Complexity International  
 

 

ISSN 1320-0682


Source:   http://www.complexity.org.au/ci/vol06/wuensche/wuensche.html   Received: 01/07/1998
Vol 6:   Copyright 1998   Accepted for publication: 15/10/1998

Discrete Dynamical Networks and their Attractor Basins

Andrew Wuensche
Santa Fe Institute, 1399 Hyde Park Road,
Santa Fe, New Mexico 87501 USA,
Email: wuensch@santafe.edu
WWW: www.santafe.edu/~wuensch/

Abstract:

A key notion in the study of network  dynamics is that state-space is connected into basins of attraction . Convergence in attractor  basins correlates with order -complexity -chaos  measures on space-time patterns. A network's ``memory'', its ability to categorize, is provided by the configuration of its separate basins, trees and sub-trees. Based on computer simulations using the software Discrete Dynamics Lab [19],  this paper provides an overview of recent work describing some of the issues, methods, measures, results, applications and conjectures.

1    Introduction

  Processes consisting of concurrent networks of interacting elements which affect each other's state over time are central to a wide range of natural and artificial systems drawn from many areas of science; from physics to biology to cognition; to social and economic organization; to computation and artificial life; to complex systems  in general. The dynamics of these ``decision making'' networks depends on the connections and update logic for each element, resulting in complex feedback webs that are difficult to treat analytically. Understanding these systems depends on numerical simulations of idealized computer models known as discrete dynamical networks.

Cellular automata  (CA) are a powerful yet simple class of network, characterized by a homogeneous rule and uniform nearest neighbour connections, providing models to study processes in physical systems such as reaction-diffusion [15], and self-organization  by the emergence of coherent interacting structures[18]. By contrast, random Boolean networks  (RBN) provide models for biological systems such as neural  [3] and genetic [11] networks, where connections and rules must be less constrained. In addition, the idealized networks themselves hold intrinsic interest as mathematical/physical systems.

A key notion underlying network behavior is that state-space is organized into a number of basins of attraction, connecting states according to their transitions, and summing up the network's global dynamics, analogous to Poincaré's  ``phase portrait'' which provided powerful insights in continuous dynamics.

 

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Figure 1: A basin of attraction (one of 15) of a random Boolean network (n=13, k=3) shown in figure 15. The basin links 604 states, of which 523 are garden-of-Eden states. The attractor period = 7, and one of the attractor states is shown in detail as a bit pattern. The direction of time is inwards from garden-of-Eden states to the attractor, then clock-wise.

The quality of dynamical behaviour of CA, from ordered to chaotic (1), is reflected by convergence in attractor basins, their characteristic in-degree , which influences the length of transients and attractor cycles. The in-degree of a state is its number of pre-images (predecessors). Bushy subtrees with high in-degree imply high convergence and order. Sparsely branching subtrees imply low convergence and chaos. In the case of RBN, attractor basins reveal how the network is able to hierarchically  categorize state-space into separate basins, trees and sub-trees, the network's ``memory''. Changes to the network's wiring or rules change the memory categories, providing insights into learning[17, 20].

Traditionally, network dynamics has been investigated by running networks forward from many initial states to study space-time phenomenology[15], and for statistical measures on basins of attraction[8]. More recently, exact representations of basins of attraction and sub-trees have become accessible, where algorithms directly compute the pre-images of network states, allowing the network to be run ``backwards'' to disclose all possible historical paths[16, 17, 21]. Based on computer simulations using the software Discrete Dynamics Lab (DDLab)[19], this paper provides an overview of network architecture, the characteristics of space-time patterns, the methods and algorithms for reconstructing basins of attraction, and related parameters, measures, results, applications and conjectures, placing the dynamics along particular trajectories in the context of global dynamics.

2    Network Architecture

  Discrete dynamical networks consist of a set of elements (cells) taking inputs from each other, and changing their cell-state according to some logical function on their inputs. The connectivity   is usually sparse. The cell-state ranges over a discrete alphabet, in this paper just a binary alphabet (0 or 1) is considered. The updating is generally synchronous, though updating sequentially in a preset order or partial order is also of interest. A partial order is a sequence of sets of cells, where updating within each set is synchronous.

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1d: k=0-13. The extra asymmetric cell in even k is on the right. The wiring is shown between two time-steps.

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2d: k=2-13 (k=0-1 as in 1d). Note that k=6 and k=7 define an effectively triangular grid by changing between odd and even rows. The classical von Neumann  and Moore  neighbourhoods are indicated.

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3d: k=6-13 (k=0-5 as in 2d), shown looking up into an axonometric cage.

Figure 2: 1d, 2d and 3d neighbourhood templates defined in DDLab. In 2d and 3d, to maximize symmetry, even k does not include the central target cell.

A CA is a very regular network, sometimes described as an artificial universe with its own physics. Cells take inputs from their nearest (and next nearest) neighbours (local ``wiring'') according to a fixed neighbourhood template, so issues of network geometry and boundary conditions are crucial. The same logical rule is applied everywhere. Figure 2 shows neighbourhood templates for 1d, 2d and 3d as applied in DDLab. An RBN relaxes these constraints, allowing arbitrary ``wiring'' and rules, as in figure 3. The number of input wires available to each cell may also vary. However, an RBN architecture can be biased in countless of ways, described in section 4, to constrain wiring or rules. For example an RBN wiring with a constant rule, or local wiring with mixed rules can be a generalisation of a CA.

The wiring and rules can be tailored to very specific requirements, as in models of neural networks  in the cortex [3]. The wiring can be constrained within a fixed distance from each cell, which confers meaning to network geometry and boundary conditions, whereas with completely arbitrary wiring the geometry just provides a convenient way of representing the network. A rule mix can be constrained to sample just a few rules or rules with a particular bias, as in genetic network models[5].

 

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Figure 3: Network wiring. top left: 1d, k=3, the wiring is shown between two time-steps. top right: 2d, k=5. bottom: 3d, k=7. In RBN, cells anywhere in the network are wired back to each position in the ``pseudo-neighbourhood''.

