PERGAMON

COMPUTER MODELLING

Mathematical and Computer Modelling 30 (1999) 35-59 www.elsevier.nl/locate/mcm

Morphology of Patterns of Lattice Swarms: Interval Parameterization

A. ADAMATZKY, C. MELHUISH AND 0. HOLLAND Intelligent Autonomous Systems Laboratory

DuPont Building, University of the West of England Renchay Campus, Coldharbour Lane, Bristol BS16 lQY, UK

<Andrew.Adamatzky><Chria.Melhuish><Owen.Holland>@nwe.ac.uk

(Received and accepted April 1999)

Abstract-This model deals with collectives of mobile agents (finite automata) that move on a twodimensional lattice in discrete time. In every trial, all automata start their evolution at the same lattice node. Every automaton moves from its current node to one of the randomly chosen neighbours if there is another automaton at the same node or if the number of other automata in the neighbourhood belongs to some specified interval of integers. This interval is referred to an interval of activation. All agents find their appropriate positions and stop. The stationary global pattern of resting agents is eventually formed. Such patterns form a key subject of the paper. To group all intervals of activation onto different classes based on the morphological claemcterdstics of the classes is a main task of the first part of the paper. The rest of the paper is devoted to investigation concerning the complete consistent parameterisation of the pattern formation rules of lattice swarm. @ 1999 Elsevier Science Ltd. All rights reserved.

Keywords-Lattice swarm, Pattern formation, Collectives of automata, Classification of the local rulee, F&action-diffusion, Nonlinear systems.

1. INTRODUCTION

Thii paper deals with the question: How are structures formed and in what parametric domain will they be formed? The question became a topic of great importance in many fields of natural

sciences: chemistry, physics, and biology (see, e.g., [l-4]). Usually investigations are focused on explaining such phenomena as how a given set of patterns in certain systems could be gener- ated and how the underlying developmental mechanisms functioned. The labyrinthine patterns, being especially aesthetically attractive, are the most frequent phenomena of a whole range of

distributed systems (see, e.g., [5-g]). With time, it became clear that pattern formation is a key mechanism of various practical realisations of smart matters [lo], collective robotics [11,12], nanotechnology [13,14], and distributed building [15-B]. In the case of potential practical imple- mentation, when or even where structure is formed, becomes more important than how it does.

A conventional mathematical analysis loses most of its power when matter is discrete (as it is in

This work wae supported by Hewlett Packard Laboratories, Bristol, under the External Research Programme. The work is a part of the University of the West of England Internal Project on the control of robot navigation by nonlinear media.

08957177/99/$ - see front matter. @ 1999 Elsevier Science Ltd. All rights reserved. Typ-t by 4&-W PII: SO8957177(99)00163-6

36 A. ADAMATZKY et al.

the case of formation of nano, meso, and macro structures), and therefore, computer experiments offer the only remaining instrument for the investigation of the subject.

The following personal motivations stimulated the present investigation: parameterization of excitation dynamics in discrete lattices and lattice swarm [19,20], automata models of morpho- genesis [21] and morphogenesis-like algorithms [22], and spatial sorting of physical objects in collective robotics [l&12].

In Section 2 of the paper, we formulate the models of lattice swarms and settle the global parameters (convergence time, putterrz she, and order) to aid visual classification of the patterns formed in the evolution of the swarms. The classification of the rules from the point of view of the morphology of the generated patterns is provided in Section 3. Section 4 gives the correspondences between the morphologically classified rules and the integral dynamics of the swarms. This

focuses on the issue of what degree of agent activity may be responsible for the destruction of a structure-the problem is analysed in Section 5. In Section 6, we discuss the possibility of the single parameterization of pattern formation; the threshold of activation and number of forbidden

configurations are considered as candidates. We provide some analogies and reveal the ways for further investigations in Section 7.

2. BASICS

2.1. The Model

Let ll be a set of m uniform mobile agents in which each incorporates a finite state control

device. The agents move at random on the 2D lattice IL in discrete time. At every time step, automaton i can move or not move to one of the eight neighbouring nodes of u(c~) relative to its node ci. The neighbourhood u(c~) = {V E lb : ]‘u - ce]~, = 1) of the node ci looks like the

following. 0 0 0 0 ci 0 0 0 0

The agent moves if there is another agent at the same node or if the number of agents in u(c~) lies in some specified interval. At the beginning of the trial, all agents are at the same node: c*:

v’iElI:c~=c*ElL. An agent moves if, after checking its neighbourhood, it finds that its node is occupied by

another agent or if the number of the occupied neighbouring nodes belongs to some specified interval [or, 021, 1 5 81 I 0s 5 8. The minimal low and maximal upper boundaries of the

activation interval are determined by a size of u(c:): eight lattice nodes are scanned by an agent to count the number of neighbouring agents.

In some versions of the model, agents do not check their neighbourhood regularly. In this case,

agents may not move even if its own node is occupied by one or more other agents. When agent i at node ci decides to change its position, it walks to a randomly chosen neigh-

bouring node. It does so if it decides to check its neighbourhood, and one of the following

situations takes place.

1. There is another automaton j # i at the same node, c: = ci. 2. The number of nonempty elements of I lies in the interval [Bi, 021.

