q-Transmission in simplicial complexes

Int. J. Man-Machine Studies (1982) 16, 351-377

q-Transmission in simplicial complexes J. H. JOHNSON

Centre for Configurational Studies, Design Discipline, The Open University, Milton Keynes M K 7 6 A A , England (Received 10 January 1981) Q-analysis assumes a qualitative difference between q-connectivity and ( q - 1 ) connectivity which is significant in the way changes in pattern values can be propagated over the backcloth. The way such changes (t-forces) move through the simplicial complexes of the backcloth is defined as q-transmission. It is shown there is a simple derived network structure for q-transmission, and this will facilitate computation in applications. Simplicial complexes are shown to possess various quantitative characteristics which can be expressed as transmission numbers. The concept of time which underlies q-transmission is considered and the relationship between q-transmission and Atkin's concept of p-event is investigated. This allows an algebraic description of prediction and suggests the possibility of relative social time.

1. Introduction The preliminary notation for this paper is given in the Appendix, and a fuller version can be found in Johnson (1981b). A more general background to Q-analysis can be found in Atkin (1974a, 1977b). The presentation of q-transmission necessarily involves some complicated notation which may obscure some relatively simple ideas. This section is written to motivate the more formal presentation and to give an intuitive feeling for q-transmission. For some time it has been argued that the q-connectivity structure of a relation between two sets underlies the changes in numerical values on the elements of the sets. Consider the simplicial complex KY(X, A) and a graded pattern 7r which assigns a value to every simplex of KY(X, h ). The fundamental idea underlying q-transmission

is that the numerical value on one simplex will effect the numerical value on another if they are q-near but not if they are only (q - 1)-near or less. The q-value represents a threshold for transmission and q-nearness is a necessary condition for q-transmission. It is a simple step to consider the numerical value on one simplex changing the numerical value on a second q-near simplex; the change on the second simplex causing a change to a third q-near simplex; and so on. In other words changes are "transmitted" down chains of pairwise q-near simplices, or changes are transmitted down q-connectivities. By hypothesis these changes are not transmitted down (q-1)-connectivities so that "q-transmission" can only occur within the q-connected components of KY(X, A). Once the simple idea of q-transmission is established, investigation of its properties is of great interest. Figure 1 (a) shows q-transmission along a chain of q-connection. In most complexes simplices are embedded in components: consider q-transmission starting somewhere in the "middle" of a q-connected component, at a simplex tr ~ for example [Figure l(b)]. Within the component there are many q-connectivities for the changes to be 351 0020-7373/82/040351 + 27 $03.00/0

O 1982 Academic Press Inc. (London) Limited

352

J.H. JOHNSON

transmitted "away f r o m " tr ~ When all these connectivities are considered together a kind of " f r o n t " for q-transmission is obtained [Figure l(b)]. This provides a way of simplifying the study of q-transmission. The study of q-transmission begins with the hypothesis that there is a change of numerical value (t-force) on some simplex, say or~ and this change is transmitted outwards through the q-transmission fronts of the q-connected c o m p o n e n t containing or0 as a kind of ripple in a multi-dimensional pond. Clearly the transmission fronts depend both on cr ~ the site of the original disturbance, and the value of q which depends on the particular transmission process under study.

)

(

( (

( q - Transmission fronts

(a)

(b) FIG. 1. q-Transmission fronts.

The q-transmission fronts are sets of simplices. They are represented in Figure l(b) as concentric rings of simplices around o-~ Thus a transmission front can be thought of as lying at a theoretical hierarchical level above the simplices, i.e. as a set of simplices. It can be argued that q-forces " e n t e r " a simplex by its q-faces, and after a time interval " l e a v e " by other q-faces. This idea extends to the transmission fronts: q-forces enter the simplices of a transmission front by some q-dimensional faces and leave by others. By defining suitable relations between the transmission fronts and the qdimensional faces a new structure can be obtained, the derived network. The derived network effectively lies at a higher hierarchical structural level to KY(X, h) and gives a much simplified view of the transmission process. The derived network depends on

the original source of disturbance, the simplex which first experiences change, and the q-value which underlies the transmission process. This has important implications for applications since m6st m a n a g e m e n t involves planning the backcloth and introducing q-forces on known simplices of the backcloth. The derived backcloth is obtained by an algebraic construction upon which computable algorithms can be based. This means

the theoretical results are computable, and the theory of q-transmission may be applied in practical cases.

353

q-TRANSMISSION IN SIMPLICIAL C O M P L E X E S

2. q-Transmission through the backcloth 2.1. T H E q - T R A N S M I S S I O N P R O P E R T Y

Let ~- be a pattern on the simplicial c o m p l e x K, i.e. ~" is a m a p p i n g f r o m K to a set of n u m b e r s (usually the integers). T h e inner p r o d u c t n o t a t i o n will be used so that the notation (o'p, 7r) m e a n s the value of the p-simplex o r, u n d e r the m a p p i n g ~-. ~- can be graded by the dimensions of the simplices it acts on and the following notation will be used: (o'r, 7r s) = (o'r, 7r) iff r = s, (O'r, "B"s) = 0 for r • s. This notation is e x t e n d e d so that (cry, 7r)T is the numerical value of the pattern ~r on the simplex trp at time r, and (crq, S~'), is the change in value on the simplex ~rq b e t w e e n times r and ~"- 1. T h e notation o'q ~ q : (i)

O'q ~ o'.

(ii)

Crq < crp

(iii) (~, 6~r)~ (iv)

) F. 1 0~lmp y (o'~, 67r)~+1 ~ 0,

(o',',, 6~r)T = 0J

where r and r + 1 represent instants of time. (The nature of this " t i m e " is discussed in m o r e depth later but for the present it will suffice to say that r + 1 is after ~-.) In other words a q - f o r c e at time r on o n e face of rrp will induce a q-force on a n o t h e r face of ~rp at time r + l. Let ~rr and o-~ be simplices of K with o-, r~ ~rp = trq and trp ~ trs = tr'q. T h e n if ,r has the q-transmission p r o p e r t y a q - f o r c e on o'r at time ~ ' - 1 can be transmitted t h r o u g h tr, to o's at time r + 1, even w h e n o-~ and or, share no vertices (Fig. 2).

d rt-

~rp

O's

Vq FIG. 2. q-Transmission along a chain of q-connection.

Figure 2 is misleading in its simplicity since in general o'r and o's will share a face of dimension less than q. All the 2-simplices (triangles) of the 1-connectivity in Fig. 3 share the vertex v3. A l t h o u g h this m a y be significant for 0-transmission, connectivities less than a given value of q (here q = 1) are not sufficient for q-transmission. By hypothesis q-forces are transmitted f r o m o n e q - f a c e to another, i.e. they are transmitted along q-connectivities. T h e 1-connectivity, or s e q u e n c e of 1-simplices, for the transmission of Fig. 3 is r e p r e s e n t e d in Fig. 4.