Hybrid networks can be constructed by putting an RBN within a CA or vice-versa. Networks of networks can be set up, with weak interactions so they perturb each other's dynamics. The functionality for setting up networks in these ways is present in DDLab.

The network parameters can be listed as follows:

size:
The system size n, the number of cells in the network.

connectivity:
The number of input wires per cell k, or the k-mix if the connectivity is not homogeneous. The connectivity is usually sparse, i.e. tex2html_wrap_inline1384 .

neighbourhood:
The neighbourhood template for CA, or the pseudo-neighbourhood for RBN, as in figure 2.

wiring:
For RBN, how each cell is wired relative to its pseudo-neighbourhood.

rule:
The rule for CA, or the rule scheme scheme for RBN. Rules are generally defined as look-up tables.

updating:
The updating, usually synchronous. Alternatively sequential according to a defined order or partial-order.

geometry:
The underlying geometry and boundary conditions, 1d, 2d, 3d, orthogonal or triangular, or some other geometry, for example a hypercube. This is essentially a function of the the neighbourhood template and wiring scheme. The geometry for graphically representing the network may not necessarily correspond to the underlying geometry.

A CA neighborhood, or RBN pseudo-neighborhood, of size k has tex2html_wrap1342 permutations of values. The most general expression of the Boolean function or rule is a lookup table (the rule-table) with tex2html_wrap1342 entries, giving tex2html_wrap1344 possible rules. Sub-categories of rules can also be expressed as simple algorithms, concise AND/OR/NOT logical statements (which could be implemented as combinatorial circuits), totalistic rules[14] or threshold functions.

By convention[14] the rule table is arranged in descending order of the values of neighborhoods, and the resulting bit string converts to a decimal or hexadecimal rule number. For example the k=3 rule-table for rule 30,

tabular145

The rule-table for other k values are set out in a corresponding way. tex2html_wrap_inline1392 rules are referred to by their hexadecimal rule numbers. tex2html_wrap_inline1394 rules are usually referred to by their more familiar decimal rule numbers.

For a given geometry, the behaviour space of CA depends on the size of rule-space, tex2html_wrap1344 , though rule symmetries effectively reduce this number. For example, the tex2html_wrap1346 rules in k=3 rule-space reduce to 88 equivalence classes[16]. The behaviour space of RBN is much greater, taking into account possible permutations of wiring and rule schemes, but there are also RBN equivalence classes relating to these permutations[10]. In general, the number of effectively different RBN of size n cannot exceed tex2html_wrap1347 (see section 9).

3    Trajectories and space-time patterns

  A state of a discrete dynamical network is the pattern of 0s and 1s at a given time-step. A trajectory is the sequence of states at successive time-steps, the systems local dynamics. Examples of 1d, 2d and 3d space-time patterns are shown in figures 4, 5 and 6. A time axis is only possible in representations of 1d or 2d systems. As well as showing cells as white(0) or black(1), an alternative presentation shows cells in colors (or shades) according to their look-up neighbourhood (figure 4). This allows the most frequently occurring colors to be progressively filtered to show up gliders and other space-time structures as in figure 5, which can be done interactively, on-the-fly, in DDLab for any CA. This is an alternative method to the ``computational mechanics'' approach[4].

 

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Figure 4: Space-time patterns of a CA (n=24, k=3, rule 90). 24 time-steps from an initial state with a single central 1. Two alternative presentations are shown. Left: cells by value, white=0 black=1. Right: cells colored (or shaded) according to their look-up neighbourhood. This allows filtering, and improves the clarity of space-time patterns in 2d and 3d.

  tex2html_wrap1442

Figure 5: Space-time patterns of the k=3 rule 54 (n=150) from the same initial state showing interacting gliders , which are embedded in a complicated background. Left: cells by value. Right: cells by neighbourhood lookup, with the background filtered.

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(a) 2d 100x100 triangular grid

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(b) 2d 56x56 +time

  tex2html_wrap1448

(c) 3d 20x20x20

Figure 6: Examples of 2d and 3d CA space patterns. (a) is an evolved time-step of a 2d CA on a k=7 triangular lattice with a reaction-diffusion  rule. (b) is the 2d game-of-Life   on a tex2html_wrap_inline1436 grid, but with a time dimension added, similar to a 1d space-time pattern. The initial state is set with a number of gliders . (c) is a time-step of a 3d k=7 CA with a randomly selected rule and starting from a single central 1.

3.1    Glider dynamics in CA

  A large body of literature is devoted to the study space-time patterns in CA.  ``Glider''  or particle dynamics, where coherent configurations emerge and interact, provide a striking instance of self-organization  in a simple system. Such dynamics are classified as complex,  in contrast to ordered or chaotic[14], a well know example being Conway's 2d ``game-of-Life''[1]. Because glider dynamics is relatively rare in CA rule spaces, much study has focused on the few known complex rules in 1d CA. However, an unlimited source of examples are now available, found by the methods described in sections 3.2 - 3.3.

Gliders are embedded within a uniform or periodic background or domain, and propagate at various velocities up the system's speed of light set by the neighbourhood diameter. Gliders are interpreted as dislocation in the background or as the boundary reconciling two different backgrounds[4, 18]. Gliders may absorb or eject sub-gliders (glider-guns). Compound gliders may emerge made up of sub-gliders re-colliding periodically. Figure 3.1 shows some examples.

Glider dynamics has been interpreted as occurring at a phase transition in rule-space between order  and chaos [9], relative to the rule parameters tex2html_wrap_inline1455 [9] and Z[16] (see section 6.2). Input-entropy  provides a measure on space-time dynamics that allows the automatic classification of rule-space (see below).

