Project Title: Dynamics of some Asynchronous Random
Boolean Networks with Nonconstant Number of Parents
Adviser: Dora Matache
Description: Random Boolean Networks
are networks of nodes that can be in one of two possible states ON or OFF, and
whose evolution from one time point to another is governed by given Boolean
rules. Each node’s evolution is influenced by the state of other nodes called
its parents. If all the nodes are updated at the same time then the network is
called synchronous, otherwise it is called asynchronous. Recent research has
focused on finding formulas for the probability of a node being ON at time t
and using these formulas to study the dynamics of the network.
A. Write a brief
description of important concepts of random Boolean networks governed by rule
126 of cellular automata, based on some papers of the following bibliography:
Andrecut M., Ali M. K., Chaos in a simple Boolean network,
International Journal of Modern Physics B, Vol. 15, 1 (2001), p. 17-23.
Matache M.T., Heidel J., A random Boolean network model exhibiting
deterministic chaos, Phys. Rev. E 69, 056214, 2004, 10 pages.
Matache M.T., Heidel J., Asynchronous Random Boolean Networks,
submitted.
This will help the student become familiar with the topic and
will serve as an introductory section of the final research report.
B. Get familiar
with the Matlab software in order to be able to write and run several programs
related to the study of the dynamics of the Boolean networks.
C. Actual study of the dynamics of asynchronous Boolean networks with a non-constant
number of parents for all nodes, in which nodes are updated according to
certain random processes to be selected together with the adviser who will
indicate appropriate references. The student will study fixed points of the
maps giving the probability of a node being ON at time t, bifurcation diagrams,
stability of initial values, Lyapunov exponents. The findings of this study
will become the core of the research paper.
D. Create a final
research report to be presented at the
MAM Symposium.
OTHER REQUIREMENTS: The
students interested in the project above are expected to have taken and passed
with maximal grades or close MATH4740 (Introduction to Probability and
Statistics). They should be familiar with computers, Matlab, and