354

J.H.

v2

JOHNSON

v4

Vl

v5

r-1

r

r+l

FIG, 3. Transmission along a 1-connectivity.

{
,

{(vz, v3)}

,-

{(v3, v~)}

FIG. 4. Transmission between 1-dimensional simplices. 2.2. TRANSMISSION FRONTS AND THE TRANSMISSION FRONT RULES The notion of q-transmission is analogous to that of dropping a stone in a pond with islands; waves are p r o p a g a t e d in fronts f r o m the source of disturbance and the nature of the waves depends on the topological connectivity of the transmitting medium. In the case of the p o n d the islands represent "holes" in its surface around which the waves are propagated, and in the case of q-transmission local disconnections (such as s h o m o t o p y loops) act as "holes" around which the transmission must occur. Consider a relation A between the sets A and B, A ___A x B. Let ao E A be any element of A and consider the simplex o-(a0)E K A ( B , A). For some fixed value of q let Fo(a0, q) be a set of simplices with dimension q or more, where F0(ao, q) _ KA(B, A), with Fo(ao, q) = {cr(a)[a ~ A and o ' ( a ) ~ o'(ao), dim o,(a)-> q}.

(T1)

Given Fo(ao, q), let Fi~-l(ao, q) be defined as Fi+~(ao, q) _~ KZ(B, h), with Fi§

q) = {o'(a)[a E A, o'(a) ~ Fi(ao, q), j --
3cr(a') E Fi(ao, q) such that or(a) is q-near cr(a')}.

(T2)

For reasons that will b e c o m e apparent (T1) and (T2) will be called the transmission front rules. To simplify the notation the set of simplices Fi(ao, q) will be abbreviated to Fi when ao and q are unambiguously known. If Cq(ao) is the q-connected component containing ao [but Cq(ao) excludes u n n a m e d faces until section 2.5), then: PROPOSITION 1. Cq (ao) is partitioned by the class of sets of simplices F = {Fo, F1, Fz . . . . }. By virtue of this proposition a mapping m :Cq(ao)-~ F with m(~r(a))= F~ iff cr(a)E Fi is well defined.

355

q - T R A N S M I S S I O N IN S I M P L I C I A L C O M P L E X E S

PROPOSITION 2. f f zr has the q-transmission property, an initial q-force at time r will be q-transmitted to every simplex of Fe at time "r+ i.

•Tr q o n

cr(ao)

In view of this, Fi will be called the ith q-transmission front of a0.

F3

FIG. 5. q-Transmission fronts.

2.3. T R A N S M I S S I O N C H A R A C T E R I S T I C S , T R A N S M I S S I O N N U M B E R S

By construction q-transmission can only occur within the q-connected components of a complex. Let Cq(ao) contain the simplex ~r(ao) and let Cq(ao) be partitioned into the q-transmission fronts of ao as F={Fo, F1, F2 . . . . . F.}, where

0 Fi = Cq(ao).

i=1

The integer n is uniquely determined by construction of the transmission fronts Fi. It will be called the q-transmission number of ~r(ao), and it will be denoted #q~r(ao). From this the q-transmission number of the component Cq(ao) can be defined as #qCq(ao) = min {#q~r(a)Jo'(a) ~ Cq(ao)}. The q-transmission number of Cq(ao) is an important characteristic of the component. It can be shown that o-(a) E Co(ao) implies #qO-(a) -< 2#qCq(ao), since, intuitively, the simplex with smallest transmission number will lie "in the middle" of the component Cq(a0). Note that #qCq(ao) is independent of the particular ao under

consideration, and is a fundamental property of the q-connected component. Following on from this, the q-transmission number of the complex K is defined as ~qK =

max

{:~f:qCq[Cq is a q-connected component of K}.

It follows that or(a) ~ K implies ~ q o ( a ) ~ 2:~qK and the transmission number of K is a fundamental property of the complex K.

356

J.H. JOHNSON

Thus for a pattern satisfying the q-transmission property, a q-force anywhere in the complex will be transmitted to every other q-connected simplex in less than or equal 2 *~ qK time intervals. In this context the n u m b e r 2 *~ qK could be called the q-transmission diameter of the complex K. W h e r e there is no ambiguity 4#qK will be abbreviated to #~. Let the vector T = {~o, ~:1, :~:2, :~:3. . . . .

:~dira K}

be called the transmission vector of K. The n u m b e r # K = max {~,lq = 0, 1 . . . . . dim K} will be called the q-transmission number of K. This single n u m b e r gives the maximum n u m b e r of time intervals for a q-force to be transmitted from any simplex in K to any other q-connected simplex in K, regardless of the value of q. In other words # K is a global measure of the speed of q-force transmission in K. Although all of the preceding definitions have been given in terms of ~r(ao) where ao ~ A, they can easily be generalized to an arbitrary simplex or ~ K as follows. Let A be a u g m e n t e d by an element a' and extend A by a'Ab iff (b)~cr. Then set a ' = a0 so that cr = cr(a') = or(a0) and the argument goes as before. W h e n K is not defined by a relation these definitions can still be applied: let the vertex set of K be the set B, and define an element a~ for each eccentric simplex of K. 2.4. THE DERIVED BACKCLOTH, KF(B, ~') AND KB(F, ~.T) Consider the q-connected c o m p o n e n t C,(ao) for s o m e a o ~ A . Let F be the set of q-transmission fronts of ao. L e t ~'(ao, q) be defined as a new relation, ~'(ao, q) ~ F • B, with (F~, b s) ~ ~'(ao, q)

iff

3tr(ak) E Fi with (ak, bi) ~ h.

~'(ao, q) will be abbreviated to ~" when ao and q are known. By construction PROPOSITION 3. For i
A,,

---~B

FIG. 6. The relation ~(ao,q) ~-F • B.

357

q-TRANSMISSION IN SIMPLICIAL C O M P L E X E S

In general this will not be the case for values smaller than q. The dimension of tr(b) in KB(F, srT) indicates the number of q-transmission fronts related to b, and hence the importance of b in the transmission process. 2.5. T H E D E R I V E D B A C K C L O T H , KA(G, ~:) W I T H KG(A, ~'r)

The set G(ao, q) is to be a cover of the set of q-simplices in the q-connected component Cq(ao) ~ KA(B, h), where henceforth Cq(ao) also contains faces of named simplices. Let G(ao, q) = {Go(ao, q), Gl(ao, q), G2(ao, q) . . . . }, where Go(ao, q) = {o-q[o-q~
q)

lit

3o'q ~ Gj such that ~q ~
By construction if ~r(a) ~ C a (ao) there exists no Gj(a0, q) g(a0, q)-related to a. If it were otherwise ~r(a) would share a q-simplex with a member of Cq(ao) and hence be a member of Cq(ao). Let Kq(ao, q) be all the q-dimensional faces of the simplices of Cq(ao), and let g: Kq(ao, q ) ~ G with g(trq)=G i iff t r q e G i. The relation ~r q) is illustrated in Fig. 7. sC(ao, q) will be abbreviated to sr when ao and q are unambiguously known. G

A,,

,-B

FIG. 7. The relation ~r

q).