>
tex2html_wrap1461

(a)7e8696de

       tex2html_wrap1463

(b)89ed7106

       tex2html_wrap1465

(c)89ed7106

       tex2html_wrap1467

(d)b51e9ce8

       tex2html_wrap1469

(e)b51e9ce8

 

Figure 7: Gliders, glider  guns and compound gliders in k=5 1d CA . (c) is a compound glider made up of two independent gliders locked into a cycle of repeating collisions. (d) is a glider with a period of 106 time-steps. (e) is a compound glider-gun.

3.2    Input entropy

  Keeping track of the frequency of rule-table look-ups (the k-block frequency, or ``look-up frequency'') in a window of time-steps, provides a measure, the variance of input-entropy  over time, which is used to classify 1d CA automatically for a spectrum of ordered, complex and chaotic dynamics[22]. The method allows screening out rules that support glider dynamics and related complex rules, giving an unlimited source for further study. The method also shows the distribution of rule classes in the rule-spaces of varying neighbourhood sizes. The classification produced seems to correspond to our subjective view of space-time dynamics, and to global measures on the ``bushiness'' of typical sub-trees in attractor basins, characterized by the distribution of in-degree sizes in their branching structure.

The look-up frequency can be represented by a histogram (figure 8) which distributes the total of tex2html_wrap_inline1483 lookups among the tex2html_wrap1342 neighbourhoods (shown as the fraction of total lookups), where n=system size, w=the window of time-steps defined and k=neighbourhood size. The Shannon entropy of this frequency distribution, the ``input-entropy'' S, at time-step t, for one time-step (w=1), is given by, tex2html_wrap_inline1497 , where tex2html_wrap_inline1499 is the look-up frequency of neighbourhood i at time t. In practice the measures are smoothed by being taken over a moving window of time-steps (w=10 in figure 8).

 

tex2html_wrap1519

Figure 8: Typical 1d CA Space-time patterns showing ordered, complex and chaotic dynamics (n=150, k=5, rule numbers shown in hex). Alongside each space-time pattern is a plot of the input-entropy, where only complex dynamics (centre) exhibits high variance because glider collisions make new gliders.

Figure 8 shows typical examples of ordered, complex and chaotic dynamics in 1d CA, with input-entropy plots and a snapshot of the lookup frequency histogram alongside. In ordered dynamics the entropy quickly settles at a low value with low or zero variance. In chaotic dynamics the entropy settles at a high value, but again with low variance. Both ordered and chaotic dynamics have low input-entropy variance. By contrast, in complex dynamics the entropy fluctuates erratically both up and down for an extended time, because glider collisions produce new gliders, often via a temporary zone of chaotic dynamics. Complex rules can be recognized by eye, subjectively. Input-entropy variance provides a non-subjective measure for recognizing complex rules automatically.

 

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Figure 9: Entropy-density scatter plot. Input-entropy is plotted against the density of 1s relative to a moving window of time-steps w=10. k=5, n=150. Plots for a number of complex rules from the automatic sample (section 3.3) are show superimposed, each of which has its own distinctive signature, with a marked vertical extent, i.e. high input-entropy variance. About 1000 time-steps are plotted from several random initial states for each rule.

A related method of visualizing the entropy-variance is to plot input-entropy against the density of 1s relative to a moving window of time-steps. Each rule produces a characteristic cloud of points which lie within a parabolic envelope because high entropy is most probable at medium density, low entropy at either low or high density. Each complex rule produces a plot with its own distinctive signature, with high input-entropy variance. Chaotic rules, on the other hand, will give a flat, compact cloud at high entropy (at the top of the parabola). For ordered rules the entropy rapidly falls off with very few data points because the system moves rapidly to an attractor.

3.3    Automatically classifying rule-space

   

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View k5 image in VRML

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View k6 image in VRML

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View image k7 in VRML

Figure 10: top: Classifying a random sample of k=5 rules by plotting mean entropy against standard deviation of the entropy, with the frequency of rules within a tex2html_wrap_inline1528 grid shown vertically. middle and bottom: Equivalent plots for samples of k=6 and 7 rules.

To distinguish ordered, complex and chaotic rules automatically, the mean input-entropy taken over a span of time-steps is plotted against the standard deviation of the input entropy. Figure 10 summarizes how random samples of k=5, 6 any 7 rules where classified by this method. For each rule, the data was gathered from 5 runs from random initial states, for 430 time-steps, discounting the first 30 to allow the system to settle, with w=5 as the size of the moving window of time-steps.

Chaotic rules are concentrated in the top left corner ``tower", ordered rules on the left with lower entropy. Complex rules have higher standard deviation, and are spread out towards the right. There is a fairly distinct boundary between ordered and chaotic rules, but a gradual transition from both towards the complex rules. As the standard deviation decreases glider interactions either become more frequent, transients longer, tending towards chaos, or less frequent, transients shorter, tending towards order. The plots for k=6 and k=7 rules indicate a greater frequency of chaotic rules at the expense of ordered and complex rules for greater k. The decrease in ordered rules is especially marked.

To check whether the expected dynamics (recognized subjectively) corresponds to the measures as plotted, the dynamics of particular rules at different positions on the plots can be easily examined in DDLab, for example with a mouse click on the scatter plot. Preliminary scans confirm that the expected behaviour is indeed found, but further investigation is required to properly demarcate the space between ordered, complex and chaotic rules and to estimate the proportion of different rule classes for different k.

Input entropy is a local measure on the space-time patterns of typical trajectories. The distribution of the rule samples according to these local measures may be compared with global measures on convergence in attractor basins, G-density and the in-degree frequency, described in section 8. Preliminary results indicate a strong relationship between these global measures and the rule sample input-entropy plots.

4    RBN space-time patterns

   

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Figure 11: Space-time patterns for intermediate 1d architecture, from CA to RBN. n=150, k=5, 150 time-steps from a random initial state. (a) Starting off as a complex CA (rule 6c1e53a8 as in figure 8), 4% (30/750) of available wires are randomized at 30 time-step intervals. The coherent pattern is progressively degraded. (b) A network with local wiring but mixed rules, vertical features are evident. (c) RBN, random wiring and mixed rules, with no bias, shows maximal chaotic dynamics.

