1,071 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
On the Role of Mobility for Multi-message Gossip
We consider information dissemination in a large -user wireless network in
which users wish to share a unique message with all other users. Each of
the users only has knowledge of its own contents and state information;
this corresponds to a one-sided push-only scenario. The goal is to disseminate
all messages efficiently, hopefully achieving an order-optimal spreading rate
over unicast wireless random networks. First, we show that a random-push
strategy -- where a user sends its own or a received packet at random -- is
order-wise suboptimal in a random geometric graph: specifically,
times slower than optimal spreading. It is known that this
gap can be closed if each user has "full" mobility, since this effectively
creates a complete graph. We instead consider velocity-constrained mobility
where at each time slot the user moves locally using a discrete random walk
with velocity that is much lower than full mobility. We propose a simple
two-stage dissemination strategy that alternates between individual message
flooding ("self promotion") and random gossiping. We prove that this scheme
achieves a close to optimal spreading rate (within only a logarithmic gap) as
long as the velocity is at least . The key
insight is that the mixing property introduced by the partial mobility helps
users to spread in space within a relatively short period compared to the
optimal spreading time, which macroscopically mimics message dissemination over
a complete graph.Comment: accepted to IEEE Transactions on Information Theory, 201
Principal Patterns on Graphs: Discovering Coherent Structures in Datasets
Graphs are now ubiquitous in almost every field of research. Recently, new
research areas devoted to the analysis of graphs and data associated to their
vertices have emerged. Focusing on dynamical processes, we propose a fast,
robust and scalable framework for retrieving and analyzing recurring patterns
of activity on graphs. Our method relies on a novel type of multilayer graph
that encodes the spreading or propagation of events between successive time
steps. We demonstrate the versatility of our method by applying it on three
different real-world examples. Firstly, we study how rumor spreads on a social
network. Secondly, we reveal congestion patterns of pedestrians in a train
station. Finally, we show how patterns of audio playlists can be used in a
recommender system. In each example, relevant information previously hidden in
the data is extracted in a very efficient manner, emphasizing the scalability
of our method. With a parallel implementation scaling linearly with the size of
the dataset, our framework easily handles millions of nodes on a single
commodity server
Bounds on the Voter Model in Dynamic Networks
In the voter model, each node of a graph has an opinion, and in every round
each node chooses independently a random neighbour and adopts its opinion. We
are interested in the consensus time, which is the first point in time where
all nodes have the same opinion. We consider dynamic graphs in which the edges
are rewired in every round (by an adversary) giving rise to the graph sequence
, where we assume that has conductance at least
. We assume that the degrees of nodes don't change over time as one can
show that the consensus time can become super-exponential otherwise. In the
case of a sequence of -regular graphs, we obtain asymptotically tight
results. Even for some static graphs, such as the cycle, our results improve
the state of the art. Here we show that the expected number of rounds until all
nodes have the same opinion is bounded by , for any
graph with edges, conductance , and degrees at least . In
addition, we consider a biased dynamic voter model, where each opinion is
associated with a probability , and when a node chooses a neighbour with
that opinion, it adopts opinion with probability (otherwise the node
keeps its current opinion). We show for any regular dynamic graph, that if
there is an difference between the highest and second highest
opinion probabilities, and at least nodes have initially the
opinion with the highest probability, then all nodes adopt w.h.p. that opinion.
We obtain a bound on the convergences time, which becomes for
static graphs
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