By construction: PROPOSITION 4 tr(a)eFo_c KA(B, A)=),O-o(a) = (Go)

in KA(G, so),

t r ( a ) e F i __KA(B, h)z:),trl(a) =(GI-1, Gi)

in KA(G, ~).

358

J. H. JOHNSON

T h e definition of a simplicial mapping X: Cq(a0)~ K A ( G , ~:) may be based on this proposition: let X be defined by X: o ' ( a ) ~ KA(B, h ) ~ o-(a)c K A ( G , ~:), with o-(a) e F o f f X(o-(a)) = (Go) and

o-(a) ~ Fi=> X(o'(a)) ~T q Fo - ~ - - I I P ' ~ F

(Gi-1, Gi).

=

8~r q I--

8>T q

~.~



F2

~



-

~

>

<~2> r

.I'('~rq)o

F3-


<~>



--



E

X ('~'q)o

X(sFq )o

FIG. 8. q-Transmission becomes 0-transmission through the derived network, KA(G, ~:). X(87r) ~ which represents the 0-force derived f r o m cr~rq will not be given a precise definition here. It is likely that category theory could help sort out these complicated structures and mappings (see also Fig. 10, below). By the foregoing definitions q-forces are transmitted through mutually exclusive sets of q-simplices, viz. the G j: PROPOSITION 5. (Simplex partition theorem.) Let zr have the q-transmission property, and let or(a)~ Fi. The q-faces of o'(a ) in KA(B, h) are partitioned by Gi-1 and Gi - Gi-1. 8zr q on tr(ao) at time ~" will be q-transmitted to be experienced at the q-faces of cr(a) in Gi-1 at time "r+ i - 1, and the q-faces of o'(a) in G~ at time 9 + i. In other words forces are q-transmitted through the Gj, and every simplex o-(a) in Fi has its q-faces partitioned into those through which the q-forces " e n t e r " at time z + i - 1, and those through which the q-forces " l e a v e " at time z + i.

Corollary 6. (Vertex partition theorem.) Let o-(a)~ Fi. The set of vertices of tr(a) can be partitioned to give two simplices tr' and tr" such that tr = o-'. o-" in the exterior algebra (Atkin, 1974b). T h e r e exists a unique vertex partition such that, given a q-force on tr(a0) at time r, it is q-transmitted to every simplex o'q ~ tr' at time z + i - 1, and it is q-transmitted to every simplex crq ~ tr" at time r + i. 2.6. THE DERIVED BACKCLOTH KF(G, 77) AND KG(F, w), THE DERIVED NETWORK For given F and G a relation ~(a0, q) m a y be defined as (Fi, G j ) ~ 7 ( a o , q)

iff

j=i

or

/=i+1.

q-TRANSMISSION

IN S I M P I . I C I A L

359

COMPLEXES

KF(G)

KF(G) o"

o"

~(ao,q) q

~' G

A"'

"a FIG. 9.

From this definition it follows that: PRoPosrrioN 7. Both KF(G, 77) and KG(F, r/ ') are 1-dimensional complexes and fori>_ I o'x(Fi) =

(Gi-1,

Gi>

o't(G/) =

in KF(G, r/), in KG(F, r/T).

This means a q-force on ~(ao) in KA(B, h) will be transmitted through two conjugate networks determined by a0 and the value of q, and these structures will be called the derived network. (Fig. 10).

X ( 6",r q ) tl

O'o(Fo)

=

X ( 8-n.q)o

~,(F~)

=

9

~

(Go>

it, ( 6 ~ q')o

o~,(F2)

=

X (&n'q) ~

,:r~(F3)

~

~

(G,>

Xl,~q)O

r

~

(G2>

~ - --.

(G3>

(G4>

KF(G, r/) 777--/7

xo

xo

cr,(Go)

(Fo>

~

(FI>

o%(G,)

~

~o

o',(G2)

(F2>

~

~rz(G3)

(F3>

KG(F, r/"r) 11/////1

FIG. 10. T h e d e r i v e d n e t w o r k for q - t r a n s m i s s i o n .

(F4>

360

J. H. JOHNSON

2.7, TYPES OF q-TRANSMISSION, DRIVING FUNCTIONS

L e t 7r have the q-transmission p r o p e r t y and let the force 8~r originate on or(a0) at time r. L e t or(a) = or'. o'" such that (or', 81r)T+i-a r 0 but (or", 8r = 0. W e shall say that or(a) is 8-activated at time r + i - 1 and 8-responds at time r + i. T h e q-transmission process will be called even iff a positive (negative) 8-activation results in a positive (negative) 8-response. It will be called odd iff a positive (negative) 8-activation results in a negative (positive) 8-response. In this context it can be imagined that s o m e t h i n g is driving the q-transmission, and this could be r e p r e s e n t e d by a driving function D i : ~'tr~.-~ ~"/T~-+ 1 which could have the properties:

increasing

(o', 67r~.) < (or, Di6"h%),

constant

(or, B0r,) = (o', D~Brr,),

decreasing

(or, Brr,) > (o', DiSrr,).

It is also possible of course to consider the cases with D discrete, continuous, or even multiple valued. 2.8. DIRECTED q-TRANSMISSION

G i v e n o-(a) = or'. at" as above, it can be the case that (Oft, (~"Wr)96 0 ~ (or', ~"/Tr+l) ~a~0 ~ (r

87Tr+2) # 0,

so that the q - t r a n s m i s s i o n occurs b o t h f o r w a r d and b a c k w a r d . By definition all q-transmission is directed forward t h r o u g h the transmission fronts. In the case that (o", 81r~) does d e t e r m i n e (or', 87r~+2) via (or", 81rT+l) it will be said that the q-transmission is directed backwards. W h e n the q-transmission process can occur both f o r w a r d s and b a c k w a r d s it will said to be bi-directional.

R9

FIG. 11. Partisans p~ authorized to be in regions R i.

361

q - T R A N S M I S S I O N IN S I M P L I C I A L C O M P L E X E S

In the case that D is odd, decreasing and bi-directional the value of (or(a), ~-)

oscillates between (o'(a), 7rT) and (co(a), 7rT~), i.e. between (or(a), zr~_t)+(6r(a), 8~'T) and (or(a), 7r,-1)+ (o-(a), 8 ~ - , ) - (or(a), 8zr~, 1), where tr(a) is activated at time z. In some cases the q-transmission may represent a qualitative change, for example the acquisition of information: (or(a), 7r)= 1 if person a has certain information, (or(a), z r ) = 0 otherwise. In this case a person is activated when he acquires the information and, assuming he does not forget it, remains activated over time: (or(a), 7r~)= 1 ~ (o'(a), 7r~ 1) = 1, so that backward transmission is irrelevant. In other cases administrative rules may forbid backward transmission, for example people may be q-transmitted through the d e p a r t m e n t s of an organization to "learn the ropes" and there may be a rule that a person cannot return to the d e p a r t m e n t he came from previously. Such rules, or constraints on the driving function, may be located at specific parts of the backcloth. If ~r(a) has the property that it is 8-activated at time r, 8-responds at time ~-+ 1, but (o-(a), 87r~+2) = 0 it will be called a q-transmission

valve.

