337 research outputs found
On the Shannon entropy of the number of vertices with zero in-degree in randomly oriented hypergraphs
Suppose that you have colours and mutually independent dice, each of
which has sides. Each dice lands on any of its sides with equal
probability. You may colour the sides of each die in any way you wish, but
there is one restriction: you are not allowed to use the same colour more than
once on the sides of a die. Any other colouring is allowed. Let be the
number of different colours that you see after rolling the dice. How should you
colour the sides of the dice in order to maximize the Shannon entropy of ?
In this article we investigate this question. We show that the entropy of
is at most and that the bound is tight, up to a
constant additive factor, in the case of there being equally many coins and
colours. Our proof employs the differential entropy bound on discrete entropy,
along with a lower bound on the entropy of binomial random variables whose
outcome is conditioned to be an even integer. We conjecture that the entropy is
maximized when the colours are distributed over the sides of the dice as evenly
as possible.Comment: 11 page
The social network model on infinite graphs
Given an infinite connected regular graph , place at each vertex Pois() walkers performing independent lazy simple random walks on simultaneously. When two walkers visit the same vertex at the same time they are declared to be acquainted. We show that when is vertex-transitive and amenable, for all a.s. any pair of walkers will eventually have a path of acquaintances between them. In contrast, we show that when is non-amenable (not necessarily transitive) there is always a phase transition at some . We give general bounds on and study the case that is the -regular tree in more details. Finally, we show that in the non-amenable setup, for every there exists a finite time such that a.s. there exists an infinite set of walkers having a path of acquaintances between them by time .EPSRC grant EP/L018896/1
On the Shannon entropy of the number of vertices with zero in-degree in randomly oriented hypergraphs
Suppose that you have colours and mutually independent dice, each of which has sides. Each dice lands on any of its sides with equal probability. You may colour the sides of each die in any way you wish, but there is one restriction: you are not allowed to use the same colour more than once on the sides of a die. Any other colouring is allowed. Let be the number of different colours that you see after rolling the dice. How should you colour the sides of the dice in order to maximize the Shannon entropy of ? In this article we investigate this question. It is shown that the entropy of is at most and that the bound is tight, up to a constant additive factor, in the case of there being equally many coins and colours. Our proof employs the differential entropy bound on discrete entropy, along with a lower bound on the entropy of binomial random variables whose outcome is conditioned to be an even integer. We conjecture that the entropy is maximized when the colours are distributed over the sides of the dice as evenly as possible
A Network Coding Approach to Loss Tomography
Network tomography aims at inferring internal network characteristics based
on measurements at the edge of the network. In loss tomography, in particular,
the characteristic of interest is the loss rate of individual links and
multicast and/or unicast end-to-end probes are typically used. Independently,
recent advances in network coding have shown that there are advantages from
allowing intermediate nodes to process and combine, in addition to just
forward, packets. In this paper, we study the problem of loss tomography in
networks with network coding capabilities. We design a framework for estimating
link loss rates, which leverages network coding capabilities, and we show that
it improves several aspects of tomography including the identifiability of
links, the trade-off between estimation accuracy and bandwidth efficiency, and
the complexity of probe path selection. We discuss the cases of inferring link
loss rates in a tree topology and in a general topology. In the latter case,
the benefits of our approach are even more pronounced compared to standard
techniques, but we also face novel challenges, such as dealing with cycles and
multiple paths between sources and receivers. Overall, this work makes the
connection between active network tomography and network coding
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
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