4,095 research outputs found
Gossip vs. Markov Chains, and Randomness-Efficient Rumor Spreading
We study gossip algorithms for the rumor spreading problem which asks one
node to deliver a rumor to all nodes in an unknown network. We present the
first protocol for any expander graph with nodes such that, the
protocol informs every node in rounds with high probability, and
uses random bits in total. The runtime of our protocol is
tight, and the randomness requirement of random bits almost
matches the lower bound of random bits for dense graphs. We
further show that, for many graph families, polylogarithmic number of random
bits in total suffice to spread the rumor in rounds.
These results together give us an almost complete understanding of the
randomness requirement of this fundamental gossip process.
Our analysis relies on unexpectedly tight connections among gossip processes,
Markov chains, and branching programs. First, we establish a connection between
rumor spreading processes and Markov chains, which is used to approximate the
rumor spreading time by the mixing time of Markov chains. Second, we show a
reduction from rumor spreading processes to branching programs, and this
reduction provides a general framework to derandomize gossip processes. In
addition to designing rumor spreading protocols, these novel techniques may
have applications in studying parallel and multiple random walks, and
randomness complexity of distributed algorithms.Comment: 41 pages, 1 figure. arXiv admin note: substantial text overlap with
arXiv:1304.135
Push is Fast on Sparse Random Graphs
We consider the classical push broadcast process on a large class of sparse
random multigraphs that includes random power law graphs and multigraphs. Our
analysis shows that for every , whp rounds are
sufficient to inform all but an -fraction of the vertices.
It is not hard to see that, e.g. for random power law graphs, the push
process needs whp rounds to inform all vertices. Fountoulakis,
Panagiotou and Sauerwald proved that for random graphs that have power law
degree sequences with , the push-pull protocol needs
to inform all but vertices whp. Our result demonstrates that,
for such random graphs, the pull mechanism does not (asymptotically) improve
the running time. This is surprising as it is known that, on random power law
graphs with , push-pull is exponentially faster than pull
Finding Rumor Sources on Random Trees
We consider the problem of detecting the source of a rumor which has spread
in a network using only observations about which set of nodes are infected with
the rumor and with no information as to \emph{when} these nodes became
infected. In a recent work \citep{ref:rc} this rumor source detection problem
was introduced and studied. The authors proposed the graph score function {\em
rumor centrality} as an estimator for detecting the source. They establish it
to be the maximum likelihood estimator with respect to the popular Susceptible
Infected (SI) model with exponential spreading times for regular trees. They
showed that as the size of the infected graph increases, for a path graph
(2-regular tree), the probability of source detection goes to while for
-regular trees with the probability of detection, say ,
remains bounded away from and is less than . However, their results
stop short of providing insights for the performance of the rumor centrality
estimator in more general settings such as irregular trees or the SI model with
non-exponential spreading times.
This paper overcomes this limitation and establishes the effectiveness of
rumor centrality for source detection for generic random trees and the SI model
with a generic spreading time distribution. The key result is an interesting
connection between a continuous time branching process and the effectiveness of
rumor centrality. Through this, it is possible to quantify the detection
probability precisely. As a consequence, we recover all previous results as a
special case and obtain a variety of novel results including the {\em
universality} of rumor centrality in the context of tree-like graphs and the SI
model with a generic spreading time distribution.Comment: 38 pages, 6 figure
Strong Robustness of Randomized Rumor Spreading Protocols
Randomized rumor spreading is a classical protocol to disseminate information
across a network. At SODA 2008, a quasirandom version of this protocol was
proposed and competitive bounds for its run-time were proven. This prompts the
question: to what extent does the quasirandom protocol inherit the second
principal advantage of randomized rumor spreading, namely robustness against
transmission failures?
In this paper, we present a result precise up to factors. We
limit ourselves to the network in which every two vertices are connected by a
direct link. Run-times accurate to their leading constants are unknown for all
other non-trivial networks.
We show that if each transmission reaches its destination with a probability
of , after (1+\e)(\frac{1}{\log_2(1+p)}\log_2n+\frac{1}{p}\ln n)
rounds the quasirandom protocol has informed all nodes in the network with
probability at least 1-n^{-p\e/40}. Note that this is faster than the
intuitively natural factor increase over the run-time of approximately
for the non-corrupted case.
We also provide a corresponding lower bound for the classical model. This
demonstrates that the quasirandom model is at least as robust as the fully
random model despite the greatly reduced degree of independent randomness.Comment: Accepted for publication in "Discrete Applied Mathematics". A short
version appeared in the proceedings of the 20th International Symposium on
Algorithms and Computation (ISAAC 2009). Minor typos fixed in the second
version. Proofs of Lemma 11 and Theorem 12 fixed in the third version. Proof
of Lemma 8 fixed in the fourth versio
A process of rumor scotching on finite populations
Rumor spreading is a ubiquitous phenomenon in social and technological
networks. Traditional models consider that the rumor is propagated by pairwise
interactions between spreaders and ignorants. Spreaders can become stiflers
only after contacting spreaders or stiflers. Here we propose a model that
considers the traditional assumptions, but stiflers are active and try to
scotch the rumor to the spreaders. An analytical treatment based on the theory
of convergence of density dependent Markov chains is developed to analyze how
the final proportion of ignorants behaves asymptotically in a finite
homogeneously mixing population. We perform Monte Carlo simulations in random
graphs and scale-free networks and verify that the results obtained for
homogeneously mixing populations can be approximated for random graphs, but are
not suitable for scale-free networks. Furthermore, regarding the process on a
heterogeneous mixing population, we obtain a set of differential equations that
describes the time evolution of the probability that an individual is in each
state. Our model can be applied to study systems in which informed agents try
to stop the rumor propagation. In addition, our results can be considered to
develop optimal information dissemination strategies and approaches to control
rumor propagation.Comment: 13 pages, 11 figure
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