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Algorithmic Approach to the Extinction Probability of Branching Processes

Algorithmic Approach to the Extinction Probability of Branching Processes,10.1007/s11009-009-9141-7,Methodology and Computing in Applied Probability,S

Algorithmic Approach to the Extinction Probability of Branching Processes   (Citations: 5)
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The extinction probability of a branching process is characterized as the solution of a fixed-point equation which, for a fairly general class of Markovian branching processes, is vector quadratic. We address the question of solving that equation, using a mixture of algorithmic and probabilistic arguments. We compare the relative efficiency of three iterative methods based on functional iteration, on the basis of the probabilistic interpretation of the successive iterations as well as on the basis of traditional rate of convergence analysis. We illustrate our findings through a few numerical examples and conclude by showing how they extend to more complex systems.
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    • ...This equation relies on the assumption of independence between individuals and may be solved numerically: linear algorithms have been developed in Bean, Kontoleon, and Taylor[ 3 ] and in Hautphenne, Latouche, and Remiche[ 9 ], and quadratic algorithms have been proposed in Hautphenne, Latouche, and Remiche[ 8 ] and in Hautphenne and van Houdt[ 12 ]...

    Sophie Hautphenneet al. Markovian Trees Subject to Catastrophes: Transient Features and Extinc...

    • ...This family of random processes has been extensively analytically characterized by Hautphenne et al. [8] among others...
    • ...The proofs of these relations are given in [8]...
    • ...It has been established in [8] that the matrix M respects the following identity:...
    • ...In [8], three different algorithms are proposed to numerically evaluate q. This vector gives us the probability that the dissemination process may eventually stop at a certain time...

    Kenji Leibnitzet al. Modeling the Extinction in the Information Diffusion Process in Wirele...

    • ...Bean et al. (2008) and in Hautphenne et al. (2010), or with the quadratic Newton algorithm in Hautphenne et al. (2008)...

    Sophie Hautphenneet al. The Markovian binary tree applied to demography

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