In this paper, we consider three versions of the lattice swarm which are distinct in how often an agent checks the number of other agents in its vicinity. They are

l serial swarm, S-swarm, where agents checks their neighbourhoods in a cycle of period m, i.e., an agent i checks its neighbourhood every such step t such that i = t modm, i = 0 ,..., m-l,andt=O,l,...,

l parallel swarm, P-swarm, where all agents check their neighbourhoods every step, and

37

t

0 4

1 0

2 0

3 0

4

5 i

6 0

7 0

8 0

9

10 i

11 0

sswarm

i 0

O :

0 0 0 0

0

0

0

; 0 0 0 il

i 0 0 0 0 0

0 i

;

0

0 0

0 0 0 :

; 0 0 0 0 0 0

Interval Parameterization

P-swarm A-swarm

Figure 1. Examples of activity of swarm of five agents during the first 12th step of the evolution. Agents sre named by integers, 0. . + 5, from left to right; time, 0. . . 11, goes up down.. An agent of A-swarm is activated at step t with the probability 0.3. The symbol J at the intersection of ith column and tth row means that agent i

decides to check its neighbourhood at step t.

l some kind of stochastically activated swarm, A-swarm, where each agent checks its neigh- bourhood every time step with some specified probability.

Some examples of the activities in the classes of swarm are shown in Figure 1. We represent nodes of IL by their integer coordinates and use additional notations

l ai = j{j E I[ : ci = $}I is the number of agents at the node ci,

l /3: = l{j E I[ : ci E u(c:)}~ is the number of agents at the nodes of u(c~), l d is a random variable from j-1,0,1} x (-l,O, l}, i.e., the vector of translation, l C(a:, /3,“, 131, 0s) is a decision function (it will be explained below).

The behaviour of an agent i in all three models is governed by the equation of motion

ct+’ = c: + da c (a:, /3;, &,8,) .rj.

The function 7 is responsible for the decision to check the neighbourhood, whereas the func-

tion <(a) participates in the final decision to move. The function C(e) is defined below:

c (4, P,“, b, 0,) = 1, if o: > 0 or pi” E [e,,e,]; C (c&p,“, or, e2) = 0, otherwise.

Agents of all classes use the same function c(e).

The function 7 is a key item which makes a difference between three classes of lattice swarm (the importance is discussed later):

0 P-swarm: r) is a constant 1, 0 S-swarm: r] = q(i,t) = 1, if i = tmodm; ~(i, t) = 1, otherwise, l A-swarm: q = p, where p is a random variable that takes a value of 1 with probability

E 5 p, < 1, e is very small, and it takes value of 0 with probability 1 - p,; E = 0.001 in

computer experiments.

The more pP, the more agents decide to check their neighbourhood, and possibly, change their positions at each time step. We can say that agents have random independent delays of switching. There is, therefore, a probabilistic continuum between S-swarm and P-swarm, where S-swarm is at one end of the continuum and P-swarm is at the other end; A-swarm occupies everything between them. In the next section of the paper, we will focus mainly on S- and P-swarms, since

38 A. ADAMATZKY et al.

they are extreme cases of the activity scale of individual agents. The A-swarm’ will be considered later in connection with “degree of order”-“degree of activity” dependences.

A small remark concerning the boundaries of [c9i, &] should be made. Actually, the range of 0 5 81 5 82 < m will also be correct because several agents can be at the same node of the lattice. However, for 81 = 0, the automata pool will never be at rest because every automaton will

change its position every step of the evolution time. Therefore, we made 01 2 1. The reason for choosing the upper boundary 8 is also made explicit. If there are more than eight agents in the neighbourhood of another agent then at least two of the neighbouring agents occupy the same lattice node and they will change their positions at the next step of simulation time. Only the

condition 1 5 01 5 82 5 8 guarantees that every node contains at most one agent in the final stationary configuration of a swarm.

2.2. Convergence Time

For all S-, P-, and A-classes of swarm, a condition is reached when there is no further movement of any agent, i.e., 3t, < cmVi E IIVt 2 t, : ct = c?+l. That is, after some step t, of the evolution, , I all automata will be at rest. We will call t, the convergence time. The convergence time is finite, due to finite m, and it is determined by the particular values of 81 and 02, and obviously, the size m of a swarm.

Figure 2 illustrates the process of convergence for different rule classes of S-swarm.

em

200

0 0 2al400 so0

TYe lcal lam 1400 lml

Figure 2. Examples of “activity” of automata in the evolutions of swarms from different clsssea. The vertical axis represents the number of automata which change their positions at current step of the evolution. Time is measured in the horisontal axes. The labels indicate the rule classes which will be discussed later.

2.3. .Morphological Characteristics

A morphology of the stationary rest configurations of swarm agents is one of the main subjects of the paper. Hereinafter, we use the term pattern which refers to the final mapping of the stationary agents on the nodes of the lattice. Formally, the pattern is ‘X? : L + (0, l}, where for every CC E L T(z) = 1, if there is an agent i such that c,” = 5 for t > t, and T(z) = 0, otherwise.

Here we deal mainly with the morphological properties of the patterns and try to split the set of all possible intervals 1 5 81 5 02 5 8 onto the equivalence classes with respect to the morphological characteristics of the patterns they generate in their evolution.

The morphological classification of patterns is based on the visual classification employing the

three following characteristics. The first one is the size T of a pattern, which measures a diameter of set of nonzero entries of T(L). In other words, the diameter of agents pattern is calculated as maxijer{c> - cp}, i.e., the maximum distance between any two agents.

Interval Parameterization 39

We use term w-order to describe the degree of structure implicit in each of the final configu- rations of the swarm. The w-order is calculated as w-order as w = max{wH, WV}, where WH and WV denote the number of local configurations of the form

0 l 0 0 0 0

o @ o and l o l , 0 l 0 0 0 0

respectively, (agents are shown by es and OS and lattice nodes without agents are blank; the agent based at the central node of neighbourhood is shown by 0). For any pattern formed by m agents, 0 5 w 5 m - 2. The measure takes its maximum on the line of agents (where agents are arranged along the column or the row); and it takes its minimum on either densely packed or sparse patterns. Later we will discuss that w-order takes its maximum (amongst all patterns) on fingerprint patterns generated by S-swarms. Such patterns are suggested to be highly structured. Every agent in the fingerprint pattern has exactly two neighbouring agents which are always in the same column or the same row.

