108 research outputs found
The coalescing-branching random walk on expanders and the dual epidemic process
Information propagation on graphs is a fundamental topic in distributed
computing. One of the simplest models of information propagation is the push
protocol in which at each round each agent independently pushes the current
knowledge to a random neighbour. In this paper we study the so-called
coalescing-branching random walk (COBRA), in which each vertex pushes the
information to randomly selected neighbours and then stops passing
information until it receives the information again. The aim of COBRA is to
propagate information fast but with a limited number of transmissions per
vertex per step. In this paper we study the cover time of the COBRA process
defined as the minimum time until each vertex has received the information at
least once. Our main result says that if is an -vertex -regular graph
whose transition matrix has second eigenvalue , then the COBRA cover
time of is , if is greater than a positive
constant, and , if . These bounds are independent of and hold for . They improve the previous bound of for expander graphs.
Our main tool in analysing the COBRA process is a novel duality relation
between this process and a discrete epidemic process, which we call a biased
infection with persistent source (BIPS). A fixed vertex is the source of an
infection and remains permanently infected. At each step each vertex other
than selects neighbours, independently and uniformly, and is
infected in this step if and only if at least one of the selected neighbours
has been infected in the previous step. We show the duality between COBRA and
BIPS which says that the time to infect the whole graph in the BIPS process is
of the same order as the cover time of the COBRA proces
Rescaled Density Processes of Voter Model Perturbations on r-Regular Random Graphs converge to Fellerâs Branching Diffusion
Voter model perturbations can be viewed as voter model (neutral competition) plus asmall perturbation rate. Cox (2017) showed that the biased voter model, viewed as a voter model perturbation, converges to Fellerâs branching diffusion under mild mixing condition. We extend this result to a general class of perturbation functions on the setting of r-regular random graphs where the nearest-neighbor voting kernel has a strong mixing property, and prove a low-density diffusive limit of which the convergence of biased voter model is considered as a special case. The other special case considered is the q-voter model whose high-density ODE limit on torus for q close to 1 has been proved by Agarwal, Simper and Durrett (2021). We will introduce the low-density approach we use and show that a mean-field simplification occurs
Coalescing and branching simple symmetric exclusion process
Motivated by kinetically constrained interacting particle systems (KCM), we
consider a reversible coalescing and branching simple exclusion process on a
general finite graph dual to the biased voter model on . Our main
goal are tight bounds on its logarithmic Sobolev constant and relaxation time,
with particular focus on the delicate slightly supercritical regime in which
the equilibrium density of particles tends to zero as .
Our results allow us to recover very directly and improve to -mixing,
, and to more general graphs, the mixing time results of Pillai and
Smith for the Fredrickson-Andersen one spin facilitated (FA-f) KCM on the
discrete -dimensional torus. In view of applications to the more complex
FA-f KCM, , we also extend part of the analysis to an analogous process
with a more general product state space.Comment: 19 pages, minor change
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
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
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