337 research outputs found

    On the Shannon entropy of the number of vertices with zero in-degree in randomly oriented hypergraphs

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    Suppose that you have nn colours and mm mutually independent dice, each of which has rr 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 XX 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 XX? In this article we investigate this question. We show that the entropy of XX is at most 12log(n)+O(1)\frac{1}{2} \log(n) + O(1) 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

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    Given an infinite connected regular graph G=(V,E)G=(V,E), place at each vertex Pois(λ\lambda) walkers performing independent lazy simple random walks on GG simultaneously. When two walkers visit the same vertex at the same time they are declared to be acquainted. We show that when GG is vertex-transitive and amenable, for all λ>0\lambda>0 a.s. any pair of walkers will eventually have a path of acquaintances between them. In contrast, we show that when GG is non-amenable (not necessarily transitive) there is always a phase transition at some λc(G)>0\lambda_{c}(G)>0. We give general bounds on λc(G)\lambda_{c}(G) and study the case that GG is the dd-regular tree in more details. Finally, we show that in the non-amenable setup, for every λ\lambda there exists a finite time tλ(G)t_{\lambda}(G) such that a.s. there exists an infinite set of walkers having a path of acquaintances between them by time tλ(G)t_{\lambda}(G).EPSRC grant EP/L018896/1

    On the Shannon entropy of the number of vertices with zero in-degree in randomly oriented hypergraphs

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    Suppose that you have nn colours and mm  mutually independent dice, each of which has rr 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 XX 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 XX? In this article we investigate this question. It is shown that the entropy of XX is at most 12log(n)+12log(πe)\frac{1}{2} \log(n) + \frac{1}{2}\log(\pi e) 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

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    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

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    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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