The third characteristic related to morphology is a density distribution D = (Q)sliss, Di = l{j E II : I(2 E JI : 1s; - sjl = 1}1 = i}l. G iven a stationary pattern of agents, density distribution (Di)olig is a vector, ith entry of which represents the number of agents each of which has exactly i other agents in its neighbourhood.

1

2

3

81 4

5

6

e2

1 2 3 4 5 6 7 8

*

Qa

e

r)

*

a

*

Figure 3. Interval + representative pattern correspondence for S-swarm.

1 2

A. ADAMATZKY et al.

02

3 4 5 6 7 8

2

3

5

6

7

8

al

e

e

Figure 4. Interval + representative pattern correspondence for S-swarm.

3. MORPHOLOGICAL CLASSIFICATION

The morphological classification provided in the section is based on the computer experiments with S- and P-swarms. It is shown that the patterns generated in the evolution of A-swarms will belong, to some degree, to the classes of patterns generated by S- or P-swarms. The swarming rules, with various ranges 01 and 0s of activation intervals, can be subdivided into several classes based on the morphological characteristics of the generated patterns. The rules belong to the

same class if, when they are applied to the agents of the swarm, the same pattern is formed in the evolution of the agents. An attempt is made to measure the degree of similarity using metric of distribution D as well as the w-order measure of structure.

The correspondences between values of 191 and 19 2, 1 5 Bi 5 132 < 8, and the representative patterns generated in S- and P-swarms are shown in Figures 3 and 4, respectively.

3.1. The Classes

The following seven classes are identified in both S-swarm and P-swarm (the classification of intervals [Or, 021 is shown in Figure 5). Further we will refer to the rules of a class in the sense of the activation intervals [&,&]; the exact values of the intervals, and therefore, rules, can be found in Figure 5. Thus, for example, rules of the C-class correspond to the intervals [l, l], [l, 21,

Interval Parameterization 41

1 ccusssss 1 yyssssss

2 cusssss 2 yssssss

3 CLLLLL 3 cLLLLL

01 4 7-l L L L 1: 01 4 L L L L L

5 L 3 3 3 5 L L L L

6 L 3 3 6 c L L

7 P P 7 P P

8 P 8 P

82 02

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

(4 04

Figure 5. Interval + class correspondence for S-swarm (a) and P-swarm (b).

[2,2], and [3,3] in S-swarm; the rules of the P- class have activation intervals [7,7], [7,8], and [8,8] in both S- and p-swarms (Figure 5).

3.2. C-Class

The rules generate condensed patterns (Figure 8). Almost every agent has its neighbourhood

fully packed with other agents (Figure 6), i.e., almost every agent has exactly one other agent in each of eight neighbouring nodes. There are very few agents with an empty neighbourhood. Distribution D is preserved for all intervals [13i, es] belonging to the class with slight spreading

toward lower densities when the transition from intervals [l, 21 to [3,3] occurs. The patterns are solid-like. Such patterns are typical only for S-swarm.

3.3. S-Class

The rules generate sparse patterns (Figure 9). The members of the class, 191 E {1,2}, 4 5 132 5 8 for S-swarm and 81 E {1,2}, 3 I 02 I 8 for P-swarm, do not depend on the degree of activity. For both S- and P-swarms, the class can be subdivided into two subclasses following density distributions D (Figures 6 and 7, [1,4], . . . , [1,8] and [2,4],. . . , [2,8]). Thus, for 01 = 1, almost

all agents have an empty neighbourhood. In contrast, for 01 = 2, more than half of the agents have exactly one other agent in their neighbourhood, while others have none. That is, the increasing of the lower bound 8i of activation interval from Bi = 1 to 8i = 2 causes an overall “compression” effect of the patterns in S-class.

3.4. ‘H-Class

The rules generate so-called halo patterns (Figure lo), i.e., patterns that have a condensed

core with a sparse halo around it. The patterns can be generated only in S-swarms. The vast majority of the agents have a fully packed neighbourhood. The halo is formed from the agents that have an empty neighbourhood (for [l, 31 interval), 1 or 2 (for [2,3] interval), or 2 or 3 (for [4,4] interval) other agents in their vicinity (Figure 6). The patterns are evocative of melting ice; so, we may think of them as two-phase transitional structures. Actually they are transitional, because they are at the boundary between condensed, C-, and sparse, S-, classes on the 81-02 plane (Figure 5).

3.5. P-Class

The rules generate so-called porous patterns (Figure 11). The patterns appear both in S- and P-swarms (Figure 5). The most expressive representative looks like a mesh with the space of

42 A. ADAMATZKY et al.

PP71 I I

IWI

[3,41, . . . t [3,31

15751 rm

[1,4], . . . ,!1,61

[WI, . . . ,I‘481 a,

Figure 6. Examples of distribution D of numbers of neighbouring agents in the patterns generated in S-swarm.

Interval Parameterization

07-2-2 07-2-3

07-5-6 07-6-6 07-6-7

07-7-7 07-8-S

Figure 7. Examples of distribution D of neighbouring agents in the patterns gener- ated in P-swarm.

variable length but width of one lattice node (Figure 11). Transition from interval [7,7] to [7,8] to [8,8] shifts the maximum of the distribution D from four to five to six agents, respectively, (Fig- ure 6).