In contrast to CA, glider dynamics in general cannot occur in RBN because of their irregular architecture. Figure 11 (left) shows glider dynamics degrading as local wiring is progressively scrambled. An alternative order-chaos notion in RBN is the balance between ``frozen'', stabilized, regions and changing regions in the space-time pattern[8]. Stable regions are characteristic of RBN with low connectivity, tex2html_wrap_inline1394 , because rules which induce stability are relatively frequent in these rule-spaces. To induce stability for tex2html_wrap_inline1558 , where chaotic rules become overwhelmingly predominant, biases on rules must be imposed, low tex2html_wrap_inline1455 (see section 6.2) or a high proportion of ``canalizing''  inputs. In a rule's lookup table, an input wire is canalizing if a particular input (0 or 1) determines, by itself, the neighbourhood's output. A rule's degree of canalization can be from 0 to k, for the same output; for the network it is the percentage of all inputs that are canalizing, C. An RBN's order-chaos characteristics, for varying C, are captured by the measures illustrated in figure 12, and described below.

tex2html_wrap1600

  tex2html_wrap1602

Figure 12: Order-chaos measures for a RBN tex2html_wrap_inline1568 , k=5. C = the percentage of canalizing inputs in the randomly biased network. top left: frozen elements that have stabilized for 20 time-steps are shown, 0s-green, 1s red, otherwise white, for C=25% and 52%. top right: the log-log ``damage spread'' histogram for C=52%, sample size about 1000. bottom: the Derrida plot for C=0%, 25%, 52%, and 75%, for 1 time-step, tex2html_wrap_inline1580 =0-0.3, interval = 5, sample for each tex2html_wrap_inline1580 = 25.

The ``Derrida plot''[2] , is analogous to the Lyapunov exponent  in continuous dynamics, and measures the divergence of trajectories based on normalized Hamming  distance H, the fraction of bits that differ between two patterns. Pairs of random states separated by tex2html_wrap_inline1580 , are independently iterated forward by one (or more) time-steps. For a sample of random pairs, the average tex2html_wrap_inline1588 is plotted against tex2html_wrap_inline1580 , and the plot is repeated for increasing tex2html_wrap_inline1580 (from 0 to 0.3 in figure 12). A curve above the main diagonal indicates divergent trajectories and chaos, below -- convergence and order. A curve tangential to the main diagonal indicates a balance. A related measure is the distribution of ``damage spread'' resulting from a single bit change at a random position in a random state, for a sample of random states. The size of damage is measured once it has stabilized, i.e. not changed for say 5 time-steps. A histogram (figure 12) is plotted of damage size against the frequency of sizes. Its shape indicates order or chaos in the network, where a balance between order and chaos approximates to a power law  distribution. Results by these measures for k=5, indicate a balance at tex2html_wrap_inline1596 (see figure 12). There are further measures on basins of attraction as in figure 14.

These methods are applied in the context of RBN models of genetic regulatory networks[8] discussed in section 10. The conjecture  is that evolution maintains genetic regulatory networks marginally on the ordered side of the order-chaos boundary to achieve stability and adaptability in the pattern of gene expression which defines the cell type[5].

5    Basins of Attraction

   

tex2html_wrap1634
For a network size n, an example of one of its states B might be tex2html_wrap1610 . State-space is made up of all tex2html_wrap_inline1616 states, the space of all possible bitstrings or patterns.
tex2html_wrap1634
Part of a trajectory in state-space, where C is a successor of B, and A is a pre-image of B, according to the dynamics of the network.
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The state B may have other pre-images besides A, the total is the in-degree. The pre-image states may have their own pre-images or none. States without pre-images are known as garden-of-Eden states.
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Any trajectory must sooner or later encounter a state that occurred previously - it has entered an attractor cycle. The trajectory leading to the attractor a transient. The period of the attractor is the number of states in its cycle, which may be only just one -- a point attractor.
tex2html_wrap1634
Take a state on the attractor, find its pre-images (excluding the pre-image on the attractor). Now find the pre-images of each pre-image, and so on, until all garden-of-Eden states are reached. The graph of linked states is a transient tree rooted on the attractor state. Part of the transient tree is a subtree defined by its root.
tex2html_wrap1634
Construct each transient tree (if any) from each attractor state. The complete graph is the basin of attraction. Some basins of attraction have no transient trees, just the bare ``attractor''.
tex2html_wrap1634
Now find every attractor cycle in state-space and construct its basin of attraction. This is the basin of attraction field containing all tex2html_wrap_inline1616 states in state-space, but now linked according to the dynamics of the network. Each discrete dynamical network imposes a particular basin of attraction field on state-space.



Figure 13: State space and basins of attraction.

The idea of basins of attraction in discrete dynamical networks is summarized in figure 13. Given invariant network architecture and the absence of noise, a discrete dynamical network is deterministic, and follows a unique (though in general, unpredictable) trajectory from any initial state. When a state that occurred previously is revisited, which must happen in a finite state-space, the dynamics becomes trapped in a perpetual cycle of repetitions defining the attractor (state cycle) and its period (minimum one, a stable point).

These systems are dissipative. A state may have multiple ``pre-images'' (predecessors), or none, but just one successor. The number of pre-images is the state's ``in-degree''. In-degrees greater than one require that transient states exist outside the attractor. Tracing connections backwards to successive pre-images of transient states will reveals a tree-like topology where the ``leaves'' are states without pre-images, known as garden-of-Eden  states. Conversely, the flow in state-space is convergent. Measures of convergence are G-density, the fraction of states that are garden-of-Eden, and the distribution of in-degrees, described in section 8. The set of transient trees rooted on the attractor is its basin of attraction (figure 1). The local dynamics connects state-space into a number of basins, the basin of attraction field, representing the systems global dynamics (figure 15).

