Graphs and Hypergraphs in Cross-Impact Analysis

Copyright © lFAC Dynamic Modelling and Control of National Economies, Budapest, Hungary 1986

GRAPHS AND HYPERGRAPHS IN CROSS· IMPACT ANALYSIS I. Ocka R esearch Institute VU5TE, Velflikova 4, 160 61 Prague, Czechoslovakia

Abstract. Methods and techniques of Cross-Impact Analysis can be further improved by an application of the graph and hypergraph representation of the system under study. The main task of the analysis is to create, identify an adequeate of the future. The scenario itself is presented in the form of events and their mutual interactions. The probability evaluation is applied as to express the interactions intensity. The proposed approach introducting graph and hypergraph representation of the system is brought as to support the process of identification of the system of events and their mutual interactions. Keywords. Graph theory; System analysis, Cross-impact; Identification.

INTRODUCTION The recent development of prognostic methods and of methods of strategic planning has largely focused on methods and techniques for creating and analysinf alternative future scenarios. Most of these methods create and analyse scenarios as a set of events and their mutual interactions.

by consulting a professional strategic planner who is expected to identify and specify the events and who also identifies the interactions and their probabilities. In doing so, he, as a rule, relies on some type of model simulations. The main obstacle facing the identification procedures rests in the necessity of aggregating the future into a workable number of events. Such limited number of events is to represent the future - the scenario with the required accuracy. Even for an expert in the strategic planning it is often a task, of a great ~if­ ficulty to identify the interactions Iprobabilitiesl within the set of events, as among other things, the relations connecting individual events are often too numerous. IFor example, the number of simple event-to-relations is, in general, n n-1, n being the number of events. In addition to those relations, conditional interactions of

By a scenario we shall understand a statement concerning the future which is presented as a set of events and corresponding information on whether or not the events occured. Occurence of a certain event and the corresponding probability of the whole scenario realization can be obtained by means of the probability evaluation of the events mutual interactions. Methods and techniques based on the above montioned principle are widely referred to as Cross-Impact Analysis. Each of such methods can be applied

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428 a far greater number must be taken into account.1 It is therefore of a great difficulty to select the relations relevant to the analysis of the future, not to mention ascribe the corresponding probabilities to them. Many inconsistencies can be therefore observed when the consistence of expert estimates of probabilities are tested.

It is our conviction that the key element of an efficient application of the CrossImpact Analysis lies in the identification of the system of events concerned and their relations. This is in accordance with a series of refinements of the original Gordon and Hayword 4 method see~ing to support the strategic planner s effort through different and more appropriate forms of the interactions estimates and through the helping cal : ulations while testing the estimates consistency. Our paper deals with a few ideas on the improvement of the identification of the system under study. It attempts to stress certain ways of the system modelling by graphs and hypergraphs and to propose some directions in which the identifikacion of the system can be further improved. 1. DEFINITION OF A SIMPLE BASIC SYSTEM Let the future be described by n events denoted as E1 , E2 •• , En' which are from our point of view regarded as relevant for the description of the behaviour of the system under study. Let us assume that the events are irreversible, i.e. each of them can obtain only once, if ever during the observed period of time. Let us further assume that the events are mutualy dependent in a stochastic sense of the term. A state of the system-scenario can be denoted as Sk (where k stands for the number of the state) and defined as Sk = {e; , e; ••• , e+J

where

if the ith event occured, i. e. Ei occured if the ith event did not occurert, i.e. Ei occured (1.1 )

At the end of the period concerned the system will occupy one of the 2 n feasible states 1 £ k ~ 2 n • The expert is to estimate the probabilities of all events and all conditional probab i t it i es:

= 1,2, ••• ,n

P {E i ) P(Ei'E j )

i,j

1,2, ••• ,n;

t.

