2,102 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
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, and every node is only allowed to call one neighbor in each round. In this work we introduce two fundamentally new techniques in studying the rumor spreading problem:
First, we establish a new connection between the rumor spreading process in an arbitrary graph and certain Markov chains. While most previous work analyzed the rumor spreading time in general graphs by studying the rate of the number of (un-)informed nodes after every round, we show that the mixing time of a certain Markov chain suffices to bound the rumor spreading time in an arbitrary graph.
Second, we construct a reduction from rumor spreading processes to branching programs. This reduction gives us a general framework to derandomize the rumor spreading and other gossip processes. In particular, we show that, for any n-vertex expander graph, there is a protocol which informs every node in O(log n) rounds with high probability, and uses O (log n · log log n) random bits in total. The runtime of our protocol is tight, and the randomness requirement of O (log n· log log n) random bits almost matches the lower bound of Ω(log n) random bits. We further show that, for many graph families (defined with respect to the expansion and the degree), O (poly log n) random bits in total suffice for fast rumor spreading. These results give us an almost complete understanding of the role of randomness in the rumor spreading process, which was extensively studied over the past years
Noisy Rumor Spreading and Plurality Consensus
Error-correcting codes are efficient methods for handling \emph{noisy}
communication channels in the context of technological networks. However, such
elaborate methods differ a lot from the unsophisticated way biological entities
are supposed to communicate. Yet, it has been recently shown by Feinerman,
Haeupler, and Korman {[}PODC 2014{]} that complex coordination tasks such as
\emph{rumor spreading} and \emph{majority consensus} can plausibly be achieved
in biological systems subject to noisy communication channels, where every
message transferred through a channel remains intact with small probability
, without using coding techniques. This result is a
considerable step towards a better understanding of the way biological entities
may cooperate. It has been nevertheless be established only in the case of
2-valued \emph{opinions}: rumor spreading aims at broadcasting a single-bit
opinion to all nodes, and majority consensus aims at leading all nodes to adopt
the single-bit opinion that was initially present in the system with (relative)
majority. In this paper, we extend this previous work to -valued opinions,
for any .
Our extension requires to address a series of important issues, some
conceptual, others technical. We had to entirely revisit the notion of noise,
for handling channels carrying -\emph{valued} messages. In fact, we
precisely characterize the type of noise patterns for which plurality consensus
is solvable. Also, a key result employed in the bivalued case by Feinerman et
al. is an estimate of the probability of observing the most frequent opinion
from observing the mode of a small sample. We generalize this result to the
multivalued case by providing a new analytical proof for the bivalued case that
is amenable to be extended, by induction, and that is of independent interest.Comment: Minor revisio
Quasirandom Rumor Spreading: An Experimental Analysis
We empirically analyze two versions of the well-known "randomized rumor
spreading" protocol to disseminate a piece of information in networks. In the
classical model, in each round each informed node informs a random neighbor. In
the recently proposed quasirandom variant, each node has a (cyclic) list of its
neighbors. Once informed, it starts at a random position of the list, but from
then on informs its neighbors in the order of the list. While for sparse random
graphs a better performance of the quasirandom model could be proven, all other
results show that, independent of the structure of the lists, the same
asymptotic performance guarantees hold as for the classical model. In this
work, we compare the two models experimentally. This not only shows that the
quasirandom model generally is faster, but also that the runtime is more
concentrated around the mean. This is surprising given that much fewer random
bits are used in the quasirandom process. These advantages are also observed in
a lossy communication model, where each transmission does not reach its target
with a certain probability, and in an asynchronous model, where nodes send at
random times drawn from an exponential distribution. We also show that
typically the particular structure of the lists has little influence on the
efficiency.Comment: 14 pages, appeared in ALENEX'0
Quasirandom Rumor Spreading
We propose and analyze a quasirandom analogue of the classical push model for disseminating information in networks (ârandomized rumor spreadingâ). In the classical model, in each round, each informed vertex chooses a neighbor at random and informs it, if it was not informed before. It is known that this simple protocol succeeds in spreading a rumor from one vertex to all others within
O
(log
n
) rounds on complete graphs, hypercubes, random regular graphs, ErdĆs-RĂ©nyi random graphs, and Ramanujan graphs with probability 1 â
o
(1). In the quasirandom model, we assume that each vertex has a (cyclic) list of its neighbors. Once informed, it starts at a random position on the list, but from then on informs its neighbors in the order of the list. Surprisingly, irrespective of the orders of the lists, the above-mentioned bounds still hold. In some cases, even better bounds than for the classical model can be shown.
</jats:p
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
Dynamics of Rumor Spreading in Complex Networks
We derive the mean-field equations characterizing the dynamics of a rumor
process that takes place on top of complex heterogeneous networks. These
equations are solved numerically by means of a stochastic approach. First, we
present analytical and Monte Carlo calculations for homogeneous networks and
compare the results with those obtained by the numerical method. Then, we study
the spreading process in detail for random scale-free networks. The time
profiles for several quantities are numerically computed, which allow us to
distinguish among different variants of rumor spreading algorithms. Our
conclusions are directed to possible applications in replicated database
maintenance, peer to peer communication networks and social spreading
phenomena.Comment: Final version to appear in PR
- âŠ