6 research outputs found
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
Space-time percolation and detection by mobile nodes
Consider the model where nodes are initially distributed as a Poisson point
process with intensity over and are moving in
continuous time according to independent Brownian motions. We assume that nodes
are capable of detecting all points within distance of their location and
study the problem of determining the first time at which a target particle,
which is initially placed at the origin of , is detected by at
least one node. We consider the case where the target particle can move
according to any continuous function and can adapt its motion based on the
location of the nodes. We show that there exists a sufficiently large value of
so that the target will eventually be detected almost surely. This
means that the target cannot evade detection even if it has full information
about the past, present and future locations of the nodes. Also, this
establishes a phase transition for since, for small enough ,
with positive probability the target can avoid detection forever. A key
ingredient of our proof is to use fractal percolation and multi-scale analysis
to show that cells with a small density of nodes do not percolate in space and
time.Comment: Published at http://dx.doi.org/10.1214/14-AAP1052 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Viral processes by random walks on random regular graphs
We study the SIR epidemic model with infections carried by particles
making independent random walks on a random regular graph. Here we assume
, where is the number of vertices in the random graph,
and is some sufficiently small constant. We give an edge-weighted
graph reduction of the dynamics of the process that allows us to apply standard
results of Erd\H{o}s-R\'{e}nyi random graphs on the particle set. In
particular, we show how the parameters of the model give two thresholds: In the
subcritical regime, particles are infected. In the supercritical
regime, for a constant determined by the parameters of the
model, get infected with probability , and get
infected with probability . Finally, there is a regime in which all
particles are infected. Furthermore, the edge weights give information
about when a particle becomes infected. We exploit this to give a completion
time of the process for the SI case.Comment: Published in at http://dx.doi.org/10.1214/13-AAP1000 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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Information dissemination via random walks
Information dissemination is a fundamental task in distributed computing:
How to deliver a piece of information from a node of a network to some or all other nodes?
In the face of large and still growing modern networks, it is imperative that dissemination algorithms are decentralised and can operate under unreliable conditions.
In the past decades, randomised rumour spreading algorithms
have addressed these challenges.
In these algorithms, a message is initially placed at a source node of a network, and, at regular intervals, each node contacts a randomly selected neighbour.
A message may be transmitted in one or both directions during each of these communications, depending on the exact protocol.
The main measure of performance for these algorithms is their broadcast time, which is the time until a message originating from a source node is disseminated to all nodes of the network.
Apart from being extremely simple and robust to failures, randomised rumour spreading achieves theoretically optimal broadcast time in many common network topologies.
In this thesis, we propose an agent-based information dissemination algorithm, called Visit-Exchange.
In our protocol, a number of agents perform independent random walks in the network.
An agent becomes informed when it visits a node that has a message, and later informs all future nodes it visits.
Visit-Exchange shares many of the properties of randomised rumour spreading, namely, it is very simple and uses the same amount of communication in a unit of time.
Moreover, the protocol can be used as a simple model of non-recoverable epidemic processes.
We investigate the broadcast time of Visit-Exchange on a variety of network topologies, and compare it to traditional rumour spreading.
On dense regular networks we show that the two types of protocols are equivalent, which means that in this setting the vast literature on randomised rumour spreading applies in our model as well.
Since many networks of interest, including real-world ones, are very sparse, we also study agent-based broadcast for sparse networks.
Our results include almost optimal or optimal bounds for sparse regular graphs, expanders, random regular graphs, balanced trees and grids.
We establish that depending on the network topology, Visit-Exchange may be either slower or faster than traditional rumour spreading.
In particular, in graphs consisting of hubs that are not well connected, broadcast using agents can be significantly faster.
Our conclusion is that a combined broadcasting protocol that simultaneously uses both traditional rumour spreading and agent-based dissemination can be fast on a larger range of topologies than each of its components separately.Gates Cambridge Trust, St John's College Benefactors' Scholarshi