3. Examples E X A M P L E 1. S T R E T C t t E R - C A S E

B E H I N D E N E M Y LINES

Consider a resistance m o v e m e n t of partisans behind e n e m y lines in an occupied country. Suppose the country is divided into administrative regions and nationals require authorization to be in an area. Let region Ri be A-related to partisan Pi iff he is authorized to be in that area. Suppose a plane crashes and one of the t w o - m a n crew cannot walk. The partisans can lead the walking airman through e n e m y territory, and one guide leading within

(a)

(b)

FIG. 12. (a) 0-Transmission fronts for walking airman; (b) 1-transmission fronts for wounded airman on stretcher.

362

J.H. JOHNSON

his authorized area is sufficient. This airman can be 0-transmitted through the transmission fronts F 0 = {R3}, FI = {R4, R5, R7, R8}, F2 = {R6, R9, R10, R l l , R13, R14, R15, R16} and F3 = {R12}. Suppose, however, the wounded airman must be carried by two partisans. If the partisans are not to move outside of their authorized areas the wounded airman must be 1-transmitted through the transmission fronts F0={R3}, F1 ={R4, R7}, F 2 = {R5, R8}, F3 = {R10, R14} and F 4 = { R 1 1 , R15}. Thus it takes "longer" to move the wounded man, and there are regions through which he cannot be moved. EXAMPLE 2. CLASSIFICATION AND TAXONOMY Let A be a set of "things" and B a set of "characteristics" with Ai A-related to Bj if thing i can be described by characteristic j. In conventional terms A admits a "perfect" classification if it can be partitioned in such a way that a and a ' belong to the same class implies or(a) = cr(a'). In the real world such classifications are rare and taxonomy becomes the art of looking for shared characteristics while overlooking others, i.e. investigating the q-nearness of the things in KA(B, }t) (Johnson, 1978). However q-nearness is not an equivalence relation and it is possible for a to be very similar to a', a ' to be similar to a", but for a and a" to be quite different.

B2

89

0

a.4(a )

B5

~, , I I I

B6

.'4(0

,,

)

o'TCo') FIG. 13. a is very similar to a', a' is very similar to a" but a and a" are quite different.

Thus q-connectivity is necessary but not sufficient to define classes of things which are similar with respect to B. In taxonomy there is a kind of non-discrimination traffic within a class, i.e. within the q-connected components. This suggests the definition of a q-taxonomy of the elements of A with respect to B as the partition of those a e A with dimension q or more into q-connected components. Then the q-transmission number of a taxonomy is the q-transmission number of KA(B, A). In general the higher the value of q and the lower the value of # q K A , the more satisfactory will be the taxonomy. Experience in the application of Q-analysis shows that clean and useful partitions are rare in set definition and classification. Instead of looking for a partition defined by a single q-value it is likely that covers depending on a number of q-values will be more useful.

363

q-TRANSMISSION IN SIMPIJCIAL COMPLEXES

The 3-transmission diameter of the complex in Fig. 13 is 2: for a; Fo(a, q ) = { a } , F~(a, q) {a'}, F2(a, q) {a"}. =

=

EXAMPLE 3. 0-TRANSMISSION OF TYPING THROUGH DEPARTMENTAL SECRETARIES

Consider a university department which has five secretaries serving 20 members of staff. Suppose that for administrative purposes each secretary does typing for six of the staff members according to the relation A, below, between secretaries, S, and academics, A.

h $1 $2 $3 $4 $5

A1 A: A3 A4 As A6 A7 A8 A9 AloAIIA12A13AI4AlsA16A17A18A19A2o 1 0 0 0 1

1 0 0 0 1

1 0 0 0 0

1 0 0 0 0

1 1 0 0 0

1 1 0 0 0

At9

0 1 0 0 0

0 1 0 0 0

0 1 1 0 0

A2o

0 1 1 0 0

0 0 1 0 0

0 0 1 0 0

0 0 1 1 0

0 0 1 1 0

0 0 0 1 0

0 0 0 1 0

0 0 0 1 1

0 0 0 1 1

0 0 0 0 1

0 0 0 0 1

A5

A~ ........

o-(S~) A;,

.......

Ai2

a-(S3)

All

FIG. 14. KS(A, A). Suppose all the secretaries are busy and academic A 3 brings his new paper to be typed by his secretary $1. In this case the only way $1 can do the work is to shuffle some of her other work on to S: and $5, and in order to do this they in turn may need to shuffle some work onto $3 and $4. Thus in this case the typing can be 0-transmitted through the transmission fronts Fo(S~, 0)={$1}, F1($1, 0) = {$2, $5}, F:(S1, 0) = {$3, $4} and the structure has transmission number 2. EXAMPLE 4. WAGE CLAIM TRANSMISSION THROUGH

THE JOB-I.OCATION BACK-

CLOTH Let job J~ be A-related to location Lj if the job is to be found that location. Suppose that workers in each job group are well-organized and very sensitive to the preservation of the wage differentials which exist between their group and the others in their

364

J.H. JOHNSON

location. Suppose also that Trade Union representation means that a settlement of x % wage increase for a job in one location will result in claims for not less than x % increase for the same job in different locations. It is easy to show that the total wage bill at each location increases by a 0-transmission process, and the driving function D is positive increasing. J~

dz

J

3 J4

JB

JIL

-: X

Jn d14

~-(L 2)

~

J12

~ d9

~-(L I)

8wocje

Jr

~woge

o-(L 3)

~- y ) X

~

~woge

~ Z )y

> X

~

FIG. 15. 0-Transmissionof wageincreases. Although this is a simplistic description of the annual adjustment of wages and salaries, consider what happens when G o v e r n m e n t attempts to restrain percentage increase and improve the lot of the lower paid. Here the G o v e r n m e n t is attempting to reduce the numerical value of the driving function D in a way which is non-linear. It is interesting to note that while it is natural for managers to be interested in what happens within their company or group, Trades Unions have a view of industry which transcends companies. While managers are concentrating their attention on the joblocation structure of their organization (even if only intuitively) they will formulate pay structures which are locally consistent and reasonable. Such an attempt at rational management may be frustrated by 0-transmission of pay scales which make the pay structure globally inconsistent. In other words, the management can discuss the particular pay structure as the wage pattern and its particular change 6wage, but it has no control over the rules for defining 8wago which are embodied in the driving function D.

