EXTENSIONS OF GRAPH INVERSION TO ... - Semantic Scholar

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Annals of Operations Research, 21 (1989) 127-142

EXTENSIONS OF GRAPH INVERSION TO SUPPORT AN ARTIFICIALLY INTELLIGENT MODELING ENVIRONMENT Harvey J. GREENBERG I, J. Richard LUNDGREN and John S. MAYBEE

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University of Colorado at Denver, Mathematics Department, Campus Box 170, 1200 Larimer Street, Denver, CO 80204, U.S.A. University of Colorado at Boulder Boulder, CO 80309, U.S.A.

Abstract This paper extends previous results in graph inversion to enable artificially intelligent model formulation by taking partial information from the problem domain into an algebraic form-that is, a mathematical model to support economicbased decision-making. Here focus is on flow relations and economic correlations. Issues of consistency are resolved, and a concept of core models is introduced to achieve minimality for simplifying model management and use. After establishing the general results, special consideration is given to input-output structures, as in Leontief models.

1. Introduction We study a class of structural problems arising in functional modeling, which pertains to having the computer participate in model formulation from only partial information about the problem. To create an artificially intelligent environment for modeling for decision support requires such structure. In particular, we want to accept qualitatively relational information, such as economic correlation-that is, whether two economic variables, like demand quantities or supply prices, tend to move the same or opposite directions in the marketplace. Qualitative analysis of functional models dates back to the 1950's in economics concerned with causality; a comprehensive summary of results and references are given by (Maybee and Quirk, [25]). A foundation for structured modeling is given by the pioneering work of (Harary, Norman and Cartwright [15]), and (Roberts [29]) gives a wealth of applications (see also (Warfield [31]). A new look at structural problems was introduced by (Greenberg [7]) in an effort to represent concepts of correlation and flow,leading to a symposium that synthesized the state-of-the-art from different vantages (Greenberg and Maybee [14]). One of the problems was to see if rules can be developed for model simplification or construction such that it may be possible to have the computer play an intelligent role in the process. This led to the notion of graph inversion, which is the construction of a functional model represented structurally with a bipartite graph that relates one set of elements (like independent variables) to another set (like

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dependent variables or functionals). T h s was first done with no assumptions about the nature of the relations, using only undirected graph theory (Greenberg, Lundgren and Maybee [4]). One way to account for correlation between pairs of variables-that is, whether their values change in the same or opposite directions-is by signing the lines that represent relations. This representation leads naturally to a theory of signed graph znuersion (Greenberg, Lundgren and Maybee [10,12]), which characterizes the class of functional models having specified correlations. (The specification may be inconsistent, resulting in an empty set of functional models.) This was subsequently applied (Greenberg, Lundgren and Maybee [13]) to network and Leontief models of energy markets, towards automated model construction and Here we consider another way to account for information about the causal nature of the relations, which may be regarded as a flow, due to its natural correspondence with network flows. Our objective is to complement earlier results with an added theory of digraph inversion, and we shall relate these results to the signed graph inversion theory. In particular, we deal with the more complex case of mixed specification of the causal nature of the given relations. While this paper does not explicitly relate qualitative reasoning to logic-based systems, there is current research on uses of neural networks in conjunction with correlation and flows to provide a broad spectrum of computer assistance for modeling and analysis. Here focus is on problem-domain input, which may be much more vague that current rule-based expert systems assume. Related approaches may be found in (Bobrow [3]), (Herbert and Williams [16]) and (Engelke et al. [4]). We also do not explore here the connections with structured modeling (Geoffrion [5]) and logic-based aids, such as the methods of (Ma, Murphy and Stohr [21]), (Murphy and Stohr [25]) and (Krishnan, Kendrick and Lee [19]). The rest of this paper is divided in to five sections. Section 2 provides the basic terms and concepts that we shall use. Section 3 gives the main results on digraph inversion, and section 4 gives relationships with signed graph inverses. Section 5 considers some special structures that arise in social science models, notably in economics, and section 6 gives some avenues for further research.

2. Basic concepts 2.1. GRAPH-THEORETIC DEFINITIONS

This section reviews standard graph-theoretic concepts we shall use. A (finite) digraph D = [V, A] consists of a finite set of poznts V and a finite set of arcs A. Each arc is an ordered pair of points ( u , u ) which is oriented from u to v. We call u a predecessor of u, and u a successor of u. The sets of predecessors and successors of a point u are denoted, respectively, by: P ( u ) = { u E V: ( u , v ) E A ) and ~ ( u=) { w E V : ( u , w ) E A ) .