6    Computing Pre-images

   

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Figure 14: Statistical data on attractor basins for a large network; a 2d RBN 20 tex2html_wrap_inline1639 20, k=5, with fully random wiring and a fraction of canalizing inputs C=50%. The histogram shows attractor types and the frequency of reaching each type from 12,360 random initial states, sorted by frequency. 46 different attractors types where found, their periods ranging from 4 to 102, with average transient length from 21 to 113 time-steps. The frequency of arriving at each attractor type indicates the relative size of the basin of attraction.

Attractor basins are constructed with algorithms that directly compute the pre-images of network states[16, 17, 21]. This allows the network's dynamics, in effect, to be run backwards in time. Backward trajectories will, as a rule, diverge. Different reverse algorithms apply to networks with different sorts of connectivity. The most computationally efficient algorithm applies to 1d networks with local wiring, taking advantage of the regularity of connections. The wiring must be uniform, as for 1d CA, but the network may have a mix of rules. Analogous algorithms could be derived for 2d and 3d networks, but have not been implemented. An alternative algorithm is required for RBN with their non-local connections and possibly mixed k. This algorithm also applies to CA of any dimension or geometry, as CA are just a sub-class of RBN.

Provided tex2html_wrap_inline1384 , these methods are in general orders of magnitude faster than the brute force method (section 9.1), constructing an exhaustive map resulting from network dynamics, a method which rapidly becomes intractable with increasing network size and so is limited to very small systems. However, the exhaustive method may be applied to all types of network, and also allows the attractor basins of random maps to be constructed, as described in section 9. The agreement of these three independent methods, and other checks, give considerable confidence in the accuracy of the pre-image computations.

Some basic information on attractor basin structure can be found by statistical methods, first applied by Walker[13], as shown in figure 6. These are also implemented in DDLab and are appropriate for large networks. Trajectories are run forward from many random initial states looking for a repeat in the network pattern to identify the range of attractor types reached. The frequency of reaching a given attractor type indicates the relative size of the basin of attraction, and other data are extracted such as the number of basins, and the length of transients and attractor cycles.

 

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Figure 15: The basin of attraction field of a random Boolean network (n=13, k=3). The tex2html_wrap1637 states in state space are organized into 15 basins, with attractor periods ranging between 1 and 7. The number of states in each basin is: 68, 984, 784, 1300, 264, 76, 316, 120, 64, 120, 256, 2724, 604, 84, 428. Figure 1 shows the arrowed basin in more detail. Right: the network's architecture, its wiring/rule scheme.

tabular467

6.1    The CA reverse algorithm

  Consider a 1d CA of size n (indexed tex2html_wrap_inline1659 ) and neighbourhood k. To find the all pre-images of a state A, let P be a ``partial pre-image'' where at least k-1 continuous bits (on the left) up to and including tex2html_wrap_inline1669 , are known. Now find the next unknown bit to the right, tex2html_wrap_inline1671 , consistent with the rule-table. ( tex2html_wrap_inline1673 indicates known, tex2html_wrap_inline1675 unknown, bits),

tabular474

If k=3 (for example), the bitstring tex2html_wrap_inline1703 corresponds to two neighbourhood entries in the rule-table. When their outputs, tex2html_wrap_inline1705 and tex2html_wrap_inline1707 , are compared with each other and with tex2html_wrap_inline1709 there are three possible consequences. The permutation is either deterministic, ambiguous or forbidden.

  1. deterministic: if tex2html_wrap_inline1711 , then tex2html_wrap_inline1671 is uniquely determined, as there is only one valid neighbourhood with the output tex2html_wrap_inline1709 .
  2. ambiguous: if tex2html_wrap_inline1717 , then both 0 and 1 are valid solutions for tex2html_wrap_inline1671 . The partial pre-image must be duplicated, with tex2html_wrap_inline1721 in one version and tex2html_wrap_inline1723 in the other.
  3. forbidden: if tex2html_wrap_inline1725 , then tex2html_wrap_inline1671 has no valid solution.

If forbidden (3) the partial pre-image P is rejected. If deterministic or ambiguous (1 or 2) the procedure is continued to find the next unknown bit to the right. However, in the ambiguous case (2), both alternative partial pre-images must be continued. In practice one is assigned to a stack of partial pre-images to be continued at a later stage. As the procedure is re-applied to determine each successive unknown bit towards the right, each incidence of ambiguous permutations will require another partial pre-image to be added to the stack. Various refinements can limit this growth.

The procedure is continued to the right to overlap the assumed start string, to check if periodic boundary conditions are satisfied; if so the the pre-image is valid. The procedure is re-applied to each partial pre-image taken from the partial pre-image stack, starting at the first unknown cell. Each time an ambiguous permutation (2) occurs a new partial pre-image must be added to the stack, but the stack will eventually be exhausted, at which point all the valid pre-images containing the assumed start string will have been found. The procedure is applied for tex2html_wrap_inline1731 start strings, assuming the different possible values of the first k-1 bits. The reverse algorithm is applied from left to right in DDLab, but is equally valid when applied from right to left. Examples are given in [16, 21].

6.2    The Z parameter

  A by product of the CA reverse algorithm is the probability of the next unknown bit being deterministic (section 6.1(1)). Two versions of this probability are calculated from the rule-table. tex2html_wrap_inline1737 for the reverse algorithm applied from left to right, and tex2html_wrap_inline1739 for the converse. The Z parameter is the greater of these values. For Z=1 it can be shown[16] that for any system size n, the maximum in-degree, tex2html_wrap_inline1747 , because the next unknown bit is always uniquely determined, so the assumed start string of length k-1 may generate at most tex2html_wrap_inline1731 pre-images. If only one of tex2html_wrap_inline1737 or tex2html_wrap_inline1739 =1, tex2html_wrap_inline1757 , because at least one assumed start string must be forbidden (section 6.1(3)). At the other extreme, for Z=0, all state space converges on the state all-0s or all-1s in one step. For high Z, low in-degree (relative to system size n) is expected in attractor basins, growing at a slow rate with respect to n. Conversely, for low Z, high relative in-degree is expected growing quickly with respect to n. High Z predicts low convergence and chaos, low Z predicts high convergence and order.