EXAMPLE 5. 1-TRANSMISSIONOF INDUSTRIALINFORMATIONBY LORRYDRIVERS Let C be the set of companies within a trading group and let Ci be A-related to Ci if one delivers goods to the other. It is well known in the field of industrial relations that a major means of communicating information on the local industrial relations conditions exists in the drivers whose primary responsibility is moving goods. This traffic of shop-floor feeling across the group will of course effect industrial relations when there are genuine grievances. Let us assume that there are two reactions to the stories brought into a factory by the drivers from another. In the case that a story comes from a single factory it may be entertained as unsubstantiated rumour, but of course still passed on to another factory. In the case that the story comes in from drivers from two distinct factories it may be regarded as information to be acted on.

q-TRANSMISSION IN S I M P L I C I A I . C O M P L E X E S

365

Thus the 0-connected components act as the backcloth for unsubstantiated r u m o u r to be transmitted through the group, while the 1-dimensional backcloth will allow the transmission of assumed information through the group. In both cases the transmission type give an indication of how quickly information can spread through the group. It is interesting to note that the tactic of disconnecting factories to restrict the 0transmission of r u m o u r will also restrict the transmission of (probably quite legitimate) information. If it can be assumed that, sooner or later, information will be transmitted to all factories (for example in T r a d e Union publications) it might be a good tactical move toward good labour relations to m a k e this information transmission backcloth more highly connected. If rumours cause unrest it may seem paradoxical to enable them to be m o r e easily transmitted to avert unrest, but the paradox is resolved by noting that the improved backcloth connectivity will allow m o r e reliable information to be transmitted more quickly. In organizations not trying to outwit their employees a low transmission n u m b e r for information transmitting structures is presumably a good thing. E X A M P L E 6. I - T R A N S M I S S I O N O F J O K E S B E T W E E N P U B L I C H O U S E S

Consider a joke which requires two people to tell it properly, perhaps a "straight guy" and a comic. Suppose the joke is rather complicated and if a single person relates it by playing both parts it just does not work and does not seem funny. If the set of public houses H is related to the set of people according to whether a person visits a pub, the joke can only be 1-transmitted through KH(P, A) by pairs of people sharing the same pubs. One can speculate that m a n y m o r e pubs share a single customer than share two or m o r e and that the obstruction to 1-dimensional jokes is much greater than the obstruction to 0-dimensional jokes. The latter are indeed transmitted very rapidly.

4. q-Transmission and time 4.1. N O W - T I M E A N D I N T E R V A L S

It is necessary to be more precise on the nature of the " t i m e " which underlies the concept of q-transmission. Let T = { t l . . . . . tn} be a set of n o w - m o m e n t s (Atkin, 1978a). No algebraic or topological properties will initially be assumed of T beyond. Time hypothesis 1

There is a quasi-order on T, i.e. a relation -< ~ T x T with (i) ti --
366

J.H.

i \

N /

N~

! i Now3 i :

Now~ I

JOHNSON

/ //

:: : i :

Now 5

Now I

(b)

(o) FIG. 16.

As illustrated, the actual position of the bob does not determine the definition of a now-time, rather it is the intuitive observation that the direction of motion reverses. In other words the now intervals are really the observables {left-right reversal 1, left-right reversal 2, left-right reversal 3 . . . . }, and these can all be discriminated by watching the bob. If the stationary bob in Fig. 16(b) is observed there is nothing observable to define m o r e than one now-time, even though during the observation there may be a sense of duration, e.g. "for how much longer am I going to observe Now1 on this stationary p e n d u l u m ? " . In physics the now times are counted and each can be m a p p e d to its sequential n u m b e r of occurance, i.e. a mapping r can be defined with r ( N o w - t ) = t. r can be extended to a m o r e abstract mapping 7 : T x T ~ " d u r a t i o n " with ~(tj. t,)= r(t~) - .r(t,) = ] - i.

Physical scientists have m a d e the inductive leap of asserting that (with respect to pendulum clocks), this mapping captures the concept of " d u r a t i o n " and this notion of duration was appropriate for the description of motion. A particular consequence of these definitions is that duration is independent of sequential now-time ordering, and that durations are additive. This view of time still holds today, even in relativistic physics if the observer and pendulum are assumed to be in the same frame of reference (Muirhead, 1973). It is important to realize that in physics motion and change in motion are used to define time in the first place, and time is then subsequently used to define motion as velocity, acceleration, etc. In other words motion is tautologically defined by motion or, equivalently, time is tautologically defined by time. Since this approach lifted physical science out of the metaphysical doldrums, and in the absence of anything m o r e promising, it will be assumed that social processes occur in tautologically defined social time. In other words it is supposed that "time is the manifestation of relations between events" (Atkin 1978a). As in the case of the stationary pendulum a social now-time will be considered different from another only when something changes. T h e r e is no a priori reason to chose one change-entity from another, and the procedure is more one of inspired trial and error than methodological prescription.

367

q - T R A N S M I S S I O N IN SIMPI.ICIAI. C O M P L E X E S Duration

Duration

r ( Now2, Now I )

r (Now3, Now z)

=(2-])=]

r(NowI) =l

:(3-2)=~

r(Now3)

r(Nowz) =2

A

--3

,A

A

,

Now I

Now 2

Now 3 _..)

Y i i

i

Duration r(Now3,Now I ) :(3-1):2

FIG. ] 7.

The "right" change entity will be retrospectively recognized by its giving a satisfactory description of change processes, and it is worth noting that the change-entities of physics, or equivalently "clock-time", fail in this respect for social systems (Sorokin & Merton, 1937). Let K be a simplicial complex and 7r be a graded pattern on K. Given any simplex o- E K suppose two observations, o] and 02, of r are m a d e by the same observer.

Time Hypothesis 2 A single observer can order his observations as being m a d e simultaneously or asynchronously. If two asynchronous observations are m a d e one always is m a d e before the other in that the record of observation can be ordered as: (no observation),

(First observation is made'~ but not second )'

( B o t h observations~ \ are m a d e ) "

Thus, each observation is m a d e at a now-time, and the now-time of " a s y n c h r o n o u s " observations is different with respect to K and ~" if

((~, ~), o,) ~ ((or, ~'), o2), where ((or, rr), oi) means the observation oi of the value of or under lr. 4.2. C O N S T R U C T I N G A S E Q U E N C E OF N O W T I M E S

Consider a single observer making observations throughout his life, and assume he records all his observations as a finite set. For this observer let 01 be the set of observations he has made, i.e. 0i = {ojlthe observer has m a d e all the observations o i up to the ith "now"}. Through the observer's life 0i changes and a sequence of sets of observations exists as {01, 02 . . . . .

0n},

368

J . H . JOI]NSON

w h e r e Oi ~_ Oi+z, and On represents the observer's total set of r e c o r d e d observations " n o w " . F o r this o b s e r v e r define a n o w - t i m e for each Oi as {tl, t2 . . . . .

t,}.