P(EiIEj }

i,j

1,2, ••• ,n;

t. (1. 2)

The above estimates are rarely consistent in practice. They, as a rule, do not satisfy the basic relations obtainable from the definition of a probability and a conditional probability: O~P(Ei)"'1

P (E j ) ~ P (E i I Ej ). P ( Ej ) = p(Ejl Ei )· P{E i }= P(E i Ej)~P(Ei) p( Ei I Ej ). P(E j ) + P (E j ) + P (Ei '-;j).

(E):

p(Ei) i,j, = 1,2, ••• ,n In accordance with the relations shown above the consistency of the estimates (1.2) is being tested and ways of their improvement are looked for (see, e.g., the technique SMIC 74 [3]). There are methods that do not deal with probabilities, and that as a means for estimating the interactions apply so called odds. The conversion of event probabilities to odds and vice versa is accomplished with the following equations: ODDS

' ·ROB.

1 - PROBe

(1. 4)

and PROS. =

ODDS

1

+

ODDS

(1 .5)

In ordinary English, the odds of an event occurance is the ratio of the number of times it is expected to occur to the number of times it is expected not to occure if repeated trials of the future are possible. The impact from more than one event can be aggregated by multiplying the appropriate cross-impact factors Isee 12J I.

429

Graphs and H ypergraphs

Interactions can be also evaluated by conditional probab i lities based on caus a tion. If the time period for which the prognosis is being built is divided into a number of time intervals the probabilities of the following kind are to be estimated: P Ei , if the event Ej occured during the preceding time interval • Conditional probabilities based on causation then do not satisfy equations (1.3) • 2. REPRESENTATION OF THE SYSTEM BY A GRA PH A simple basic system of the kind so far discused can be represented by a graph. The vertices of the graph wilL stand for events; the set of vertices of the graph will be given by the set 1.E1,E2, ••• ,En ~ • The set of oriented edges wi ll represent the set of the con-

ditional probabilities, i.e. the set {Ei,E j I P(EiIEj) "F P(E i ) ; i,j = 1, ••• ,n ; i "F j • In constructing the graph we shall confine to only the relations Iconditional probabilit i esl represent i ng the statistical dependence of the corresponding two events. In other words, we shall require that P(EiIEj} "F P(E i ) , which is, of course, only another form how to say that the two events Ei and Ej are independent. For an illustrative purpose, let us consider an example when n = 5. Estimated probabilities P(E i ) will be ascribed to the appropr i ate vertices, the estimats of the probabilities p(E . IE.) will charakterize the corres1 J ponging edges of the graph. The following type of a graph is thus obtained Isee fig. 1/:

P(Ea/E,.)

If we have estamated P(E.) , P{E.) and 1 J one of the probabilities P(E . 1E .), 1 J P (Ej IE i ) the formule (1.3) determines the other probability. It is simplier to represent the system as it is on fig. 2.

Th i s representation of the system may enable the exper~strategic planner to work up to his estimates of the probabilities P(E.\E . ) he regards as relevant 1 J in a more comfortable way. Moreover, he is given a means as to distinguish

I. Ocka

430

A hypergraph is shown in Fig. 3. An edge Vi with IVil!lt2 is drawn by connectic.n of all vertices of Vi from one point. An edge Vi with I Vi' = 2 is drawn as a curve connecting its two vertices. An edge Vi with IVi' = '1 is drawn as a loop.

P(.I.)

the direct impact that a given event excerts upon another one from intermediate indirect impacts that are realized through third events. In a conversational mode of computing this representation gives way for expressing the feedback outcomes of the consistency tests control procedures. It appears convenient not to deal with all the relations for which P(EoIEo) ~ P(Eo) and , J , rather confine to only those for which P(E i ) - c ~ P(Ei

lE)

~ P{E i ) + c,

OF THE SYSTEM BY A HYPERGRAPH

REpn~SENT A TION

A hypergraph is a means of model representation that may bring forward a more profound insight into the analysis of the system strukture. This may be of a particular importance whenever the requirements of the selection and identification of the events are concerned. To begin with, let us define some basic notions. Reffering to l1)p. 389 we define. Let E = {E 1 ,E 2 , ••• , En} be a finite set, and let '17'= (Vi I i (, I) be a fami ly of subsets of E. The fami ly 1T i s said to be a hypergraph on E if Vi

~ fJ

U V I '

0

i

=E

I

The incidence matrix of a hypergraph is a matrix consisting of elements a ji , where

a ji =

where c is an appropriately chosen constant. 3.