The tex2html_wrap_inline1775 neighborhoods of size k, each indexed tex2html_wrap_inline1779 , each have an output T (0 or 1) which makes up the rule-table (section 2), and may be expressed as tex2html_wrap_inline1783 . To calculate tex2html_wrap_inline1737 from the rule table, let tex2html_wrap_inline1787 be the count of rule-table entries belonging to deterministic pairs, such that,

tex2html_wrap_inline1789 and tex2html_wrap_inline1791 (not T)

The probability that the next bit is determined because of the above is given by, tex2html_wrap_inline1795 . This is a first approximation of tex2html_wrap_inline1737 .

Let tex2html_wrap_inline1799 be the count of rule-table entries belonging to deterministic 4-tuples (where `` tex2html_wrap_inline1675 '' may be 0 or 1), such that,

tex2html_wrap_inline1803 and tex2html_wrap_inline1805

The probability that the next bit is determined because of the above is given by, tex2html_wrap_inline1807 . This count is repeated if necessary for deterministic 8-tuples where tex2html_wrap_inline1809 , 16-tuples, 32-tuples, tex2html_wrap_inline1811 up to the special case of just one tex2html_wrap_inline1775 -tuple which occupies the whole rule-table. These are are independent non-exclusive probabilities that the next bit is determined. The union of the probabilities tex2html_wrap_inline1815 , is given by the following expression (the order of the probabilities makes no difference to the result),

eqnarray538

which simplifies to,

displaymath547

and may be expressed as(2) tex2html_wrap_inline1817 where tex2html_wrap_inline1819 , and tex2html_wrap_inline1821 = the count of rule-table entries belonging to deterministic tex2html_wrap_inline1823 -tuples. A converse procedure gives tex2html_wrap_inline1739 , and the Z parameter = the greater of tex2html_wrap_inline1737 and tex2html_wrap_inline1739 . Examples are given in [16, 21].

By virtue of being a convergence parameter, Z is also an order-chaos parameter varying from 0(order) - 1(chaos). Z can be compared with Langton's[9] well known tex2html_wrap_inline1455 parameter (3). tex2html_wrap_inline1455 is an order-chaos parameter for CA which may have values greater than binary, and measures the density of ``non-quiescent'' outputs in a rule-table, so for just binary CA, tex2html_wrap_inline1855 where c=the count of 1s a rule-table on k inputs. tex2html_wrap_inline1455 varies between 0 (order) - 0.5 (chaos) - 1 (order). To allow Z and tex2html_wrap_inline1455 to be compared, a normalized version of binary tex2html_wrap_inline1455 is defined[16], tex2html_wrap_inline1869 where tex2html_wrap_inline1871 is the count of 0s or 1s in the rule-table, whichever is less. tex2html_wrap_inline1875 then varies from 0 (order) - 1 (chaos) just as Z.

Plots of G-density against both the tex2html_wrap_inline1875 and Z parameters, showing the discrepancies as well as similarities, are shown in figure 6.2, for the 256 k=7 totalistic rules, which reduce to 136 non-equivalent rules in 72 clusters (having equal tex2html_wrap_inline1875 and Z). Points plotted in the top right corner of the tex2html_wrap_inline1875 graph represent tex2html_wrap_inline1875 values that do not correspond to behaviour as expected.

 

tex2html_wrap1905

Figure 16: G-density against both tex2html_wrap_inline1875 and Z for the set of k=7 totalistic rules, n=16, for tex2html_wrap_inline1903 . The complete basin of attraction field was generated for each rule and garden-of-Eden states counted.

6.3    The RBN reverse algorithm

   

tex2html_wrap1976

Figure 17: Computing RBN pre-images. The changing size of a typical partial pre-image stack at successive elements. n=24, k=3.

Consider an RBN of size n. Find all pre-images of a state A, tex2html_wrap_inline1916 tex2html_wrap_inline1918 ). Each network element tex2html_wrap_inline1709 , has a pseudo-neighbourhood size tex2html_wrap_inline1922 (assuming a mixed k network), indexed tex2html_wrap_inline1926 , a wiring scheme tex2html_wrap_inline1928 , tex2html_wrap_inline1930 ), where tex2html_wrap_inline1932 is a number between n-1 and 0, the position of the wire connection from the jth branch of the pseudo-neighbourhood, and a rule-table tex2html_wrap_inline1938 .

To find the all pre-images of A, let P, tex2html_wrap_inline1944 , be a candidate pre-image consisting of empty network elements, as yet unassigned to either 0 or 1. Starting with an element of A, tex2html_wrap_inline1709 , assign bits from all valid pseudo-neighbourhoods in the rule-table tex2html_wrap_inline1938 , i.e. that are consistent with tex2html_wrap_inline1709 , to separate copies of P according to the wiring scheme tex2html_wrap_inline1928 . As there will be a mix of 0s and 1s in tex2html_wrap_inline1938 , only some of the tex2html_wrap_inline1962 possible pseudo-neighbourhoods will be valid. This will produce a stack of ``partial pre-images'' with some bits allocated and the remainder empty.

Now repeat the procedure for another element of A, say tex2html_wrap_inline1966 , but this time independently for each partial pre-image previously created. If the allocation of a bit to a given partial pre-image conflicts with the bit already assigned, then the partial pre-image is rejected. Otherwise, the partial pre-image is added to the next generation of partial pre-images in a new stack. The allocation will be valid if it is made to an ``empty'' element, or to an allocated element with an equal bit. Valid allocation increases the size of the partial pre-image stack, conflicts reduce the size of the stack.

This procedure is repeated in turn for the remaining network elements of A. If the stack size is reduced to zero at any stage A has no pre-images. The algorithm works for any ordering of elements in A, though to minimizes the growth of the partial pre-image stack, the order should correspond to the greatest overlap of wiring schemes. The changing size of the stack at successive elements can be displayed in DDLab, an example is shown in figure 17. When the procedure is complete, the final pre-image stack may still have empty network elements, which did not figure in any wiring scheme. These are duplicated so that all possible configurations at empty element positions are represented. The resulting pre-image stack is the complete set of pre-images of A without duplication.