3.6. JM%ss

The rules generate an overall porous patterns with some condensed regions, large holes of an empty lattice nodes inside, and a small halo outside (Figure 12). They vary in appearance but a meristema-like structure is common for all of them. They are typical for P-swarm only and occupy the same place on 191-82 plane as P-rules of S-swarms. The distribution D has its maxima at five and six agents (Figure 7).

44 A. ADAMATZKY et al.

Figure 8. C-pattern. Figure 9. S-pattern.

l . . l : .‘. , . . . . . ..,.’ .

.: .’ l

.

. : : ’

..‘.’ .

.

; : .

. :

. I : . .

=:

*

. - .

. .

.

. . . :

.

0 . : . :

.

’ : , . . ,

..’ .

. . . : :

- . . . .‘. . m . . . . ’

.

Figure 10. H-pattern. Figure 11. ‘P-pattern.

Figure 12. Y-pattern. Figure 13. C-pattern.

Figure 14. F-pattern.

3.7. L-class

The rules generate labyrinthine patterns (Figure 13). They look like classical labyrinthine patterns obtained in the laboratory and computer experiments with diffusive systems, magnetic domains, chemicals on the surface of catalysts, etc. The chains of resting agents, that form the walls of a labyrinth, have many turns and free-standing ends. By the varying of boundaries of the activation interval [&,&], we can achieve a shift of the distribution toward increasing the number of agents in the neighbourhood of an agent (Figure 6). The patterns generated by the

Interval Parameterization 45

rules of the class are most varied in structure amongst other patterns. However, they are very typical for both S- and P-swarms (Figures 3 and 4).

3.8. F-Class

The rules generate fingerprint-like patterns (Figure 14). They can be found only in S-swarms (Figure 3). Every pattern is subdivided into the small number of large domains with either vertical or horizontal orientation of agent chains. They are the domains of the walled corridors. In general, there are few instances when the boundary between one domain and another is not closed. Usually, we cannot easily move from one domain to another. The examples shown in this paper demonstrate that more than half of all agents have exactly two neighbours each (Figures 14 and 6, [6,8]). They are the agents that have a neighbourhood configuration of the following form.

0 0 0

0 l 0

0 0 0

Other agents have either three, the corresponding neighbourhood configuration is

0 l 0

l l 0

0 0 0

or four neighbouring agents, the corresponding local configuration is as follows.

0 l 0

0 0 0

The agents being on the boundaries of the pattern make an income to other small parts of the density distribution D (Figure 6).

As one can see from the examples, the classification is far from perfect, but the transition classes are found, for example, ‘%clsss.

4. TIME, SIZE, AND ORDER

Size, convergence time, and degree of structure are the essential global characteristics which, more or less, successfully describe various aspects of the behaviour of spatio-temporal systems, They are attributes of the swarm in general. An interval [f&, ,921, in contrast, is an agent based, local parameter. How do they relate to each other? How do changes in the local parameter space affect global characteristics of the spat&temporal system?

As a result of computer experiments (m = 1,. . . , 1000, 1 5 81 5 82 5 8, for both S- and P-swarms; 10 trials for every set of experiments are done), we found that the size r = r(m) of the pattern generated in evolution of m agents, m = 1,. . . , 1000, is approximated as r(m) = ,/$Yii, 7,. = 1;.(8i,&). The convergence time tC = tC(m) is approximated by tC(m) = -ytm, +n = “(t(&, 02). The values of order w were obtained in the experiments with S- and P-swarms of m = 1000 agents.

To represent the results in a qualitative form, we consider an 8 x 8 matrix hereinafter referred to as e-lattice. Thii is indexed by pairs (01, ez), 1 < 81 5 02 <_ 8. When dealing with time, size, and order, we assign the corresponding values of -yt(&, ez), ^(r(Bi, 02) and w(&, 0,) to the nodes of the e-lattice. After that, we build “gradient-like fields” on the lattices. Every node z of e-lattice is assigned a vector oriented toward the neighbour y which gains the maximal value of yt(z), y,.(z), and w(z) amongst all other neighbours of z. The vector in CC has null length if z itself has the maximum value in u(z). It is possible to have more than one vector in CC if more than one node in U(Z) have the same maximum value. The sinks of the fields are called the attractors. The elements which are neither maximal nor minimal may be thought of as the sources.

46

4.1. Size

A. ADAMATZKY et al.

The y,-field is represented in Figure 15. The family of rules 61 = 1, 92 = 4, , . . ,8 of the S-class represent the perfect attractors in y,-field. They are also global attractors. It is also found that

the lower bound of activation, 81, bears the fill responsibility for the size of the generated patterns in both S-and P-swarms. This is correct for the following domains of the ~,.(f?1,&):

l S-swarm: 62 1 5, 1 5 ~91 < 8 and 132 = 4, 81 = 1, ., 3; 0 P-swarm: 133 2 3, 1 I 61 5 8.

For these domains, it was found that y,(&, 02) = clO;ca, where cl and cz equals 2.3 and 0.78 for S-swarm and 2.6 and 0.76 for P-swarm.

-92

12345678

l-r-*4.....

2 /“/“/“TTtfT

a/“/“/“trtff

01 4 0 /“f/T r tt

5oo/“/TTTT 6 o o o /“/“t f t

70 0 0 o//tt 8 0 0 0 0 0 / /” t

Figure 15. r,-field for S- and P-swarms. Values of ct (&, 02) are quite close for both types of swarm; differences do not exceed l/10.

02 e2 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

1

2

3

01 4

5

6

7

8

+ .

