243 research outputs found
A particle system with cooperative branching and coalescence
In this paper, we introduce a one-dimensional model of particles performing
independent random walks, where only pairs of particles can produce offspring
("cooperative branching"), and particles that land on an occupied site merge
with the particle present on that site ("coalescence"). We show that the system
undergoes a phase transition as the branching rate is increased. For small
branching rates, the upper invariant law is trivial, and the process started
with finitely many particles a.s. ends up with a single particle. Both
statements are not true for high branching rates. An interesting feature of the
process is that the spectral gap is zero even for low branching rates. Indeed,
if the branching rate is small enough, then we show that for the process
started in the fully occupied state, the particle density decays as one over
the square root of time, and the same is true for the decay of the probability
that the process still has more than one particle at a later time if it started
with two particles.Comment: Published at http://dx.doi.org/10.1214/14-AAP1032 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Voter Model Perturbations and Reaction Diffusion Equations
We consider particle systems that are perturbations of the voter model and
show that when space and time are rescaled the system converges to a solution
of a reaction diffusion equation in dimensions . Combining this result
with properties of the PDE, some methods arising from a low density
super-Brownian limit theorem, and a block construction, we give general, and
often asymptotically sharp, conditions for the existence of non-trivial
stationary distributions, and for extinction of one type. As applications, we
describe the phase diagrams of three systems when the parameters are close to
the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and
Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert,
Lieberman, and Nowak, and (iii) a continuous time version of the non-linear
voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first
application confirms a conjecture of Cox and Perkins and the second confirms a
conjecture of Ohtsuki et al in the context of certain infinite graphs. An
important feature of our general results is that they do not require the
process to be attractive.Comment: 106 pages, 7 figure
Randomised Algorithms on Networks
Networks form an indispensable part of our lives. In particular, computer networks have ranked amongst the most influential networks in recent times. In such an ever-evolving and fast growing network, the primary concern is to understand and analyse different aspects of the network behaviour, such as the quality of service and efficient information propagation. It is also desirable to predict the behaviour of a large computer network if, for example, one of the computers is infected by a virus. In all of the aforementioned cases, we need protocols that are able to make local decisions and handle the dynamic changes in the network topology. Here, randomised algorithms are preferred because many deterministic algorithms often require a central control. In this thesis, we investigate three network-based randomised algorithms, threshold load balancing with weighted tasks, the pull-Moran process and the coalescing-branching random walk. Each of these algorithms has extensive applicability within networks and computational complexity within computer science.
In this thesis we investigate threshold-based load balancing protocols. We introduce a generalisation of protocols in [2, 3] to weighted tasks.
This thesis also analyses an evolutionary-based process called the death-birth update, defined here as the Pull-Moran process. We show that a class of strong universal amplifiers does not exist for the Pull-Moran process. We show that any class of selective amplifiers in the (standard) Moran process is a class of selective suppressors under the Pull-Moran process. We then introduce a class of selective amplifiers called Punk graphs.
Finally, we improve the broadcasting time of the coalescing-branching (COBRA) walk analysed in [4], for random regular graphs. Here, we look into the COBRA approach as a randomised rumour spreading protocol
How to Spread a Rumor: Call Your Neighbors or Take a Walk?
We study the problem of randomized information dissemination in networks. We
compare the now standard PUSH-PULL protocol, with agent-based alternatives
where information is disseminated by a collection of agents performing
independent random walks. In the VISIT-EXCHANGE protocol, both nodes and agents
store information, and each time an agent visits a node, the two exchange all
the information they have. In the MEET-EXCHANGE protocol, only the agents store
information, and exchange their information with each agent they meet.
We consider the broadcast time of a single piece of information in an
-node graph for the above three protocols, assuming a linear number of
agents that start from the stationary distribution. We observe that there are
graphs on which the agent-based protocols are significantly faster than
PUSH-PULL, and graphs where the converse is true. We attribute the good
performance of agent-based algorithms to their inherently fair bandwidth
utilization, and conclude that, in certain settings, agent-based information
dissemination, separately or in combination with PUSH-PULL, can significantly
improve the broadcast time.
The graphs considered above are highly non-regular. Our main technical result
is that on any regular graph of at least logarithmic degree, PUSH-PULL and
VISIT-EXCHANGE have the same asymptotic broadcast time. The proof uses a novel
coupling argument which relates the random choices of vertices in PUSH-PULL
with the random walks in VISIT-EXCHANGE. Further, we show that the broadcast
time of MEET-EXCHANGE is asymptotically at least as large as the other two's on
all regular graphs, and strictly larger on some regular graphs.
As far as we know, this is the first systematic and thorough comparison of
the running times of these very natural information dissemination protocols.The authors would like to thank Thomas Sauerwald and Nicol\'{a}s Rivera for helpful discussions.
This research was undertaken, in part, thanks to funding from
the ANR Project PAMELA (ANR-16-CE23-0016-01),
the NSF Award Numbers CCF-1461559, CCF-0939370 and CCF-18107,
the Gates Cambridge Scholarship programme,
and the ERC grant DYNAMIC MARCH
A Course in Interacting Particle Systems
These lecture notes give an introduction to the theory of interacting
particle systems. The main subjects are the construction using generators and
graphical representations, the mean field limit, stochastic order, duality, and
the relation to oriented percolation. An attempt is made to give a large number
of examples beyond the classical voter, contact and Ising processes and to
illustrate these based on numerical simulations.Comment: These are lecture notes for a course in interacting particle systems
taught at Charles University, Prague, in 2015/2016 and again in the fall of
2019. Compared to the first version, a number of small typos and mistakes
have been corrected, most notably the proof of Lemma 4.18, which was wrong in
the first version. Some parts have been rephrased for greater clarit
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