The reverse algorithm for RBN works for networks with any degree of intermediate architecture between RBN and CA, including CA of any dimension. More detailed explanations of the algorithm are given in [17, 21].

7    Constructing and portraying attractor basins

  To construct a basin of attraction containing a particular state, the network is iterated forward from the state until a repeat is found and the attractor identified. The transient tree (if it exists) rooted on each attractor state is constructed in turn. Using one of the reverse algorithms, the pre-images of the attractor state are computed, ignoring the pre-image lying on the attractor itself. Then the pre-images of pre-images are computed, until all ``garden-of-Eden'' states have been reached.

In a similar way, just a subtree may be constructed rooted on a state. Because a state chosen at random is very likely to be a garden-of-Eden state, it is usually necessary to run the network forward by at least one time-step, and use the state reached as the subtree root. Running forward by more steps will reach a state deeper in the subtree so allow a larger subtree to be constructed.

For CA, a considerable speedup in computation is achieved by taking advantage of ``rotational symmetry''[16], a property of the regularity of CA and periodic boundary conditions, resulting in equivalent subtrees and basins.

Attractor basins are portrayed as state transition graphs, vertices (nodes) connected by directed edges. States are represented by nodes, by a bit pattern in 1d or 2d, or as the decimal or hex value of the state. In the graphic convention[16, 19], the length of edges decreases with distance away from the attractor, and the diameter of the attractor cycle approaches an upper limit with increasing period. The direction of edges (i.e. time) is inward from garden-of-Eden states to the attractor, and then clockwise around the attractor cycle, as shown in figure 1. Typically, the vast majority of states in a basin of attraction lie on transient trees outside the attractor, and the vast majority of these states are garden-of-Eden states.

8    Attractor basin measures

   

tex2html_wrap2030

Figure 18: The G-density plotted against system size n, for the ordered, complex and chaotic rules shown in figures 8 and 19. The the entire basin of attraction field was plotted for n=7 to 22, and garden-of-Eden states counted. The relative G-density and rate of increase with n provides a simple measure of convergence.

Measures on attractor basins include the number of attractors, attractor periods, size of basins, characteristic length of transients and the characteristic branching within trees. The last in particular gives a good measure of the convergence of the dynamical flow in state-space, where high convergence indicates ordered, and low convergence indicates chaotic dynamics.

The simplest measure that captures the degree of convergence is the density of garden-of-Eden states[18], G-density, counted in attractor basins or sub-trees, and the rate of increase of G-density with n as shown in figure 18. A more comprehensive measure is the in-degree frequency distribution, plotted as a histogram. The in-degree of a state is the number of its immediate pre-images. This can be taken on a basin of attraction field, a single basin, a subtrees, or on just part of a subtree for larger systems. Subtrees are portrayed as graphs showing trajectories merging onto the sub-tree root state.

Examples of in-degree histograms for typical sub-trees of ordered, complex, and chaotic rules are shown in figure 19. The horizontal axis represents in-degree size, from zero (garden-of-Eden states) upwards, the vertical axis represents the frequency of the different in-degrees. The system size n=50 for the complex and chaotic rules. For very ordered rules in-degrees become astronomical. The ordered rule shown is only moderately ordered, however the system size was reduced to n=40 to allow easier computation.

From the preliminary data gathered so far, the profile of the in-degree histogram for different classes of rule is as follows:

Ordered rules: Very high garden-of-Eden frequency and significant frequency of high in-degrees. High convergence.

Complex rules: Approximates a power law distribution. Medium convergence.

Chaotic rules: Lower garden-of-Eden frequency compared to complex rules, and a higher frequency of low in degrees. Low convergence.

tex2html_wrap2032 tex2html_wrap2034
Figure 19a: Ordered dynamics. Rule 01dc3610, n=40, Z=0.5625, tex2html_wrap_inline1875 =0.668. right: The complete sub-tree 7 levels deep, with 58153 nodes, G-density=0.931.

tex2html_wrap2036 tex2html_wrap2038
Figure 19b: Complex dynamics. Rule 6c1e53a8, n=50, Z=0.727, tex2html_wrap_inline1875 =0.938. right: The sub-tree, stopped after 12 levels, with 144876 nodes, G-density=0.692.

tex2html_wrap2040 tex2html_wrap2042
Figure 19c: Chaotic dynamics. Rule 994a6a65, n=50, Z=0.938, tex2html_wrap_inline1875 =0.938. right: The sub-tree, stopped after about 75 levels, with 9446 nodes, G-density=0.487.

 

9    Random Maps

  The attractor basins of discrete dynamical networks can be put into the wider context of random graph  theory. CA belong to the set of RBN which in turn belong to the set of random directed graphs with out degree one, known as random maps. This is a mapping of the Boolean hypercube of sequences of length n (i.e. a set tex2html_wrap_inline2050 of size tex2html_wrap_inline1616 comprising all binary strings of length n), a mapping from tex2html_wrap_inline2056 . The structures found in random maps correspond to those in the attractor basins of discrete dynamical networks, where each separate component of the graph is made up of trees rooted on just one closed cycle. The structures can be computed in DDLab just as the attractor basins of CA or RBN.

 

tex2html_wrap2100

Figure 20:The basin of attraction field of a typical unbiased random map, n=12. The tex2html_wrap_inline2060 states in state space are connected into 9 basins of attraction. The period (p) and size (s) of the biggest three (top row), including their percentage of state-space, are as follows: (1) p=118 s=3204=78.2%. (2) p=20 s=599=14.6%. (3) p=32 s=194=4.74%. The field's G-density=0.37, this is a low value implying chaotic dynamics.