Il/rCtcl

1

2

o/“/“t\\\\ 3

O/~~ZZZJ 014 0 O/T@\lJ 5

0 0 o/t+.+ 6

0 0 0 o/“/“TZ 7

0 0 0 0 0 / t + 8

Figure 16. -yt-fields for S-swarm (a) and P-swarm (b); ct( 1,3) = 2.65 for P-swarm, ct(l,4) = 2.8 and ct(6, 7) = 1.28 for S-swarm.

4.2. Convergence Time

The structure of the y,-field is different for S- and P-swarms (Figure 16). In the case of the S-swarm, there are two attractors: ~t(l, 4) = 2.8 and yt(6,7) = 1.28; the first one is a global attractor because yt(6,7) dominates only in the domain (0, = 5,. . . ,7; 82 = 6,. . . ,8). The node yt(l,3) = 2.65 is an attractor in yt-field for P-swarm, whereas yt(l,&) = 2.25, 82 = 6,. . . ,8

are local attractors. Both maximums correspond to rules of S-class. Therefore, we can say that the swarms that generate sparse uniform patterns have the longest transient periods. It

Interval Parameterization

2.4 I I I I I

2.2 - *+* * *

2-

1.8 -

1.6 -

1.4 - ** * * “Ir

1.2 -

l- t+k D D

0.8 - Eo% D

0.6 - lm

0.4 - *o$ O0

0.2 I I 1 I I

0 0.5 1 1:5 2 2.5 3 rt

Figure 17. Distribution of the clssses in rt - rr map for S-swarm. The following symbols are used: o-‘P-clsss, I-C-class, Q-WC&~, o-.Cclsss, ~-L-class, +-S- class.

does contradict the quite popular conjecture stating that complex patterns are generated by the systems with the long transient periods.

If we look at the 7,. - yt-map (Figure 17), we find that rules from all classes except the S-class

form quite recognizable clusters. It is especially expressive for t-, C-, and P-classes. The overall distribution of the rules shows that we can roughly split the map into three regions. The first region includes the P-, C-, and ‘H-classes (Figure 17), *yr < 0.6 and yt < 0.75). Thii is a domain of the rules which generates condensed patterns. The second region (Figure 17, 0.6 2 car 5 1) includes 3- and L-classes. This is a domain of the rules which generates structured patterns. And the last region (Figure 17, +y,. > 1.2, Tt > 1.3) consists of the rules of the S-class. These

rules generate the sparse homogeneous patterns.

e2 e2 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

1

2

3

e1 4

5

6

7

8

Figure 18. w-fields for S-swarm (a) and P-swarm (b); w(5,7) = w(6,8) = 0.69 for S=swarm, w( 1,l) = 0.1 and w(5,7) = 0.25 for P-swarm.

4.3 w-Order

There are two attractors in the w-field of S-swarm (Figure 18a). They are w(5,7) = w(6,8) = 0.69. The w(5,7) = 0.25 is the only sensible attractor for P-swarm (Figure 18b). All attrac-

tors correspond to the rules of the 3-class; the rules generate most important ordered patterns (Figures 3 and 18). The domains of W- field corresponding to S-class are sources: they gain zero values of w(&,&).

48 A. ADAMATZKY et al.

2.4

2.2

2-

1.8 -

1.6 -’

1.4 ir %

1.2 -’

l- ED D

0.8 - D D D

0.6 -’ 0 Oc8 -

‘0 a0

I I I I I I I _

0 100 200 300 400 560 600 766 wm

Figure 19. Distribution of the classes in ‘yr - w map for S-swarm. The following symbols are used: c-‘P-class, 2-Gclass, o-7+class, o--;f-class, D-L-class, +-S- class. Data obtained in experiment with S-swarms of m = 1000 agents.

W

0.5

0.4

0.3

0.2 0.4 0.6 0.8 1 PP

Figure 20. Order w vs. degree of activity p,, computed for the swarm of m = 1000

agents.

It is hard to find any clusters of the rules in f̂r - w map. Only rules of the Jr-class form an easily recognizable cluster (Figure 19).

5. CONTROLLING STRUCTURE ORDER

This section deals with controlling the order of the patterns created by P-, A-, and S-swarms by the probability of activation pp. We “smoothly” vary the probability pp in the model of A- swarm, from close to zero value to the value 1, and measure the w-order for every increment of

PP.

Among the patterns generated by S- and P-swarms, those generated by P-swarms are the least structured.

The degree of order is inversely logarithmically proportional to degree of activity (see example in Figure 20). Thus, for example, in the set of experiments with S-swarms of m = 125,250,500,1000

and with the [6,8] interval of activation, it is shown that w(pp,m) = (cl(m) ln(p,) + cz(m))/m, where cl(m) = -0.12m + 9.02 and cz(m) = 0.26m - 4.1.

The explanation of this reduction of order, correct for at least some members of the F-class, may be found in the following idealistic example of the S- swarm, 81 = 6 and 02 = 8. Consider an infinite domain with a vertical orientation, where a node with an agent is black and an empty node is blank as shown in the scheme below:

Interval Parameterization 49

. . .

. . . 0 . 0 . 0 . . . .

. .

Every empty node in the domain has six agents around it (three on the left and three on the right). At the same time, every agent in such a domain has exactly two other agents around it. Moreover, all its neighbours are in the same column as the agent under consideration. Let us “drop” one agent j into the domain. It will land on either an empty node or on the resting agent. If agent j drops onto an empty node then has six neighbours and will decide to move because its activation interval is [6,8]. Notice that in this situation, the stationary agents will rest because each of them has four neighbouring “spare” nodes which can be filled by other agents. If agent j moves onto another previously empty node, the situation will be the same again. Let it move to one of the nodes occupied by the agent i. In the S-swarm model, the agents check their neighbourhood and their own nodes in the cycle of period m, therefore, either agent i or j will decide to move, while the other agent will remain at the same node. So, the structure of the agent chain is preserved. However, the probability of the destruction of the chain will increase proportionally with the degree of activity of the agents, i.e., p,,.

