8,128 research outputs found
The Influence of Canalization on the Robustness of Boolean Networks
Time- and state-discrete dynamical systems are frequently used to model
molecular networks. This paper provides a collection of mathematical and
computational tools for the study of robustness in Boolean network models. The
focus is on networks governed by -canalizing functions, a recently
introduced class of Boolean functions that contains the well-studied class of
nested canalizing functions. The activities and sensitivity of a function
quantify the impact of input changes on the function output. This paper
generalizes the latter concept to -sensitivity and provides formulas for the
activities and -sensitivity of general -canalizing functions as well as
canalizing functions with more precisely defined structure. A popular measure
for the robustness of a network, the Derrida value, can be expressed as a
weighted sum of the -sensitivities of the governing canalizing functions,
and can also be calculated for a stochastic extension of Boolean networks.
These findings provide a computationally efficient way to obtain Derrida values
of Boolean networks, deterministic or stochastic, that does not involve
simulation.Comment: 16 pages, 2 figures, 3 table
State Concentration Exponent as a Measure of Quickness in Kauffman-type Networks
We study the dynamics of randomly connected networks composed of binary
Boolean elements and those composed of binary majority vote elements. We
elucidate their differences in both sparsely and densely connected cases. The
quickness of large network dynamics is usually quantified by the length of
transient paths, an analytically intractable measure. For discrete-time
dynamics of networks of binary elements, we address this dilemma with an
alternative unified framework by using a concept termed state concentration,
defined as the exponent of the average number of t-step ancestors in state
transition graphs. The state transition graph is defined by nodes corresponding
to network states and directed links corresponding to transitions. Using this
exponent, we interrogate the dynamics of random Boolean and majority vote
networks. We find that extremely sparse Boolean networks and majority vote
networks with arbitrary density achieve quickness, owing in part to long-tailed
in-degree distributions. As a corollary, only relatively dense majority vote
networks can achieve both quickness and robustness.Comment: 6 figure
Discrete time piecewise affine models of genetic regulatory networks
We introduce simple models of genetic regulatory networks and we proceed to
the mathematical analysis of their dynamics. The models are discrete time
dynamical systems generated by piecewise affine contracting mappings whose
variables represent gene expression levels. When compared to other models of
regulatory networks, these models have an additional parameter which is
identified as quantifying interaction delays. In spite of their simplicity,
their dynamics presents a rich variety of behaviours. This phenomenology is not
limited to piecewise affine model but extends to smooth nonlinear discrete time
models of regulatory networks. In a first step, our analysis concerns general
properties of networks on arbitrary graphs (characterisation of the attractor,
symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc).
In a second step, focus is made on simple circuits for which the attractor and
its changes with parameters are described. In the negative circuit of 2 genes,
a thorough study is presented which concern stable (quasi-)periodic
oscillations governed by rotations on the unit circle -- with a rotation number
depending continuously and monotonically on threshold parameters. These regular
oscillations exist in negative circuits with arbitrary number of genes where
they are most likely to be observed in genetic systems with non-negligible
delay effects.Comment: 34 page
Complexity and Information: Measuring Emergence, Self-organization, and Homeostasis at Multiple Scales
Concepts used in the scientific study of complex systems have become so
widespread that their use and abuse has led to ambiguity and confusion in their
meaning. In this paper we use information theory to provide abstract and
concise measures of complexity, emergence, self-organization, and homeostasis.
The purpose is to clarify the meaning of these concepts with the aid of the
proposed formal measures. In a simplified version of the measures (focusing on
the information produced by a system), emergence becomes the opposite of
self-organization, while complexity represents their balance. Homeostasis can
be seen as a measure of the stability of the system. We use computational
experiments on random Boolean networks and elementary cellular automata to
illustrate our measures at multiple scales.Comment: 42 pages, 11 figures, 2 table
Complex Networks
Introduction to the Special Issue on Complex Networks, Artificial Life
journal.Comment: 7 pages, in pres
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