A random map can be constructed by assigning a successor to each state in state-space, i.e. independently assign one successor (or image) tex2html_wrap_inline2080 also belonging to tex2html_wrap_inline2050 , chosen at random (or with some bias) to each element tex2html_wrap_inline2084 of the set tex2html_wrap_inline2050 . There are tex2html_wrap1347 possible mappings. The mapping is represented below as tex2html_wrap_inline1616 pairs of strings (or states in state-space), where each image tex2html_wrap_inline2080 represents a possibly different member of the set tex2html_wrap_inline2050 .

displaymath2046

The list of images is likely to contain repeats, and if so some other members of tex2html_wrap_inline2050 must be missing from the list. Transitions to some arbitrary element tex2html_wrap_inline2096 may thus be one-to-one or many-to-one, or may not exist. The latter is a garden-of-Eden state in the terminology of discrete dynamical networks. A representation of the particular mapping may be drawn as a basin of attraction field or fragment thereof just as for CA or RBN, and will have the same general topology of trees rooted on attractor cycles, as shown in figure 20.

Random maps provide the most general context for a discrete dynamical system and are equivalent to a fully connected RBN where k=n, (the neighbourhood = the network size). This follows because each cell in the RBN can be assigned an arbitrary output for any network state.

9.1    The Random Map Reverse Algorithm

  The ``brute force'' reverse algorithm for finding the pre-images of states in random maps can also be applied to discrete dynamical networks, RBN and CA. The method depends on first constructing an exhaustive mapping tex2html_wrap_inline2056 . For discrete dynamical networks, the mapping is defined by iterating the network forward by one step from every state in state-space and filling in the image list accordingly. A list of tex2html_wrap_inline1616 pairs, each state and its image (successor), is held in a data structure. The pre-images of an arbitrary state S are found by scanning the image list; any occurrence of S in the list gives a pre-image, the state paired with S. If S does not occur in the list it has no pre-images, a garden-of-Eden state.

10    Biological networks

  Genetic regulatory networks have been thought of as discrete dynamical networks, to explain how gene expression is able to settle into a number of distinct stable patterns or cell types, despite the fact that all eukaryotic cells  in an organism carry an identical set of genes[5, 7, 11, 23]. The gene expression pattern of a cell needs to be stable but also adaptable. Section 4 described biases to RBN to achieve such a balance, and related measures.

Cell types have been interpreted as the separate attractors or basins of attraction into which network dynamics settles from various initial states. Trajectories leading to attractors are seen as the pathways of differentiation.  The attractor basins in RBN are idealized models for the stability of cell types against mutations, and also perturbations of the current state of gene activation. Figure 21 illustrates both effects. If a particular reference state (pattern of gene activation) undergoes a 1 bit perturbation, the dynamics may return to the same subtree, the same basin, or it may be flipped to another basin, a different cell type. In this case the basin of attraction field remains unchanged. Alternatively, the network itself my undergo a mutation (in the genotype), resulting in an an altered basin of attraction field (the phenotype).

The examples in figure 21 are small so that the pattern at each node can be shown. Larger networks are affected in analogous ways. The consequences of a one bit mutation has a relatively smaller effect with increasing network size. However, a particular one bit mutation may cause drastic consequences whatever the size, such as breaking an attractor cycle. The consequences of moving a connection wire is usually greater than a one bit mutation in a rule.

10.1    Memory

  Attractors classify state-space into broad categories, the network's ``content addressable'' memory  in the sense of Hopfield[6].  Furthermore, state-space is categorized along transients, by the root of each subtree forming a hierarchy of sub-categories. This notion of memory far from the equilibrium  condition of attractors greatly extends the classical concept of memory by attractors alone[17, 20].

It can be argued that in biological networks such as neural networks in the brain or networks of genes regulating the differentiation and adaptive behavior of cells, attractor basins and subtrees, the network's memory, must be just right for effective categorization. The dynamics need to be sufficiently versatile for adaptive behavior but short of chaotic to ensure reliable behavior, and this in turn implies a balance between order and chaos in the network.

A current research topic, known as the ``inverse problem'', is to find ways to deduce network architecture from usually incomplete data on transitions, such as a trajectory. This is significant in genetics, to infer the genetic regulatory network (modeled as RBN) from data on successive patterns of gene expression in the developing embryo[12].  In pattern recognition and similar applications in the area of artificial neural networks, solutions to the inverse problem would provide ``learning'' methods for RBN to make useful categories[17, 20].

 

tex2html_wrap2121

Figure 21:The basin of attraction field of (a) The RBN (n=6, k=3) as defined in the table (above), and (b) the RBN following a 1 bit mutation to one of its rules. Some differences in the fields are evident. The result or a 1 bit perturbation to a reference state of all 1s (rs) is indicated by its 1 bit mutants (m).

tabular769

11    Conclusion

  Important insights may be gained by considering network dynamics in the context of attractor basins. Some methods of achieving this have been presented, including parameters and measures on particular trajectories that may be related to those on global dynamics. It is hoped that these methods may provide a basis for future research, both in theory and applications, in the many areas of complex systems where network dynamics plays a central role.

Acknowledgments: Thanks to Cosma Shalizi for suggestions, and to the many people who have contributed to this work over the years, notably Chris Langton and Stuart Kauffman.

Footnotes

(1)
The notion of ``chaos''  is used here by analogy only to its meaning in chaos theory, although there are many common properties, for example sensitivity to initial conditions.
(2)
Acknowledgment and thanks to Guillaume Barreau and Phil Husbands at COGS, Univ. of Sussex, for deriving this expression.
(3)
Other versions of binary tex2html_wrap_inline1455 are ``internal homogeneity'' introduced earlier by Walker[13], and the P parameter, applied for RBN, which varies between 0.5 (chaos) - 1 (order). tex2html_wrap_inline1845 where tex2html_wrap_inline1847 is the count of 0s or 1s in the rule-table, whichever is more, tex2html_wrap_inline1851 . A number of alternative order-chaos parameters have also recently been proposed, for example by Marty Zwick and Burton Voorhees.
 

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