6. IS THERE A SINGLE PARAMETER RESPONSIBLE FOR STRUCTURE CONTROL?

Since the beginning of complex systems theory, particularly for cellular automata, researchers were eager to find a single parameter that governs the space-time dynamic of system. Some achievements and failures of such a search can be found in [23-261. In this section, we focus on two possible candidates:

l threshold of activation; is well known from the theory of excitable media and neural networks,

l transition-probability-like parameter, invented for Boolean networks and cellular auto- mata.

6.1. Threshold of Activation

Let us assume that the agent i of a swarm decides to change its position if there is another agent at the same node or if a number of other agents in the neighbourhood of i exceeds some specified threshold. Let us call this the threshold of activation. In this case, it is enough to consider intervals [I&, Oz], where 1 5 01 < 8, 02 = 8 and 81 is the threshold. From Figures 3 and 4, we see that while the threshold 81 increases, the patterns generated by swarm are more and more condensed. Somewhere between the entirely sparse patterns and the condensed porous patterns, we find the highly structured labyrinthine and fingerprint-like patterns. That is, the transitions S --f L 4 F -+ P and S ---) L: --) P occurred in S- and P- swarms, respectively, when the threshold 81 is changed from 1 to 8. Therefore, in the case of our models, the threshold of activation seems to be quite a good parameter which characterises phase-like transitions between the generated patterns.

In Figure 21, we see how convergence time, size, and order of patterns change with the thresh- old of activation 01. This illustrates that increase of the convergence for the thresholds 5 and 6 (Figure 21a) reflects emergence of a structure. The size goes down smoothly (Figure 21b) fol- lowing the earlier approximation. Correspondingly, the order increases with the increase of the

50 A. ADAMATZKY et al.

threshold until the threshold reaches value of 6; after this the order then decreases. From the graphs, we can see that when threshold pammeterization is employed

l the convergence time is proportional to the order of the generated structures,

l the size of the generated structures is inversely proportional to the threshold of activation.

(b)

(cl

Figure 21. Schematic graphs of 7t(O - 1,8) (a), 7,.(&,8) (b), and w(&,~)(c), 1 5 01 5 8, horizontal axis, for S-(solid line) and P-(dash line) swarms.

6.2. Forbidden Configurations

There are certain configurations of neighbours which force the agent under condition to move. They are referred to as forbidden configurations.

The number of so-called forbidden configurations of an agent around an agent with the acti- vation interval [Or, &] was found to be

4(&, 02) = c 8! jE~ol,ezl 0-3 - 89!’

The actual number of forbidden configurations for each interval is shown in Figure 22. It may reflect the probability of an agent moving, assuming the agent was dropped at one of the randomly chosen configurations of the agents.

The correspondence between rules of the morphological classes and the numbers of forbidden configurations is shown in Figure 23 (hereinafter all data deal with S-swarms). Unfortunately, there is no obvious correlation. At least we see that the rules for the condensed and porous

Interval Parameterization 51

92

1 2 3 4 5 6 7 8

1 8 36 92 162 218 246 254 255

2 0 28 84 154 210 238 246 247

3 0 0 56 126 182 210 218 219

01 4 0 0 0 70 126 154 162 163

5 0 0 0 0 56 84 92 93

6 0 0 0 0 0 0 36 37

7 0 0 0 0 0 0 8 9

8 0 0 0 0 0 0 0 1

Y--

I I I I 1

0 50 100 150 200 250 4(~1,&)

Figure 23. Correspondence between classes and qS(., .) parameter. Values of r$(., .) for each rule of the class is drawn with circles and connected with a line to guide the

eye.

classes, C-, P-, and Y-classes, have low numbers of forbidden configurations. The rules of the S-class tend to have above average value of 40. The L-class is spread along almost all axis of 40.

The fingerprint rules, 3-class, have values of rj() ranging between 50 and 100, i.e., not low but below average. If we calculate the average 4() inside every class, we find that the rules generating unstructured patterns, P-, Y-, C-, and S-classes, can be found in the domains with either a low or

high number of forbidden configurations. In contrast, the rules leading to formation of structure, L-, ‘H-, and 3-classes, are characterised by the mean numbers of the forbidden configurations.

In Figures 24 and 25, we see the overall trend that the larger the number of forbidden config- urations the larger is the convergence time and the smaller the size of the pattern. However, the dependence is highly nonlinear and cannot be easily described.

For the w - $()-map (Figure 26), we can recognize two peaks, which correspond to the laby- rinthine (C-) and fingerprint (3-) classes.

So, we can see that the number of forbidden configurations $0 cannot be considered a candidate solely responsible for the classification and prediction of pattern formation.

7. DISCUSSION

It is quite a risky business to draw out any analogies based on simple automata models. Nevertheless, we will try to consider at least some of the problems which are related to the matters discussed in the paper. First we discuss some of the relationships and offer suggestions for further investigations of biological and social swarms such as social insects. Then we look at what happens in the framework of physical pattern formation, particularly reaction-diiusion systems and vibrating granular materials.

52 A. ADAMATZKY et al.

Yt

2.5

2

1.5

1

0.5

n ”

50 100 150 200 250 9w, b)

Figure 24. Tt - 4(., *)-plane. Moving average curve is for the guiding of the eye.

