Attractor Basins of Discrete Networks
Implications on self-organisation and memory
andy AT ddlab DOT org
for the degree of D.Phil at
THE UNIVERSITY OF SUSSEX
School of Cognitive and Computing Sciences
October 1996, Revised Feb 1997
The thesis consists of 239 pages with 71 figures + appendices figures.
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for a hard copy of the complete thesis - please contact:
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School of Cognitive and Computing Sciences
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Only the abstract, contents, overview and selected references
are included below:
Discrete Dynamics Lab
New tools are available for reconstructing the attractor basins of discrete
dynamical networks where state-space is linked according the network's dynamics.
In this thesis the computer software
"Discrete Dynamics Lab"
is applied to
examine simple networks ranging from cellular automata (CA) to random Boolean
networks (RBN), that have been widely applied as idealised models of physical
and biological systems, to search for general principles underlying their
dynamics. The algorithms and methods for generating pre-images for both CA
and RBN, and reconstructing and representing attractor basins are described,
and also considered in the mathematical context of random directed graphs.
RBN and CA provide contrasting notions of self-organisation. RBN
provide models of hierarchical categorisation in biology, for example memory
in neural and genomic networks. CA provide models at the lower level of
emergent complex pattern. New measures and results are presented on CA
attractor basins and how they relate to measures on local dynamics and
the Z parameter, characterising ordered to "complex" to chaotic behaviour.
A method is described for classifying CA rules by an entropy-variance measure
which allows glider rules and related complex rules to be found automatically
giving a virtually unlimited sample for further study.
The dynamics of RBN and intermediate network architectures are examined
in the context of memory, where categorisation occurs at the roots of subtrees
as well as at attractors. Learning algorithms are proposed for "sculpting" the
basin of attraction field. RBN are proposed as a possible neural network model,
and also discussed as a model of genomic regulatory networks, where cell types
have been explained as attractors.
2. Basins of Attraction
Attractor Basins of Random Maps
Self-Organisation in One-D Cellular Automata
Memory in Random Boolean Networks
6. Conclusions and open questions
Discrete Dynamical Networks are central to our current perception of a wide
range of natural and artificial phenomena drawn from many areas of science;
from physics to biology to cognition; to social and economic organisation;
to parallel computation and Artificial Life.
It can be argued that the dynamics of non-equilibrium parallel
processing systems composed of discrete interacting components create
and maintain much of the complex biological phenomena that dominate our
world. Various types of discrete dynamical networks are increasingly being
applied as idealised models of physical and biological phenomena. For this
reason, and also because of their intrinsic interest as mathematical/physical
systems, finding general principles underlying the dynamics of the idealised
networks themselves has become an important task. Their behaviour is difficult
to describe by classical mathematics using its tools of partial differential
equations and continuous dynamics.
This thesis applies computer simulation tools developed by the author,
"Discrete Dynamics Lab" (DDLab) and available on the Internet
(Wuensche 1996), to examine a range of simple discrete dynamical networks
ranging from cellular automata (CA) to random Boolean networks (RBN). In
particular local dynamics is placed in the context of the global dynamics
of the network represented by its basin of attraction field or fragment
thereof. The terminology is borrowed from the notion of the "phase portrait"
which proved so powerful in continuous dynamics. Constructing and visualising
sub-trees, basins of attraction and the entire basin of attraction field
of discrete dynamical networks (referred to collectively as attractor basins),
and relating their characteristics to local dynamical trajectories was
introduced in the author's book
"The Global Dynamics of Cellular Automata"
(Wuensche and Lesser 1992a). This thesis extends the investigation into
CA and embarks into new areas involving RBN. New results of computer
experiments are presented, some of which are preliminary based on work
in progress, and lines of future enquiry are described. The thesis also
allows itself some interpretations, conjectures and speculations relating
to the results, and how these might provide insights into self-organisation
and memory. It is hoped that the distinction between hard results and more
general discussion will be clear.
The capability of constructing attractor basins depends on reverse
algorithms invented by the author for directly computing the
predecessors (known as pre-images) of network states. Different reverse
algorithms apply to CA and RBN, though the RBN method encompasses
These methods are in general orders of magnitude faster than the brute
force method, constructing an exhaustive map resulting from network
dynamics. This exhaustive method rapidly becomes computationally
intractable with increasing network size so is limited to small
systems. It applies to all network types and also allows the attractor
basins of random maps to be constructed. These three independent methods
together form a valuable reality check on the correctness of the
computed pre-images and attractor basins.
The CA reverse algorithm forms the basis for a trajectory convergence
parameter, named the Z parameter (Wuensche and Lesser 1992a, Wuensche
1994a), a sort of inverse Liaponov exponent, which predicts the
bushiness of sub-trees in attractor basins. Measures of bushiness are
captured by "garden-of-Eden" density and more generally by the
distribution of in-degree. It is shown that these measures correlate
with the quality of dynamics, ordered to chaotic, where order turns out
to be the most bushy, chaos the least. A histogram of the in-degree
distribution shows contrasting profiles for order and chaos, for complex
rules the distribution seems to follow a power law.
