161 research outputs found
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
Fast Graphical Population Protocols
Let be a graph on nodes. In the stochastic population protocol model,
a collection of indistinguishable, resource-limited nodes collectively
solve tasks via pairwise interactions. In each interaction, two randomly chosen
neighbors first read each other's states, and then update their local states. A
rich line of research has established tight upper and lower bounds on the
complexity of fundamental tasks, such as majority and leader election, in this
model, when is a clique. Specifically, in the clique, these tasks can be
solved fast, i.e., in pairwise interactions, with
high probability, using at most states per node.
In this work, we consider the more general setting where is an arbitrary
graph, and present a technique for simulating protocols designed for
fully-connected networks in any connected regular graph. Our main result is a
simulation that is efficient on many interesting graph families: roughly, the
simulation overhead is polylogarithmic in the number of nodes, and quadratic in
the conductance of the graph. As a sample application, we show that, in any
regular graph with conductance , both leader election and exact majority
can be solved in pairwise
interactions, with high probability, using at most states per node. This shows that there are fast and
space-efficient population protocols for leader election and exact majority on
graphs with good expansion properties. We believe our results will prove
generally useful, as they allow efficient technology transfer between the
well-mixed (clique) case, and the under-explored spatial setting.Comment: 47 pages, 5 figure
On the Inherent Anonymity of Gossiping
Detecting the source of a gossip is a critical issue, related to identifying
patient zero in an epidemic, or the origin of a rumor in a social network.
Although it is widely acknowledged that random and local gossip communications
make source identification difficult, there exists no general quantification of
the level of anonymity provided to the source. This paper presents a principled
method based on -differential privacy to analyze the inherent
source anonymity of gossiping for a large class of graphs. First, we quantify
the fundamental limit of source anonymity any gossip protocol can guarantee in
an arbitrary communication graph. In particular, our result indicates that when
the graph has poor connectivity, no gossip protocol can guarantee any
meaningful level of differential privacy. This prompted us to further analyze
graphs with controlled connectivity. We prove on these graphs that a large
class of gossip protocols, namely cobra walks, offers tangible differential
privacy guarantees to the source. In doing so, we introduce an original proof
technique based on the reduction of a gossip protocol to what we call a random
walk with probabilistic die out. This proof technique is of independent
interest to the gossip community and readily extends to other protocols
inherited from the security community, such as the Dandelion protocol.
Interestingly, our tight analysis precisely captures the trade-off between
dissemination time of a gossip protocol and its source anonymity.Comment: Full version of DISC2023 pape
Best-of-Three Voting on Dense Graphs
Given a graph of vertices, where each vertex is initially attached an
opinion of either red or blue. We investigate a random process known as the
Best-of-three voting. In this process, at each time step, every vertex chooses
three neighbours at random and adopts the majority colour. We study this
process for a class of graphs with minimum degree \,, where
. We prove that if initially
each vertex is red with probability greater than , and blue
otherwise, where for some , then with high
probability this dynamic reaches a final state where all vertices are red
within steps.European Research Counci
Large Scale Stochastic Dynamics
The goal of this workshop was to explore the recent advances in the
mathematical understanding of the macroscopic properties which emerge on large space-time scales from interacting microscopic particle systems. There were 55 participants,
including postdocs and graduate students, working in diverse
intertwining areas of probability and statistical mechanics. During
the meeting, 29 talks of 45 minutes were scheduled and an evening
session was organised with 10 more short talks of 10 minutes, mostly by younger participants.
These talks addressed the following topics :
randomness emerging from deterministic dynamics,
hydrodynamic limits, interface growth models and slow convergence to
equilibrium in kinetically
constrained dynamics
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