By construction ti = tj iff 0i = 0 i and j = i + 1 m e a n s no new observations are made b e t w e e n ti and tj. T h u s the o b s e r v e r ' s time changes only as he m a k e s observations of things changing. Things seem to be constantly changing a b o u t us and most of us have a p e r m a n e n t feeling that time is passing since an individual's set of n o w - t i m e s is based on all his observations w h e t h e r formal or informal. This is the time the individual needs to describe his p e r s o n a l life, indeed this time actually is his life, but for most purposes it is too detailed. This applies to b o t h physical and social sciences and the definition of time is usually m a d e with respect to a small subset of formally made observations w h e n " b e i n g empirical scientists". In o t h e r words, the c o n c e p t of time is defined with respect to selected traffic on selected parts of the backcloth. In t e r m s of this discussion it is meaningful to hypothesize a set of now-times {tx . . . . . tn} with respect to a p a t t e r n ~- on a c o m p l e x K m e a n i n g that n observations of ~r were m a d e and these observations can be o r d e r e d 1 to n. In future the expression (o'. (tO, or) will d e n o t e the value of ~r on the simplex tr ~ K at n o w - t i m e t;. Let the set of n o w - t i m e s be written

To = {(tj)}, and let T1 = {(t~, tj)l(t,), (tj) ~ To}. T h e expression (or. (ti, tj), S~r) will d e n o t e the c h a n g e in value of ~- on cr b e t w e e n the n o w - t i m e ti and the n o w - t i m e t~. In this context it is meaningful to introduce the algebraic relationship

(t,, tj) = -l(tj, t,), w h e r e K • (To w T1) is interpreted as a chain c o m p l e x with the integers as coefficients. Thus

(,~. (t~, tj), 8~r) = (,~. (tj), ~')- (~. (t,), It). If c~ represents the usual b o u n d a r y o p e r a t o r of algebraic t o p o l o g y (Hilton & Wylie, 1965) with

a(t,, tj) = (tj)- (t,), we have

(o,. (t,, t~), a~r) = (or. ~(t,, t,), ~-). F o r any simplex trp e K a c h a n g e in the p a t t e r n 7r b e t w e e n tz and t2 is thus represented as p + 2 traffic (Fig. 18). In this respect zr o+z is the n o w - p a t t e r n and 7/"p+2 is the interval pattern. DEFINITION (Atkin, 1 9 7 8 a ) . On any given backcloth S(N) time is a specific traffic, viz.

(i) the traffic which consists of a total ordering of all p-simplices in S(N);

q-TRANSMISSION IN SIMPLICIAL COMPLEXES < tI >

<

369 t2

---~- (o-p.< t I ,t 2 >, #r) = (~-p.< t~,>,w)-(wp, <

(~o. < t t >, r )

tI

>,~-)

(~a.< tz > ,.-) FIG. 18.

(ii) the time pattern of this traffic is of the form 7. = ~ ( r V + ' r p + l ) = ~ Y V ; p~O

p

(iii) each YP contains the pattern r p and ~.p+l where ~'P is the now pattern a n d r p ~1 is the interval pattern. F u r t h e r m o r e A t k i n ( 1 9 7 8 a ) suggests T h e c o n c e p t of " e v e n t " is a p e c u l i a r l y s t r u c t u r a l c o n c e p t . S o m e e v e n t s a r e 0simplices, s o m e will b e 3 - s i m p l i c e s ; g e n e r a l l y an e v e n t will b e t h e r e c o g n i t i o n of a p - s i m p l e x in S(N). 4.3. q-TRANSMISSION AND p-EVENTS I n t u i t i v e l y the p + 1 e v e n t o'p. (tl) m e a n s t h a t o-~ is r e c o g n i z e d at n o w - m o m e n t ti. If p is a " r e c o g n i t i o n p a t t e r n " t a k i n g v a l u e 1 for r e c o g n i t i o n a n d 0 for n o n - r e c o g n i t i o n the o b s e r v a t i o n (o',,.
is a p + 1 event. S u p p o s e a n o t h e r p a t t e r n ~- is m e a s u r e d on ~rp " a t n o w - m o m e n t ti". T h e observation consists of t h e p a i r ((o'p. (t,), p), (crp. (t,), zr)) = (1, x), m e a n i n g t h e s i m p l e x o-,. (t~) t o o k t h e v a l u e x u n d e r 7r w h e n cr0 . (t~) was r e c o g n i z e d , or t h a t ~rp. (t~) t o o k the v a l u e x u n d e r ~- at e v e n t crp. (ti). F o r e x a m p l e , let crp = (s), w h e r e s is a l o c a t i o n . T h e n ((s, ti), p ) = 1 m e a n s (s, ti) was recognized and

(((s, ti), p), ((s, ti), 7r) = (1, x) m e a n s that at p o s i t i o n s a n d t i m e - m o m e n t tg t h e p a t t e r n ~r t o o k t h e v a l u e x. Since the r e c o g n i t i o n p a t t e r n is i m p l i c i t in m o s t o b s e r v a t i o n s it is c o n v e n i e n t to define for all p a t t e r n s an e x t e n s i o n "rr ~ 7r. p,

370

J.H. JOHNSON

meaning that ~- takes the value 0 on events which have not been recognized. A consequence of this is that the extension of 7r is non-zero only on the recognized structure-time backcloth: patterns are only measured as non-zero for p-events. 4.4. TRANSMISSION EVENTS

Let 0-(1) and 0-(2) be q-near by sharing the face 0-q. rI

r2

r3

\

.ci;

(2) %

FIG. 19.

Underlying the idea of q-transmission is the possibility that a change on o'(1) at the time rt can induce a later change on 0"(2) by virtue of the shared face 0"q. In other words it is postulated that (0"(1). (rl, r2), 3rr) # 0

implies

(0"(2). (7"2, 7"3), ~TT) ~ 0,

since 0-(1) ~ 0-(2) = 0-q. To see what this means: (0-(1). (ra), rr) = a~]

(0-(1) (7.2), ~') = ~2 /

f i r ( l ) . (7"1, r2), & r ) = a 2 - a , implies

(r (r~), 7/') (0-(2) (~'~), ~') =/33

=1321~

(r

(7"2, 7"3}, ~ ' / 7 " ) ~ ' ~ 3 - / 3 2 -

The essence of q-transmission lies in the assertion that changes must be transmitted through the temporal-structural backcloth. In particular the change on 0-(1) is transmitted to r through the shared q + 1 event 0-q. (7"2). With these definitions q-transmission can be viewed as a sequence of (q + 1)- and (q +2)-events. In a sense the q-transmission fronts can be seen as sets of "simult a n e o u s " p-events for p = q + 1, q + 2. 4.5. D E R I V E D T R A N S M I S S I O N EVENTS

By the simplex partition t h e o r e m of section 2 every simplex in a complex K belonging to q-transmission front Fi has its q-faces partitioned into G i - i and Gi through which q-transmitted faces enter and leave. In this context define the set of derived events as ((O,). % ) , O) = 1,

q-TRANSMISSION

IN SIMPLICIAL

371

COMPLF.XES

where p is defined with respect to a single observer meaning that all the Gi are in the "same frame of reference". With this definition the transmission can be viewed as taking place on two conjugate derived temporal-backcloth 2-dimensional complexes as 1- and 2-dimensional events. r2

z"3





(xb'~'*) z

T4



,~G 3 >

2- ( x ~ m ) z .

