## 2006.32: The Effects of State Dependent and State Independent Probabilistic Updating on Boolean Network Dynamics

2006.32:
Natasha Saint Savage
(2005)
*The Effects of State Dependent and State Independent Probabilistic Updating on Boolean Network Dynamics.*
PhD thesis, University of Manchester.

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## Abstract

We study semi-synchronous Boolean networks with robabilistic updating schemes and various topologies (tree, loop, and random). As well as state independent probabilistic updating we investigate a state dependent scheme which allows us to control the `accuracy' of nodes. A node is accurate at $n$ if it has been updated at $n$, or if its state is as it would be if it had updated.

The state dependent re-evaluation probabilities are determined by the `accuracy heuristic': a stochastic equation which depends on the estimation of a distribution; we look at ways of estimating this distribution and derive variance expressions for the estimators.

Through our work on random Boolean trees we observe that (in general) the output of a Boolean function with correlated inputs, becomes less correlated as the number of inputs is increased. We also discover that the correlation of a Boolean function's output directly affects the ability of the heuristic to achieve the node's target accuracy.

Deterministic random Boolean network dynamics are viewed in a new way, via the distribution of node output distributions (the probability a node's state is $1$ or $0$). This view shows the `activity' of nodes across the network. We find that as in-degree is increased the topology has less effect on the activity and the distribution of the Boolean functions dominates. We present a theoretical result to support this theory.

To understand the dynamics of probabilistically updating Boolean networks we use a numerical approximation to Flyvbjerg's frozen component.

The concept of stability in probabilistically updating Boolean networks is addressed and investigated. For the loop topology the dynamics of active loops fall into two categories: those with an odd number of inversion nodes and those with an even number. We discuss the stability of a fixed point in both cases. For the random topology we derive an annealed approximation which indicates a phase transition similar to that previously found in the deterministic networks.

Item Type: | Thesis (PhD) |
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Subjects: | MSC 2000 > 37 Dynamical systems and ergodic theory |

MIMS number: | 2006.32 |

Deposited By: | Dr N Savage |

Deposited On: | 14 March 2006 |

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