2.4 , I I I I I 1

2.2

2

1.8

1.6

1.4

1.2

1

0

A ?l

0.8

t

V

0.6 0 o VW

0

0

00

50 100 150 200 250 4(el,e2)

Figure 25. *fr - qS(., .)-plain. The moving average curve is shown to guide the eye.

50 100 150 200 250 cw, 02)

Figure 26. w - qS(., .)-plane. Moving average curve is shown to guide the eye.

Interval Parameterization 53

[2,4] D [I,21 h61 D [4,41

. rn. . . . .

.

:

I)

.

.

. .

. . l - .

k&61 D 11921 [4,61 D [I, 31

1. s ‘: .‘. .

. . . .: . . . . . . . ’ .’ =

1.. mm-..

. . . 1 .*

. :-:

.: .‘.

. ‘. . .

. . .

*

. .

.* . .

.:

0.: .:

- .’ ..- . . . . . .: . . .

. =. . . . . ’ -1. .*:..

Figure 27. Examples of patterns generated in D composition in S-swarm. The pat- terns areobtainedfor thefollowingcompositions: [2,4]D[l,2], [2,4]D[l,3], [4,6]D[1,2],

[4,6] D [L3], (6961 D [4,4].

Figure 28. A simplification Aranson et al. [48] phase diagram (left) and the diagram derived from our results (right).

Figure 29. Pattern formed in swarming of 1000 agents in 3D lattice. The activation interval is [20,23]. One-quarter of the pattern is removed. Agents are shown by the balls. The edge agents are shown in dark color.

54 A. ADAMATZKY et al.

7.1. Biological Swarms

One of the most successful interpretations of natural swarming in terms of physical systems is done in the analysis of pedestrian dynamics [27,28]. Namely, the principles of molecular dynamics lead to the paradigms of a social force and the concept of active walkers. Thus, for example, the movements of pedestrians are described by the following equation of motion [28]:

vi = fi(t) + fluctuation,

where vi(t) is some kind of velocity of a pedestrian i and fi(t) is a socalled social force, or perhaps more appropriate, a motivation to act [28]; a fluctuation term is responsible for the stochastic components of pedestrian behaviour. Helbing et al. [28] indicate four key features of social forces:

(i) action does not equal reaction, (ii) no conservation of energy or momentum,

(iii) agents are active, (iv) information transfer substitutes momentum transfer.

The first three principles were implicitly used in our investigations. The fourth principle was not employed because we assumed that agents do not communicate either directly or indirectly, and they do not leave or read any markings in the simulation space (our agents even do not detect any “gradients” of other agents at all). It would be useful to consider consequences of the fourth principle in further computer experiments.

The lattice swarm concepts were successfully applied to the simulation of pattern formation by mobile cells in bacterial colonies [29,30]; the relationship between local cell interactions and global patterns was obtained. Some kind of simple parameterization can be derived from [31], where active walkers move with a constant absolute velocity and assume the average direction of motion of their neighbours subject to noise influence. On a “density of walkers-level of noise” plane, a domain of low density and low noise corresponds to the sizeable mobile clusters of the walkers. The domain of low noise and high density is responsible for the completely ordered motion of all walkers. The noise and density parameters from other slots of the plane lead to the various types of random movement of walkers (with some correlations).

Quite good analogies are done in [8,32] between reaction-diffusion systems, morphogenesis and pattern formation with mobile agents (namely social insects). Despite the fact that most of the agent related results of [8] are simple reinterpretation of some results of the well-established field of social dynamics and active walkers (see e.g., [27,28,33-35]), the viability of the simplification of the biological agents are proved again.

The models presented in the paper were partly inspired by the question: how can we keep swarm swarming? That is, how do we make the growth of a swarm cluster bounded. Exactly the same question led to the design of Game-of-Life rules for cellular automata with binary cell states [36,37]. The matter was also slightly touched on in our previous paper dealing with the interval parameterisation of excitable swarm [19]. If we reject the long-distance interactions and field effects as well as diffusion of any substances, a stop state may be the only answer to this question. Sooner or later, after the beginning of evolution, the agents have to stop, otherwise the diameter of the cluster will grow infinitely. In our models, the agents stop moving if there are no other agents in their vicinities. We suspect that this is the only way for minimalist systems with local interactions. Alternatively, the agents can move in cycles or the substrate space could have reflecting or periodic boundaries (see e.g., [17,38]) or a swarm is assumed to be moving along a cylinder.

The distributed building in collectives of social insects was another source of inspiration. Again,

as with pedestrian dynamics, the space and time dynamics of the insect collectives may be de- scribed in terms of physical systems. Thus, for example, in the experiments with leptothoracine

Interval Parameteriaation 55

ants, it was demonstrated [15] that when the ants build a circular wall around their nest, they sim- ply pick up and drop the grains of sand and actually imitate the pressure of the gas molecules on the walls of the elastic ball that causes extension of the ball [15] (compare this with otir computer experiments on the approximation on the tessellation of a lattice by pebble automata [22]). Some attempts to parameterise a space of local building rules are done in [8,16,17]. Three conjectures

were proposed there.

(i) Structureless architectures are more likely to be produced by an arbitrarily chosen building

rule. (ii) Rules close to each other in the rule space produce similar architectures. (iii) It is necessary to switch building rules in the subsequent time intervals (see [17]).