)-(xs~'q) z

FIG, 20. The derived-temporal backcloth for transmission events (see also Fig. 21). 4.6. q-TRANSMISSION, p-EVENTS AND PREDICTION Given a q-connectivity we might ask when a t-force at one end will be q-transmitted to the other, The answer can be given precisely as the number of transmission fronts along the chain. ro

rl

r2

Fo

FI

F2

;'-3

rn-I

Fn -I

rn

Fn

FIG. 21. ?J~rcan be predicted to arrive at cr at time-moment n. This gives the tautological prediction that a t-force will arrive at a simplex o- precisely at the m o m e n t it arrives. Those of us steeped in Western traditions will ask impatiently, "but when will that be'?" and we will be irritated that no clock-time or calendar date is forthcoming. When they are not trying to force the social world into a physics-like system of laws administrators always work in social time. For example, consider a building due to be electrically wired in February and decorated in April. If an industrial strike delays the wiring the decorating will be put off: the important now-moment being "the wiring is d o n e " and not "the calendar shows 31 March". This is one of the reasons why time-series analysis which make observations against clock-time must fail to give meaningful predictions in many cases.

372

J.H. JOHNSON

H o w e v e r the p r o b l e m of co-ordinating events remains: how do administrators make events occur together? In our social lives we m a k e use of clocks and calendars to achieve this, "let's m e e t at 11.00 a.m. next Friday" usually achieves the objective. But this is merely the beginning, few people would continue with their arrangements as " a n d we will talk for 18 rain about A, 27 rain about B, 42.5 rain about C, etc.". Since clock-time can only guarantee the p-event of meeting, it cannot g u a r a n t e e / p r e dict the subsequent t-forces since these are measured by the n o w - m o m e n t s "we have finished talking about A " , "we have finished talking about B", etc. It appears that a m o r e useful attitude to prediction would involve specifying the p-events necessary to achieve a future objective. On the basis of this one can at least predict when an objective is impossible given the existing backcloth. It also allows the prediction that we shall know what to do at n o w - m o m e n t t whenever it occurs in calendar/clock-time. However, clock-time need not be thrown away altogether since it offers a way of knowing when various n o w - m o m e n t s are synchronous. 4.7. R E L A T I V E q-TRANSMISSION A N D C L O C K - T I M E

T h e n o w - m o m e n t s underlying q-transmission have been constructed with respect to a single observer. As such each observer will have his own set of times, and this would differ from most other observers. This immediately brings up the relativist question " h o w do the observations of one observer based on his n o w - m o m e n t s relate to the observations of another observer based on his different set of n o w - m o m e n t s ? " . Indeed we can formulate cases of observers socially moving away from each other and these observers having different perceptions of the " s a m e " thing. Consider a backcloth with three vertices (Job), (Shop), (Bank), and two timem o m e n t s 7"1 and r2. 7" I

v

Job

Shop

FIG. 22.

Let observer A and observer B agree that ((Job). (~.AB), ~r~) = s

wages,

((Shop). ( r ~ ) , 71"1) = - s

spending,

((Bank). (r AB ), r 1) = - s

saving,

and that ~_AB= ~.~ = r~. Using the coface operator A we have ~2rrl = r 3 = s

q-TRANSMISSION IN S I M P L I C I A L C O M P I . E X E S

373

i.e. w a g e s = s p e n d i n g + s a v i n g for these observers. S u p p o s e o b s e r v e r B negotiates a 10% pay rise at t i m e - m o m e n t ~'z, i.e. he observes

((Job). (r~,

ran),

a,n-1) = s

with ((Job). (Tan), ~r 1) = s Suppose also that ((Shop). (ran), ~r 1) = - s

for o b s e r v e r B and

( ( B a n k ) . (r~, ran), (5~"z) = - s B ;"2

AB

rl

Job

Shop

FIG. 23.

Nothing has c h a n g e d for o b s e r v e r A not k n o w i n g of B's a d v a n c e m e n t and he is financially at r AB, while B is at ran : the two o b s e r v e r s are m o v i n g a p a r t in social space. S u p p o s e wages and prices are related as / wage per m a n \ (prices)v, oc / . . . . / \ p r o d u c t i o n per man/~,'

zi >- ri,

and assume that B ' s wage increase does not involve a productivity element. Then s o m e t i m e after B's zan, say r3, there will be a change in prices as, say,

((Shop). <~), ~.1) = s Then B can m a k e the o b s e r v a t i o n that ((Job, S h o p ) . (r3n), ~.2) = s 1 6 3

= s

but the only o b s e r v a t i o n A can m a k e is ((Job, S h o p ) . (r3n), ~.2) = s 1 6 3

= s

Thus despite the fact that prices rise on the s a m e calendar d a y for b o t h A and B, both are experiencing different 2-events. A and B are living at different time-moments, even on the s a m e day! It is a very real question as to h o w A and B can observe each other. Is it an exaggeration to say, for e x a m p l e , that rich and p o o r cannot contemplate what the world looks like f r o m the o t h e r s ' point of view? E a c h o n e of us has an individual view of things and we genuinely live in a social s p a c e - t i m e world of our own. T h e c o m m u n i c a t i o n necessary for society to function can b e c o m e a real p r o b l e m and the a p p a r e n t l y socially indifferent clocks and calendars can be useful in providing points of reference. F o r any p e r s o n let C T be the clock-time

374

J.H. JOHNSON T2

r~

dob

Shop

FIG. 24. Observers A and B moving away from each other in social time-space.

or calendar-time mapping with CT: {time-moment 1, t i m e - m o m e n t 2 . . . . }-~ calendar-clock-time. Without worrying too much of the properties of CT or clock-calendar-time (topology of the latter, continuity, etc. of the former) most of our social events can be tied down to a clock-calendar time interval or instant. Given CT individuals can begin to c o m p a r e their observations, and indeed find points of a g r e e m e n t such as noting that both observed the passing of a certain clock-time. With patience the various p-events can be shuffled backwards and forwards relative to clock-time allowing observers to retrospectively admit time-moments. This is presumably what education and scholarship are all about, the former authoritatively giving t i m e - m o m e n t s the student could not experience, the latter attempting to authenticate t i m e - m o m e n t s not agreed by all. Clock-time is a resource to be used intelligently like any other. It can be used in the thoughtless and dry way of the economist and time-sequence curve-fitter with the unfortunate social effects we observe all too clearly. It can be used better and there is plenty of indication how. Certainly the biological-physical aspect of h u m a n beings makes some calendar-clock-time events coincide with social events. Atkin (1978b) has showed how the dimensionality of p-events may give an indication of their relationship with calendar-clock-time events.