Whereas RBN provide models in biology because of their non-local
connections and heterogeneous rules, CA provide models in physics with
its notions of continuous space and universal laws. Contrasting notions
of self-organisation are apparent between the two systems. RBN provide
models of hierarchical categorisation in biology, for example memory in
neural, genomic, and immune networks. CA provide models of
self-organisation at the lower level of emergent interacting space-time
configurations, particles or "gliders", metaphors for the emergence of
life from the stuff of inanimate physics by a process of
In general, coherent space-time configurations cannot emerge in RBN
because of their non-locality. On the other hand, a CA's ability to
flexibly categorise distributed patterns is subject to severe symmetry
constraints. Different and contrasting "edge-of-chaos", or phase
transition, notions are seen to apply to CA and RBN. In the case of CA,
this is the phase in rule-space where glider behaviour might emerge,
quantified by rule parameters such as the lambda parameter (Langton
1990) and the author's Z parameter, measures on the dynamics such as the
"input-entropy" and its variance, and by measures on the topology of
attractor basins and sub-trees, such as the distribution of in-degree
In RBN, as applied to gene regulatory networks (Kauffman 1969, Somogyi
and Sniegoski 1996), the phase transition
represents the phase in the wiring/rule setup where categorisation of
patterns of gene activation, the network's memory, is sufficiently
versatile for adaptive behaviour but short of chaotic to ensure reliable
behaviour. Tuning the degree of "canalisation" at the level of rules
(Kauffman 1984, Harris et al 1997) moves behaviour across the
transition, though the characteristics of network connections also plays
a role. A measure at the level of dynamics is given by the Derrida plot
(Derrida and Stauffer 1986), analogous to the Liaponov exponent in
continuous dynamics. There are further measures such as damage
spreading, the percolation of frozen regions, input-entropy over time
relating to single genes, and the transient/frozen skeleton/attractor
The body of the thesis goes on to discuss these ideas in depth, based on
published work by the author and work in progress. The introductory
chapter describes and defines simple discrete dynamical networks ranging
from CA to RBN and explains how network dynamics can be understood
globally in terms of attractor basins. Comparisons are made with
attractor basins in classical continuous dynamics. Previous work and
applications in the field are discussed. Ideas of emergence, phase
transitions and various measures and parameters relating to CA and RBN
dynamics are introduced.
Chapter 2 looks more closely at the methods for constructing and
portraying attractor basins.
Chapter 3 considers discrete
dynamical networks and their attractor basins in the mathematical
context of directed maps in random graph theory. Discrete dynamical
networks are a subclass of random maps. Attractor basins of random maps,
possibly incorporating some bias in the random mapping, are amenable to
probabilistic analysis, which can be checked against a numerical
DDLab. This is work in progress in collaboration
with Christian Reidys (Reidys and Wuensche 1997).
Chapter 4 focuses on CA and develops ideas first presented in the paper
"Complexity in One-D Cellular Automata" (Wuensche 1994a) and subsequent
work. The reverse algorithm for generating pre-images, and the Z
parameter which derives from this procedure, are reviewed. The chapter
looks in particular at new measures and results on attractor basins and
how they relate to measures on local dynamics, characterising ordered to
"complex" to chaotic behaviour, where complex behaviour corresponds to
Wolfram's class 4 (Wolfram 1984a). A method is described for
classifying CA rules by a measure of the variance of input-entropy over
time. The method allows glider rules that support coherent interacting
configurations and related complex rules to be found automatically
giving a virtually unlimited sample for further study. Glider dynamics
in CA is of prime interest as an example of self-organisation and
emergence in simple systems, where all aspects of the system can be
Chapters 5 focus on RBN, and develops ideas first presented in the
papers "The Ghost in the Machine" (Wuensche 1993a) and "The Emergence of
Memory" (Wuensche 1994b). Recent collaborative work on RBN as models of
genomic regulatory networks is also described.
The reverse algorithm for generating pre-images in RBN is explained.
The dynamics of RBN and intermediate network architectures are examined, in
particular in the context of memory. The notion of memory far from equilibrium,
categorisation by the roots of subtrees as well as by attractors is proposed.
Algorithms for learning by "sculpting" the basin of attraction field are
described. The inverse problem of finding the set of minimal RBN that will
satisfy a predetermined set of transitions is discussed. Alternative methods
for solving the inverse problem have recently been formulated independently
by Manor Askenazi
(1996) and John Myers (1996). The biological implications
of memory far from equilibrium, and RBN or networks of RBN as the basis of a
neural network model is discussed.
Chapter 5 goes on to discuss RBN as applied to model genomic
regulatory networks, where cell types are explained as attractors or
"frozen skeletons" in the dynamics of the network.
is being used
as a simulation platform for these models, and to extract various measures
distinguishing dynamics between order and chaos. The results and methods
are described. This is work in progress in collaboration with Stuart Kauffman
and others. (Harris et al 1997, Somogyi
and Sniegoski 1996).
The thesis concludes with an overview of future work and open questions.