It follows from our results that the two first conjectures are not entirely correct because they both put the main stress on how the sampling of rules space is affected and how we define the distance between the rules in a rule space. The problem is actually similar to those discussed

in our papers on the morphology of excitation rules [19,20]: the full rule space exhibits some kind of “edge-of-chaos” parameterization [23], but in the subspace of the interval-sensitive rules, the standard parameterization does not work. Thus, concerning the first conjecture, in S-swarms

about l/6 of all rules produce patterns with high degree of w-order. Using uncertainty of the term stmctureless, we can easily take into account the P- and L-classes and increase, therefore, a ratio of structure forming rules to l/2. As to the second conjecture, we can assure that rules “close” to each other in the rule space do not necessarily produce similar architectures. There are usually several phase transitions in the rule space, and hence, several classes. This is demonstrated in the papers dealing with continuous and discrete models of pattern formation. However, inside

a class, there may be a smooth transition between the patterns built, and therefore, close rules produce close patterns if the rules are in the same class. The third conjecture is extremely valuable. In a majority of real situations, we certainly need to switch the rules to produce the

required architecture. The nonuniformity of the building is usually determined as a number of different rules between which the building process has to be switched (see e.g., hierarchies of two-dimensional cellular automata that simulate grows of inflorescence-like patterns [21]).

The rule switching can be easily incorporated to our models. To do it, we define the serial com-

position D of the interval rules 11 and I~,11 D 12 : l 3 4, 2 4,. In other words, the pattern Prl is generated by the rule with interval [&, 021 = 11. After the swarm reaches its stationary state, the rule with interval 12 is applied to the agents. The agents begin their movements again and after some period of perturbations once again become stationary. The pattern 5, is formed. The classes of patterns can be generated as a result of such serial composition. Some of the examples are shown in Figure 27.

Finally, we want to discuss how one of the biology inspired concepts relates to our results. This is the concept of tensigrity derived in [39] which asserts that aesthetically complex stable patterns arise through tension and local compression, i.e., global attraction and local repulsion. The

validity of the concept was demonstrated on the examples of biological structure formation [39] and it is definitely true for some models of mobile swarms [29,30,40,41]. As to our model, the rules of swarming have implicit compression-all automata start evolution at the same site of a lattice and an automaton does not move if there are no neighbours. There is also explicit tension-an automaton does not share the same site of the lattice with ahy other automaton and the automaton does not stay at a node if the node is surrounded by a certain number of other automata.

7.2. Physical Swarms

Examples of models investigated in the paper are similar to diffusive systems, sand pile models, and externally driven granular material.

56 A. ADAMATZKY et al.

lattice inside the swarm cluster are shown by the balls.

Figure 31. Representatives of patterns generated in iS-swarm.

The study of a formation of the complex patterns by Hagberg et al. [42] is one of the recent examples related to our results. They considered nongradient reaction-diffusion system with two

scalar fields, u and w

ut = u - u3 - v + V2u and wt = c(u - aizl - ac) + SV2v.

They investigated regimes of pattern formation in the E - S-plane from the points of view of bifurcation and transverse instability. The instabilities are found to form boundaries between domains responsible for the formation of different patterns. Thus, for example, in the Ising regime, the stationary ordered strips appear for the parameters below the transverse instability,

whereas the labyrinthine patterns emerge above the instability [42]. In the Bloch regime, the propagating fronts coexist with strips and spiral waves. There are transitive regions as well (e.g., those with the replicating spots) [42].

Interval Parameterization

Figure 32. Representatives of patterns generated in iP-swarm

In the sandpile models [43], grains of sand at the lattice node topple down if the height of the

sand at the node is above the number of neighbours. This is similar to our model except that only certain randomly chosen neighbours would receive sand grains. Thus, some neighbours can receive more than one grain from their neighbours at the same time step. This is the principle difference. The so-called hot sand piles [44] have much in common with our models: even a node

with height h of sand grains below the number of neighbours k can topple the grains of sand with probability e-(k-h)lT, where T is an analogue of temperature; the fallen grain will be gained by one of the neighbours chosen at random. We have not found any results on pattern formation in either sandpile or hot sandpile models.

Transition from sand piles to granular materials is quite natural. Pattern formation in driven granular materials is another example close to our model (see e.g., [45-471). In the experimental setup described in [45,46], small metal balls are put into the flat container to form a layer of several balls. The container is vibrated vertically with various vibration rates. Roughly speaking, the

vibration rate can be considered as a parameter which can be changed to produce transitions between square patterns, strip patters, hexagons, and labyrinthine patterns.

Let us consider one example [48-501. If we assume high frequency of the vibration of granular material, the usual coupling between complex amplitude II, of the parametric layer and friction of material vanishes and the following model can taken into consideration [48]:

%ti = rlCt* - (1 - iw)$ + (1+ ib)V21c, - ]$]2$,

where grains oscillate with frequency w, and b is a parameter to reproduce correct wave number m at a given frequency, and y is a vibration amplitude.

58 A, ADAMATZKY et al.

The simplified phase map is shown in Figure 28 (left). It is probably impossible to set up any common-sense analogies between 81 and 6’s on one side and 7 and w on another one. However, the resultant similarity between the phase portraits is sophisticated (compare Figure 28 (left) and Figure 28 (right)). This similarity “proves” an intuitive approach chosen in our paper.

7.3. More Dimensions and More Intervals

Does the offered parameterization work for the swarms in three-dimensional lattices? Definitely. A complete analysis is beyond the scope of this discussion and only one typical example is considered. The pattern formed by three-dimensional swarm with activation rule lying roughly in the domain corresponding to the L-class is shown in Figure 29; the empty nodes inside the “corridors” of labyrinthine patterns are shown in Figure 30.

What if we consider the interval of rest instead of the interval of activation? That is, we slightly change the strategy of the swarming and prevent agents from moving if there is no other agent at the same node and if the number of agents in its neighbourhood belongs to the interval [&, es]. In this case, the function C(.) is shown below:

The correspondence “interval of rest -+ generated pattern” obtained in computer experiments is shown in (Figure 31 and 32).

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