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A (directed) path is a sequence of points (u,, u,, . . . , u,,) such that u,,, E ~ ( v ,for ) i = 0, 1 , . . ., n - 1. We say that the path contains the points v,, u, , . . . , it contains (n, including repetitions); the arc length is the number of arcs it traverses ( n - 1, including repetitions). Point u is a descendant of point u, and an ancestor of v, if there exists a path (v,, v, ,.. . , u,) with u, = u and v,, = u. This is denoted u + u. A digraph is bicomplete with respect to a partition X U Y if, for every x E X and y E Y, ( x , y ) is in its arc set. This is denoted

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v=

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w e say D' = [V', A'] is a subdigraph of D = [V, A] if V' c V and A' c A. A directed clique, or simply diclique, between X and Y is the subdigraph with y ' = X U Y that is bicomplete with respect to X and Y. The union of two digraphs D, = [V,, A,] and D, = [V,, A,] is the digraph D = [V, U V,, A, u A,]. A collection of dicliques of a digraph D is said to cover D if their union equals D.A cover is minimal if it contains no proper subcover. Given a digraph D = [V, A], its associated (undirected) graph is G(D) = [V, El, where E is the disorientation of the arcs in A. 2.2. GRAPHS OF MATRICES

Here we review the graphs associated with an m X n qualitative matrix, M = [MI,], where MI, = 0, 1 or -1. The fundamental bigraph of M is a bipartite graph B ( M ) = [R, C, El, where R corresponds to the rows of M, and C corresponds to the columns of M. The set of edges, E , corresponds to the nonzeroes of M, where (i, j ) E A if, and only if, MI, Z 0. The row graph of M is RG(M) = [R, E,], where (i, k ) E ER if, and only if, there exists j E C such that ( i , j ) and ( k , j ) are edges in E; equivalently, two row points are adjacent if they have a nonzero in a common column ( j). The column graph of M is CG(M) = [C, E,], where ( J , k ) E E, if, and only if, there exists i in R such that (i, J ) and ( I , k ) are in E; equivalently, two column points are adjacent if they have a nonzero in a common row (i). Note denotes the transpose of M. , that RG(M) = C G ( M ~ )where One inversion problem is to find M such that RG(M) = G, where G is a given graph. This is always possible. A more difficult inversion problem is when we are given two graphs G, and G, for which we seek M that satisfies RG(M) = G, and CG(M) = G,. This is not always possible, and the key is a characterization of clique covers. (See (Greenberg, Lundgren and Maybee [9]) for details of characterization and (Greenberg, Lundgren and Maybee [10,13]) for methods and related applications.) The signed bigraph of M, denoted B + ( M ) , is the fundamental bigraph with each edge in E signed according to its associated nonzero: s((i, j ) ) = sgn( MI,). The row graph is signable if s((i, j ) ) . s ( ( k , j ) ) has the same sign for all j for which (i, j ) and ( k , j ) are each edges in B + ( M ) ; in this case the signed row

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graph is denoted by RG+(M), and s((i, k)) = s(i, j ) ) s((k, j ) ) for (any) j for which (i, j ) and (k, j ) are each in E. Similarly, the column graph is signable if s((i, j ) ) s((i, k)) has the same sign for all i for which (i, j ) and (i, k ) are in E, in which case s ( ( j , k)) = s((i, k ) ) for any i for which (i, j ) and (i, k ) are in E. This is denoted CG+(M). It is not difficult to show that the row graph is signable if, and only if, the column graph is signable, in which case RG'(M) = One signed graph inversion problem is to find a qualitative matrix M for which RG+(M) = G+, where G + is a given signed graph. While this too has at least one solution, it is generally not sufficient to find an unsigned inverse, then assign signs to its nonzeroes. The fundamentals for this and the analogous 2-graph inversion problem also rely upon clique covers, but some applications also rely on notions of balance. Another way to represent sign patterns in a qualitative matrix M i's with its fundamental digraph D ( M ) = [R, C, A]. This is isomorphic to B(M), but the edges are oriented to comprise the arc set as follows. For i E R and j E C define arc (i, j ) if Mjj = - 1 and define arc ( j , i ) if Mjj = + 1. The row digraph of M is RD(M) = [R, A,], where (i, k ) E A, if there exists j E C for which (i, j ) and ( j , k ) are in A; equivalently, two row points are adjacent if they have oppositely signed nonzeroes in a common column, where the predecessor point is the one with the negative value. Similarly, CD( M ) = RD( - M T). Unlike the signed graph representation, row and column digraphs always exist, but they need not be isomorphic to the row and column graphs; in particular, R D ( M ) and CD(M) have no arcs if M is a nonnegative (or nonpositive) matrix. The digraph inversion problems are analogous to the other inversion problems, but the usual clique covering results do not apply. We shall develop the requisite theory and relate this to signed graph inverses to provide a foundation for mapping problem-oriented terms like subsitute, complement, and flow, to a precise structural representation, automating completion of model formulation. At a minimum, the computer can detect structural inconsistency; beyond may lie

