‚їrst enters a node b before returning into starting node a. Regenerative properties of the random walk allow using an accelerated method for the cycles simulation based on the splitting technique. The results of numerical experiments confirm the advantages of the proposed method
The Cost of Uncertainty in Curing Epidemics
Motivated by the study of controlling (curing) epidemics, we consider the
spread of an SI process on a known graph, where we have a limited budget to use
to transition infected nodes back to the susceptible state (i.e., to cure
nodes). Recent work has demonstrated that under perfect and instantaneous
information (which nodes are/are not infected), the budget required for curing
a graph precisely depends on a combinatorial property called the CutWidth. We
show that this assumption is in fact necessary: even a minor degradation of
perfect information, e.g., a diagnostic test that is 99% accurate, drastically
alters the landscape. Infections that could previously be cured in sublinear
time now may require exponential time, or orderwise larger budget to cure. The
crux of the issue comes down to a tension not present in the full information
case: if a node is suspected (but not certain) to be infected, do we risk
wasting our budget to try to cure an uninfected node, or increase our certainty
by longer observation, at the risk that the infection spreads further? Our
results present fundamental, algorithm-independent bounds that tradeoff budget
required vs. uncertainty.Comment: 35 pages, 3 figure
Temporal-varying failures of nodes in networks
We consider networks in which random walkers are removed because of the
failure of specific nodes. We interpret the rate of loss as a measure of the
importance of nodes, a notion we denote as failure-centrality. We show that the
degree of the node is not sufficient to determine this measure and that, in a
first approximation, the shortest loops through the node have to be taken into
account. We propose approximations of the failure-centrality which are valid
for temporal-varying failures and we dwell on the possibility of externally
changing the relative importance of nodes in a given network, by exploiting the
interference between the loops of a node and the cycles of the temporal pattern
of failures. In the limit of long failure cycles we show analytically that the
escape in a node is larger than the one estimated from a stochastic failure
with the same failure probability. We test our general formalism in two
real-world networks (air-transportation and e-mail users) and show how
communities lead to deviations from predictions for failures in hubs.Comment: 7 pages, 3 figure
Estimating and interpreting secondary attack risk: Binomial considered harmful
The household secondary attack risk (SAR), often called the secondary attack
rate or secondary infection risk, is the probability of infectious contact from
an infectious household member A to a given household member B, where we define
infectious contact to be a contact sufficient to infect B if he or she is
susceptible. Estimation of the SAR is an important part of understanding and
controlling the transmission of infectious diseases. In practice, it is most
often estimated using binomial models such as logistic regression, which
implicitly attribute all secondary infections in a household to the primary
case. In the simplest case, the number of secondary infections in a household
with m susceptibles and a single primary case is modeled as a binomial(m, p)
random variable where p is the SAR. Although it has long been understood that
transmission within households is not binomial, it is thought that multiple
generations of transmission can be safely neglected when p is small. We use
probability generating functions and simulations to show that this is a
mistake. The proportion of susceptible household members infected can be
substantially larger than the SAR even when p is small. As a result, binomial
estimates of the SAR are biased upward and their confidence intervals have poor
coverage probabilities even if adjusted for clustering. Accurate point and
interval estimates of the SAR can be obtained using longitudinal chain binomial
models or pairwise survival analysis, which account for multiple generations of
transmission within households, the ongoing risk of infection from outside the
household, and incomplete follow-up. We illustrate the practical implications
of these results in an analysis of household surveillance data collected by the
Los Angeles County Department of Public Health during the 2009 influenza A
(H1N1) pandemic.Comment: 25 pages, 8 figure
FLEET: Butterfly Estimation from a Bipartite Graph Stream
We consider space-efficient single-pass estimation of the number of
butterflies, a fundamental bipartite graph motif, from a massive bipartite
graph stream where each edge represents a connection between entities in two
different partitions. We present a space lower bound for any streaming
algorithm that can estimate the number of butterflies accurately, as well as
FLEET, a suite of algorithms for accurately estimating the number of
butterflies in the graph stream. Estimates returned by the algorithms come with
provable guarantees on the approximation error, and experiments show good
tradeoffs between the space used and the accuracy of approximation. We also
present space-efficient algorithms for estimating the number of butterflies
within a sliding window of the most recent elements in the stream. While there
is a significant body of work on counting subgraphs such as triangles in a
unipartite graph stream, our work seems to be one of the few to tackle the case
of bipartite graph streams.Comment: This is the author's version of the work. It is posted here by
permission of ACM for your personal use. Not for redistribution. The
definitive version was published in Seyed-Vahid Sanei-Mehri, Yu Zhang, Ahmet
Erdem Sariyuce and Srikanta Tirthapura. "FLEET: Butterfly Estimation from a
Bipartite Graph Stream". The 28th ACM International Conference on Information
and Knowledge Managemen
Monte Carlo algorithms are very effective in finding the largest independent set in sparse random graphs
The effectiveness of stochastic algorithms based on Monte Carlo dynamics in
solving hard optimization problems is mostly unknown. Beyond the basic
statement that at a dynamical phase transition the ergodicity breaks and a
Monte Carlo dynamics cannot sample correctly the probability distribution in
times linear in the system size, there are almost no predictions nor intuitions
on the behavior of this class of stochastic dynamics. The situation is
particularly intricate because, when using a Monte Carlo based algorithm as an
optimization algorithm, one is usually interested in the out of equilibrium
behavior which is very hard to analyse. Here we focus on the use of Parallel
Tempering in the search for the largest independent set in a sparse random
graph, showing that it can find solutions well beyond the dynamical threshold.
Comparison with state-of-the-art message passing algorithms reveals that
parallel tempering is definitely the algorithm performing best, although a
theory explaining its behavior is still lacking.Comment: 14 pages, 12 figure
Universality of trap models in the ergodic time scale
Consider a sequence of possibly random graphs , ,
whose vertices's have i.i.d. weights with a distribution
belonging to the basin of attraction of an -stable law, .
Let , , be a continuous time simple random walk on which
waits a \emph{mean} exponential time at each vertex . Under
considerably general hypotheses, we prove that in the ergodic time scale this
trap model converges in an appropriate topology to a -process. We apply this
result to a class of graphs which includes the hypercube, the -dimensional
torus, , random -regular graphs and the largest component of
super-critical Erd\"os-R\'enyi random graphs
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