5, Research suggestions 5.1. M U L T I - O R I G I N TRANSMISSION

The presentation of q-transmission in this paper is in terms of a single originating change in pattern on an element a0. In applications it is possible and likely that the "originating" changes will occur on m o r e than one ai within a q-connected component. In this case the transmission fronts of each origin will eventually merge and the derived network will naturally depend on the site of the originating changes. A possible research p r o b l e m involves multi-origin transmission involving m o r e than one transmission process and associated with this different time sequences. The many possible ways of investigating this situation have not been laid out here because they would require too much space. Out of the sets of now m o m e n t s one would have to decide which were to be considered "synchronous" and this would effect the trans-

q-TRANSMISSION IN SIMPI.ICIAI. COMPI.EXES

375

mission fronts. D e c i d i n g which identities are the m o s t interesting and relevant is best done in the context of real applications, and there the m a t t e r will rest for this paper. 5.2. SHOMOTOPY OBJECTS AND OBSTRUCTION TO q-TRANSMISSION S h o m o t o p y objects will clearly act as obstructions to q-transmission and it seems likely that the existence of objects will increase the transmission n u m b e r of a q - c o n n e c t e d component. It is interesting to speculate on t-forces being transmitted r o u n d objects, but to investigate this it would be necessary to m o d i f y the definition of transmission fronts so that they do not partition the elements of A. A g a i n this is a d e v e l o p m e n t best left until it can be considered in the context of practical applications. 5.3. a-/3-FORCE TRANSITION An a - f o r c e can result in a fl-force w h e r e a ~ f l , w h e r e for q-transmission this is implicitly assumed to h a p p e n on particular simplices according to the structure of the coface operator. If it is a s s u m e d that a is less t h a n / 3 the possibility of an a - f o r c e being t r a n s f o r m e d into a / 3 - f o r c e represents a p r o b l e m for q-transmission, since it is not clear which value to take for q. T h e obvious choice of taking the lowest value is consistent with the o b s e r v a t i o n that m a n y social processes a p p e a r to be capable of transmission t h r o u g h 0-dimensional complexes. A g a i n practical applications are required to p u r s u e this possibility and u n d e r s t a n d its consequences. I am grateful to Dr C. Earl and Dr Y. Ho, for pointing out some inadequacies in the first draft of this paper, and to Dr A. Gatrell of Salford University for bringing a serious manuscript error to my attention.

References and bibliography ATKIN, R. H. (1974a). Mathematical Structure in Human Affairs. London: Heinemann Educational Books.

ATKIN, R. H. (1974b). An algebra for patterns on a complex, I. International Journal of Man-Machine Studies, 6, 285-307.

ATKIN, R. H. (1977a). Research Report X. University of Essex: Regional Research Project. ATKIN, R. H. (1977b). Combinatorial Connectivities in Social Systems. Basel: Birkhauser. ATKIN, R. H. (1978a). Decision making as an Event Search: traffic on a multi-dimensional structure. Mimeo. University of Essex (March). ATKIN, R. H. (1978b). Time as a pattern on a multi-dimensional structure. Journal of Biological Structures 1978(1), 281-295. ATKIN, R. H. (1978c). Q-analysis: a hard language for the soft sciences. Futures, 492-499 (December). ATKIN, R. H. (1980). How to study corporations by using concepts of connectivity. Mimeo. Essex University (March). ATKIN, R. H. (1981). Multi-dimensional Man. Harmondsworth: Penguin. GOULD, P. (1980). Q-analysis, or a language of structure: an introduction for social scientists, geographers and planners. International Journal of Man-Machine Studies, 13, 169-199. HII.TON, P. J. & WYLIE,'S. (1965). Homology Theory. Cambridge University Press. JOHNSON, J. n . (1975). A multi-dimensional analysis of urban road traffic. Ph.D. Thesis, University of Essex. JOHNSON, J. H. (1976). The Q-analysis of road intersections. International Journal of M a n Machine Studies, 8, 531-548. JOHNSON, J. H. (1977). A study of road transport. University of Essex: Department of Mathematics.

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KILMINSTER,C. W. (1970). Special Theory of Relativity. Oxford: Pergamon Press. LEFSCHETZ, S. (1949). Introduction to Topology. Princeton University Press. MUIRHEAD, M. (1973). The Special Theory of Relativity. London: MacMillan. SOROKIN, P. A. & MERTON, R. K. (1937). Social time: a methodological and functional analysis. American Journal of Sociology, 42(5), 615-629.

Appendix: Preliminary notation Let I be a relation b e t w e e n two sets A and B, d e n o t e d h c- A x B , such that a is h-related to b itt (a, b ) e h. For each a e A define an abstract p-dimensional simplex, p-simplex, trp(a) = (b . . . . . . .

b~,),

w h e r e b~, is a vertex of ~rv(a), i.e. (b~,) is a O-dimensionalface of trp(a), if and only if a is h-related to b~,. This is symbolically r e p r e s e n t e d as

(b~,)~trv(a)

iff

(a,b~,)~h.

W h e n it is convenient not to specify the dimension, trp(a) will be written as ~ ( a ) . In general tr ~< tr~ m e a n s tr is a [ace of crp. T h e set of all simplices {crl3a ~ A such that tr ~ o'p(a)} is a simplicial complex on the vertex~set B and will be d e n o t e d K A ( B , h ). T h e simplicial c o m p l e x similarly o b t a i n e d f r o m the relation h a'c_ B x A , (a, b) ~ h iff (b, a) e h T, is d e n o t e d K B ( A , hT). T h e s e complexes, a b b r e v i a t e d to K A and KB, respectively, when u n a m b i g u o u s , are said to be conjugate to each other. A simplex trq is a q-dimensional face of a simplex ~rp if every vertex of crq is a vertex of crp. T h e face relation is written trq ~ o'p. T w o simplices are said to be q-near in a c o m p l e x K itI they share a q - d i m e n s i o n a l face in K. T h e relation of being q - n e a r on the p-simplices of K with p-> q is reflexive (each tr, is q - n e a r itself) and symmetric (err q - n e a r ~s implies o's is q - n e a r o'r). T w o simplices are q-connected in K if there is an intermediate s e q u e n c e of q-near simplices in K. In o t h e r words the relation of being q - c o n n e c t e d on the p-simplices of K with p - q is the transitive closure of the relation of being q-near. Trivially for q > 0: PROPOSITION 1

cr~ q-near tro implies ~r~(q -- D-near crt3, tr~ q-connected ~ro implies cr~ (q - 1)-connected cro. T h e q-connectivity relation is w e a k e r than the q-nearness relation. Thus:

q - T R A N S M I S S I O N IN S I M P L I C I A L C O M P L E X E S

377

PROPOSITION 2. q-Connectivity is an equivalence relation on K~ where Kq = {o'p ~

Kip -> q}.

DEFINITION 1. The equivalence classes of Kq determined by the q-connectivity relation

are called the q-connected components of K.