If the edges Ei are distinct and if IEil = 2 for all i, then 11 is a simple graph without isolated vertices and vice versa.

( 3.1 )

< o

if

(3.3)

with m rows represent th e edges and n columns th3t represent the vertices of hYfle r graph.

A dual hy pergraph H corresponding to the hypergraph Ii is o bt aoined as a hypergraph to which corresponds the transpose of the incidenc e matrix of the hype rgrap h H. ~ n e XRmp le mJ Y ; llustrate the de fin 1 t ion. ,.;<> t I' i x ,r. , i n 9 e 11 era l !'. neEd not be a squ a r e ma trix. vertices

o1

11 1 0 ( o 0 o0

0 0 0 1 0 U) 1 0 0 e dg e s 1 1 0 0 1 1

(3.4)

represents the hypergraph /graph/ of Fig. 2, while matrix AT

(3.2)

vertices

4i

The couple H = (E,"') is called a hypergraph. IEI = n is called the order of this hypergraph. The elements E ,E 2 , ••• , 1 En are called the vertices and the sets V1 ,V 2 "",V m are called the edges.

if

1

.~

T

=

(i ~ ~ ~ ~) o o

1 0 1 1 0 0 0 1

edges

(3. 5 ~ represents the dual hypergraph in Fig. 3.

Graphs and Hypergraphs

The procedure leading to a hypergraph representation of the system of events and interactions is equal to the procedure of section 2. Each graph without isolated vertices is obvionsly a hypergraph and we shall therefore assume that throughout our analysis no events will lack interaction with other events. what benefit can be offered by the application of the concept of a hypvrgraph in analysing the system of events and their interactions? The focus of our treatment of the system, analogously to the system analysis approach, rests in emphasizing the relations of the system, i.e., in our case, the inter-

Fig. 5

431

actions among events. This provide the methods of the cross-impact analysis with the basis for creating individual scenarios and for evaluating their occurance in the form of probabilities. By establishing the dual graph we have established a further possibility to analyse individual events in a greater depth. For the expert-strategic planner it is furthermore important that he may apply a new viewpint upon the system he seeks to identify. It can be seen in the following example of 10 events Isee fig. 4/. The corresponding dual hypergraph is shown in Fig. 5.

432

1. Ocka

Clearly, dual hypergraphs representing the systems described in section 1 have the following property

Ei

i=1, ••• ,n Vk' E.1 Vl '

Step 1

k, l; kj! l jH,l E.1

The procedure of events and interactions identification will therefore consist of the following four steps: The experts-strategic planners identify the events E1 , ••• En •

V. J

(3.6)

Step 2

Put otherwise, each vertice of a dual hypergraph is contained in exactly two relations of the dual hypergraph. One more characteristic of the hypergraph representation of the system may be noteworthy. The expert is often uncapable of identifying the relevant interactions in themselves, i.e. as pure impact axcerted by the event Ei upon the event Ei • It is therefore more convenient for the expert to determine the relevant interaction for a given set of events as a whole. This type of an expert solution leads to a hypergraph representation of the system. Then, further steps can be proceeded as to create the standard model described in section 1. 4. IDENTIFICATION OF THE SYSTEM INTEnACTIONS IN THE FORM OF A SET OF EVENTS The main thesis of the procedure proposed here states that the identif i cation of the relevance of the individual interactions connecting events may be carried out separately from the probability estimation. Therefore, in the first phase of the procedure the interactions can be treated in the form of a set of events.

