94 research outputs found

### On the number of attractors in random Boolean networks

The evaluation of the number of attractors in Kauffman networks by Samuelsson
and Troein is generalized to critical networks with one input per node and to
networks with two inputs per node and different probability distributions for
update functions. A connection is made between the terms occurring in the
calculation and between the more graphic concepts of frozen, nonfrozen and
relevant nodes, and relevant components. Based on this understanding, a
phenomenological argument is given that reproduces the dependence of the
attractor numbers on system size.Comment: 6 page

### The properties of attractors of canalyzing random Boolean networks

We study critical random Boolean networks with two inputs per node that
contain only canalyzing functions. We present a phenomenological theory that
explains how a frozen core of nodes that are frozen on all attractors arises.
This theory leads to an intuitive understanding of the system's dynamics as it
demonstrates the analogy between standard random Boolean networks and networks
with canalyzing functions only. It reproduces correctly the scaling of the
number of nonfrozen nodes with system size. We then investigate numerically
attractor lengths and numbers, and explain the findings in terms of the
properties of relevant components. In particular we show that canalyzing
networks can contain very long attractors, albeit they occur less often than in
standard networks.Comment: 9 pages, 8 figure

### Number and length of attractors in a critical Kauffman model with connectivity one

The Kauffman model describes a system of randomly connected nodes with
dynamics based on Boolean update functions. Though it is a simple model, it
exhibits very complex behavior for "critical" parameter values at the boundary
between a frozen and a disordered phase, and is therefore used for studies of
real network problems. We prove here that the mean number and mean length of
attractors in critical random Boolean networks with connectivity one both
increase faster than any power law with network size. We derive these results
by generating the networks through a growth process and by calculating lower
bounds.Comment: 4 pages, no figure, no table; published in PR

### Perturbation propagation in random and evolved Boolean networks

We investigate the propagation of perturbations in Boolean networks by
evaluating the Derrida plot and modifications of it. We show that even small
Random Boolean Networks agree well with the predictions of the annealed
approximation, but non-random networks show a very different behaviour. We
focus on networks that were evolved for high dynamical robustness. The most
important conclusion is that the simple distinction between frozen, critical
and chaotic networks is no longer useful, since such evolved networks can
display properties of all three types of networks. Furthermore, we evaluate a
simplified empirical network and show how its specific state space properties
are reflected in the modified Derrida plots.Comment: 10 pages, 8 figure

### The dynamics of critical Kauffman networks under asynchronous stochastic update

We show that the mean number of attractors in a critical Boolean network
under asynchronous stochastic update grows like a power law and that the mean
size of the attractors increases as a stretched exponential with the system
size. This is in strong contrast to the synchronous case, where the number of
attractors grows faster than any power law.Comment: submitted to PR

### Critical Kauffman networks under deterministic asynchronous update

We investigate the influence of a deterministic but non-synchronous update on
Random Boolean Networks, with a focus on critical networks. Knowing that
``relevant components'' determine the number and length of attractors, we focus
on such relevant components and calculate how the length and number of
attractors on these components are modified by delays at one or more nodes. The
main findings are that attractors decrease in number when there are more
delays, and that periods may become very long when delays are not integer
multiples of the basic update step.Comment: 8 pages, 3 figures, submitted to a journa

### Vertex routing models

A class of models describing the flow of information within networks via
routing processes is proposed and investigated, concentrating on the effects of
memory traces on the global properties. The long-term flow of information is
governed by cyclic attractors, allowing to define a measure for the information
centrality of a vertex given by the number of attractors passing through this
vertex. We find the number of vertices having a non-zero information centrality
to be extensive/sub-extensive for models with/without a memory trace in the
thermodynamic limit. We evaluate the distribution of the number of cycles, of
the cycle length and of the maximal basins of attraction, finding a complete
scaling collapse in the thermodynamic limit for the latter. Possible
implications of our results on the information flow in social networks are
discussed.Comment: 12 pages, 6 figure

### Mutual information in random Boolean models of regulatory networks

The amount of mutual information contained in time series of two elements
gives a measure of how well their activities are coordinated. In a large,
complex network of interacting elements, such as a genetic regulatory network
within a cell, the average of the mutual information over all pairs is a
global measure of how well the system can coordinate its internal dynamics. We
study this average pairwise mutual information in random Boolean networks
(RBNs) as a function of the distribution of Boolean rules implemented at each
element, assuming that the links in the network are randomly placed. Efficient
numerical methods for calculating show that as the number of network nodes
N approaches infinity, the quantity N exhibits a discontinuity at parameter
values corresponding to critical RBNs. For finite systems it peaks near the
critical value, but slightly in the disordered regime for typical parameter
variations. The source of high values of N is the indirect correlations
between pairs of elements from different long chains with a common starting
point. The contribution from pairs that are directly linked approaches zero for
critical networks and peaks deep in the disordered regime.Comment: 11 pages, 6 figures; Minor revisions for clarity and figure format,
one reference adde

### Finite size corrections to random Boolean networks

Since their introduction, Boolean networks have been traditionally studied in
view of their rich dynamical behavior under different update protocols and for
their qualitative analogy with cell regulatory networks. More recently, tools
borrowed from statistical physics of disordered systems and from computer
science have provided a more complete characterization of their equilibrium
behavior. However, the largest part of the results have been obtained in the
thermodynamic limit, which is often far from being reached when dealing with
realistic instances of the problem. The numerical analysis presented here aims
at comparing - for a specific family of models - the outcomes given by the
heuristic belief propagation algorithm with those given by exhaustive
enumeration. In the second part of the paper some analytical considerations on
the validity of the annealed approximation are discussed.Comment: Minor correction

### Computational core and fixed-point organisation in Boolean networks

In this paper, we analyse large random Boolean networks in terms of a
constraint satisfaction problem. We first develop an algorithmic scheme which
allows to prune simple logical cascades and under-determined variables,
returning thereby the computational core of the network. Second we apply the
cavity method to analyse number and organisation of fixed points. We find in
particular a phase transition between an easy and a complex regulatory phase,
the latter one being characterised by the existence of an exponential number of
macroscopically separated fixed-point clusters. The different techniques
developed are reinterpreted as algorithms for the analysis of single Boolean
networks, and they are applied to analysis and in silico experiments on the
gene-regulatory networks of baker's yeast (saccaromices cerevisiae) and the
segment-polarity genes of the fruit-fly drosophila melanogaster.Comment: 29 pages, 18 figures, version accepted for publication in JSTA

- â€¦