156 research outputs found
Polynomial tuning of multiparametric combinatorial samplers
Boltzmann samplers and the recursive method are prominent algorithmic
frameworks for the approximate-size and exact-size random generation of large
combinatorial structures, such as maps, tilings, RNA sequences or various
tree-like structures. In their multiparametric variants, these samplers allow
to control the profile of expected values corresponding to multiple
combinatorial parameters. One can control, for instance, the number of leaves,
profile of node degrees in trees or the number of certain subpatterns in
strings. However, such a flexible control requires an additional non-trivial
tuning procedure. In this paper, we propose an efficient polynomial-time, with
respect to the number of tuned parameters, tuning algorithm based on convex
optimisation techniques. Finally, we illustrate the efficiency of our approach
using several applications of rational, algebraic and P\'olya structures
including polyomino tilings with prescribed tile frequencies, planar trees with
a given specific node degree distribution, and weighted partitions.Comment: Extended abstract, accepted to ANALCO2018. 20 pages, 6 figures,
colours. Implementation and examples are available at [1]
https://github.com/maciej-bendkowski/boltzmann-brain [2]
https://github.com/maciej-bendkowski/multiparametric-combinatorial-sampler
Stochastic approximation on non-compact measure spaces and application to measure-valued Pólya processes
Our main result is to prove almost-sure convergence of a stochasticapproximation algorithm defined on the space of measures on a noncompact space. Our motivation is to apply this result to measure-valued Pólya processes (MVPPs, also known as infinitely-many Pólya urns). Our main idea is to use Foster-Lyapunov type criteria in a novel way to generalize stochasticapproximation methods to measure-valued Markov processes with a noncompact underlying space, overcoming in a fairly general context one of the major difficulties of existing studies on this subject. From the MVPPs point of view, our result implies almost-sure convergence of a large class of MVPPs; this convergence was only obtained until now for specific examples, with only convergence in probability established for general classes. Furthermore, our approach allows us to extend the definition of MVPPs by adding "weights"to the different colors of the infinitelymany- color urn. We also exhibit a link between non-"balanced"MVPPs and quasi-stationary distributions of Markovian processes, which allows us to treat, for the first time in the literature, the nonbalanced case. Finally, we show how our result can be applied to designing stochasticapproximation algorithms for the approximation of quasi-stationary distributions of discrete- and continuous-time Markov processes on noncompact spaces
Stochastic approximation on non-compact measure spaces and application to measure-valued Pólya processes
Our main result is to prove almost-sure convergence of a
stochastic-approximation algorithm defined on the space of measures on a
non-compact space. Our motivation is to apply this result to measure-valued
P\'olya processes (MVPPs, also known as infinitely-many P\'olya urns). Our main
idea is to use Foster-Lyapunov type criteria in a novel way to generalize
stochastic-approximation methods to measure-valued Markov processes with a
non-compact underlying space, overcoming in a fairly general context one of the
major difficulties of existing studies on this subject.
From the MVPPs point of view, our result implies almost-sure convergence of a
large class of MVPPs, this convergence was only obtained until now for specific
examples, with only convergence in probability established for general classes.
Furthermore, our approach allows us to extend the definition of MVPPs by adding
"weights" to the different colors of the infinitely-many-color urn. We also
exhibit a link between non-"balanced" MVPPs and quasi-stationary distributions
of Markovian processes, which allows us to treat, for the first time in the
literature, the non-balanced case.
Finally, we show how our result can be applied to designing
stochastic-approximation algorithms for the approximation of quasi-stationary
distributions of discrete- and continuous-time Markov processes on non-compact
spaces
Degrees in random -ary hooking networks
The theme in this paper is a composition of random graphs and P\'olya urns.
The random graphs are generated through a small structure called the seed. Via
P\'olya urns, we study the asymptotic degree structure in a random -ary
hooking network and identify strong laws. We further upgrade the result to
second-order asymptotics in the form of multivariate Gaussian limit laws. We
give a few concrete examples and explore some properties with a full
representation of the Gaussian limit in each case. The asymptotic covariance
matrix associated with the P\'olya urn is obtained by a new method that
originated in this paper and is reported in [25].Comment: 21 pages, 5 figure
Dynamical models for random simplicial complexes
We study a general model of random dynamical simplicial complexes and derive a formula for the asymptotic degree distribution. This asymptotic formula generalises results for a number of existing models, including random Apollonian networks and the weighted random recursive tree. It also confirms results on the scale-free nature of complex quantum network manifolds in dimensions d>2, and special types of network geometry with Flavour models studied in the physics literature by Bianconi and Rahmede [Sci. Rep. 5 (2015) 13979 and Phys. Rev. E 93 (2016) 032315]
Contagion Source Detection in Epidemic and Infodemic Outbreaks: Mathematical Analysis and Network Algorithms
This monograph provides an overview of the mathematical theories and
computational algorithm design for contagion source detection in large
networks. By leveraging network centrality as a tool for statistical inference,
we can accurately identify the source of contagions, trace their spread, and
predict future trajectories. This approach provides fundamental insights into
surveillance capability and asymptotic behavior of contagion spreading in
networks. Mathematical theory and computational algorithms are vital to
understanding contagion dynamics, improving surveillance capabilities, and
developing effective strategies to prevent the spread of infectious diseases
and misinformation.Comment: Suggested Citation: Chee Wei Tan and Pei-Duo Yu (2023), "Contagion
Source Detection in Epidemic and Infodemic Outbreaks: Mathematical Analysis
and Network Algorithms", Foundations and Trends in Networking: Vol. 13: No.
2-3, pp 107-251. http://dx.doi.org/10.1561/130000006
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