On the set of events the experts identify the set of interactions V1 ,V 2 , ••• ,V m in the sense of the definition. For example the interactions on the set of events E" ••• ,E 7 may take up the following form: V1 = { E2 , E3 , Ed , V2 {E 3 , ES' E6 , E7 } V3 = fE 2 , E3 },fV 4 = E1 , Ed Vs

tEs' E7 }

, (4.1

)

The outcome of the two steps is a system of events and interactions which can be represented by a hypergraph. An example is shown in Fig. 6. This system contains our initial conception of the events and their interactions, so far without the probability evaluation of the relations intensities.

De fin it ion: Let {E 1 ' E2' ••• , En) be a set of events. Then a no~empty subset V of this set is called an interaction, if 'VI~2. The definition, of course, corresponds to what has been referred to as a relation of a hypergraph. The additional condition lvl ? 2 has been introduced as to define an interaction only as a relation between at least two events. Apparently, if for all elements of an arbitrary set of interactions {V j,j= = 1 ,2, ••• ,m holds that \ V, = 2, then this set, together with the set of events is the system described in section 1.

J

Fig. 6 Step 2 can be supported, if needed, by a test for the correctness of the events and interactions identification, as it my be overviewed by a dual hypergraph representation.

433

Graphs and H ypergraphs

Step :S

This can be done by suppLementing additionaL events. In the case depicted in Figs. 6 and 7 the events Ea and E9 have been suppLe mented.

J

V3

{E 6 , E9 ] VZd = {E7,E g {EZ,E3}, V4 = {E1 ,E 3 ],

Vs

{ES,E 7

VZc Wi t h the heLp of experts aLL interactions are converted into the interactions of the two-eLe ment set type within the third step.

J

with the set of events f~1,Ez, ••• ,E93 (4.2 ) as shown in Fig. 7. The experts seek to determine the interactions intensities estimates. In the system of section 1 the experts evaLuate probabilities (1. 2) • For the purpose of the consistency tests we can determine appropriate probabiLity estimates (1.2) for the whoLe extended system.

CONCLU SION

Fi g . 7 The additionaL even t s are as to express th e mutu a L interactions among groups of events. The experts may heLp us formuLate an adequate interpretation of the additionaL events, for exampLe as a synergic effect. An additionaL event can be aLso conceived of as a point towards which a group fo events direct the i r effects and which in return can affect the events as weLL. An extended set of events E1 , Ez, ••• ,En,En+1, ••• ,En+r can thus be obtained, where r is the number of the additionaL events. SimiLarLy we identify a new set of interactions V1,VZ, ••• ,Vm+1' ••• 'Vm+s' where s stands for the number by which the initiaL interactions have been extended. In our exampLe the system (4.1)is transformed into the system. V1 a = {E z , Ea 1 V1b V1c = {E 3 ,E a } VZa O.M.C.N.E.-O

{E 4 ,E a },

1'

{E 3 ,E 9}, VZb = {E S,E 9

A graph and hypergraph representation of the system of events and their mutuaL interactions can be viewed as a means for improving the proce d ures of the system identification appLied by methods of Cross-Impact AnaLysis. Experts are thus provided wi th a more profound view upon the syst e m under identificat i on. It appears as convenient to appLy this type of a representation during the conversationaL determinat i on of the interactions probabiLity estimates when the system is observabLe on the computer dispLay. OuaL hypergraphs appLication gives way for a deeper anaLysis of events in the system under study. ~e have attempted to demonstrate that the representation of the system events and i nteractions by a hypergraph aLLows for an introduction of a more generaL nation of the system reLations in the initiaL phase of the identification procedure.

REFERENC ES (1J Berge, c. (1973). Graphs and Hypergraphs, North HoLLand. Enzer, S. (19aO). INTER I\X - An Interactive ModeL for Studiing Future Business Environments: Part I and 11 Tehn. Forecasting and SociaL Change 17, p.141-1S9,Z11-Z4Z.

L2J

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DLlperrin, J. , Godet, St'I IC 74 - Hethod ting and Ranking Futures, voL. 7,

M. (1975). for constucScenarios, No 4, p.303-

31 2.

L4J Gordon, Hayword (1968). InitiaL experiments with the CrossImpact Matrix Hethods of Forecasting, Futures 2, voL 1.