Broadcasting in random recursive dags
Abstract: A uniform $k$-{\sc dag} generalizes the uniform random recursive tree by picking $k$ parents uniformly at random from the existing nodes. It starts with $k$ ''roots''. Each of the $k$ roots is assigned a bit. These bits are propagated by a noisy channel. The parents' bits are flipped with probability $p$, and a majority vote is taken. When all nodes have received their bits, the $k$-{\sc dag} is shown without identifying the roots. The goal is to estimate the majority bit among the roots. We identify the threshold for $p$ as a function of $k$ below which the majority rule among all nodes yields an error $c+o(1)$ with $c<1/2$. Above the threshold the majority rule errs with probability $1/2+o(1)$.
- Broadcasting on random recursive trees. The Annals of Applied Probability, 32(1):497–528, 2022.
- Coexistence in preferential attachment networks. Combinatorics, Probability and Computing, 25(6):797–822, 2016.
- Jean Bertoin. Limits of Pólya urns with innovations, 2022. URL https://arxiv.org/abs/2204.03470.
- Broadcasting in Harary-like graphs. In 2014 IEEE 17th International Conference on Computational Science and Engineering, pages 1269–1276, 2014.
- Archaeology of random recursive dags and Cooper-Frieze random networks. Combinatorics, Probability and Computing, 2023, to appear.
- Generating Random Networks and Graphs. Oxford University Press, 2017. ISBN 9780198709893.
- An urn model with random multiple drawing and random addition. Stochastic Processes and their Applications, 147:270–299, 2022.
- Broadcasting induced colourings of random recursive trees and preferential attachment trees. arXiv preprint arXiv:2110.15050, 2021.
- Broadcasting on trees and the Ising model. The Annals of Applied Probability, 10(2):410 – 433, 2000.
- A new construction of broadcast graphs. Discrete Applied Mathematics, 280:144–155, 2020.
- Svante Janson. Functional limit theorems for multitype branching processes and generalized pólya urns. Stochastic Processes and their Applications, 110(2):177–245, 2004.
- Svante Janson. Random replacements in Pólya urns with infinitely many colours. Electronic Communications in Probability, 24:1 – 11, 2019.
- Robust reconstruction on trees is determined by the second eigenvalue. Annals of Probability, 32(3B):2630–2649, 2004.
- Pólya urns via the contraction method. Combinatorics, Probability and Computing, 23(6):1148–1186, 2014.
- Two-color balanced affine urn models with multiple drawings. Advances in Applied Mathematics, 90:1–26, 2017.
- Broadcasting on random directed acyclic graphs. IEEE Transactions on Information Theory, 66(2):780–812, 2020.
- Elchanan Mossel. Reconstruction on trees: beating the second eigenvalue. The Annals of Applied Probability, 11(1):285–300, 2001.
- Elchanan Mossel. Survey: information flow on trees. In Graphs, morphisms and statistical physics, volume 63 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 155–170. Amer. Math. Soc., Providence, RI, 2004.
- Robin Pemantle. Nonconvergence to unstable points in urn models and stochastic approximations. The Annals of Probability, 18(2):698–712, 1990.
- Robin Pemantle. A survey of random processes with reinforcement. Probability Surveys, 4:9–12, 2007.
- Allan Sly. Reconstruction for the Potts model. The Annals of Probability, 39(4):1365 – 1406, 2011.
- LJ Wei. The generalized Pólya’s urn design for sequential medical trials. The Annals of Statistics, 7(2):291–296, 1979.
- Li-Xin Zhang. Convergence of randomized urn models with irreducible and reducible replacement policy. arXiv preprint arXiv:2204.04810, 2022.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.