2.3. MATRICES OF FUNCTIONAL MODELS

In this section we review qualitative (matricial) representations of functional models. The linear (or affine) model has the form, y = Ax + b. Our interest is in the relational matrix A, where we define M = sgn(A) - i.e., Mi, = s g n ( ~ , Vi, ~ ) j. M describes the structure of the model insofar as the (absolute) rates of change are increasing ( Mi, = + I), decreasing (Mi, = - 1) or zero ( Mij = 0). A nonlinear

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model, arising from a Cobb-Douglas production function in economic relations,

Here we also define M = sgn(A), but now M,, measures the relat~verate, called the elasticity, of y, with respect to x,. More generally, (Provan [26,27]) and (Maybee [22]) have described qualitative representations of functional models, having the form for which certain equilibrium conditions are studied with a structural analysis of the jacobian (possibly with sign constraints on dx). To help fix ideas consider an example for which we seek a functional model of four variables : S = supply of crude oil R = refining level to produce distillate oil D = demand for heat E = electricity generation. We are given flow relations in the form of the digraph D:

The problem is to characterize the set of models represented by a qualitative matrix M for which RD(M) = D. One solution is given by the incidence matrix

If we were to construct a linear model with x = ( S , R, D, E ) and relational matrix A such that sgn(A) = M, we would have y = A x with the restrictions: All < 0, A2, > 0, A2, < 0, A,, < 0, A,, > 0, A,, > 0, A,, > 0, A, < 0 and all other A,, = 0. The dependent variables ( y) may be regarded as net flows through nodes from ordinary transportation activities (x). Can we find a model with fewer than four dependent variables-that is, does there exist M with fewer than four columns for which RD(M) = D? As an


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We can also show there is no model with fewer than 3 column variables (i.e., there is no matrix M with fewer than 3 columns for which RD(M) = D.) That is, any M for which RD(M) = D can be reduced to exactly one of these by discarding any column that does not result in violating the given flow specification: RD(M) = D.

3. Digraph inversion In this section we establish a foundation upon which we build a meaning of model structure. The example given in the previous section will be used to illustrate how the results apply to have the computer provide modeling assistance through logical reasoning mechanisms. To begin, we associate columns C( M ) of a qualitative matrix M with dicliques. Let X, denote the set of rows of M for which Mjj = - 1 and let Y, denote the set of rows of M for which M,j = 1. Then, X, Y, is a diclique in RD(M), where we allow X, = 0 or Y, = 0 (the empty set), but not both. Thus, we refer to C ( M ) as the columns of M or dicliques of RD( M ), depending on context. This immediately yields the following.

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THEOREM 1. (FUNDAMENTAL THEOREM OF DIGRAPH INVERSION)

RD(M) = D = [V, A] if, and only if, M has #V rows and C ( M ) is a diclique cover of D.

applies to the two inverses given. The first of these is a special inverse, called the arcset inverse, denoted by A(D), for which M is the node-arc incidence matrix of D. It is always true that RD(A(D)) = D, so there is always at least one inverse for any digraph. Because A(D) is the usual model for a network problem, we regard the arcs as flow relations between pairs of variables. In relating digraph inverses to the undirected case, care must be taken. Whereas it is always true that RG(M) = G(D) whenever RD(M) = D, it is not

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true that any inverse of G(D) can be signed to form an inverse of D. In our example G(D) is given by:

An undirected inverse is given by:

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Suppose Ma is a signing of M (i.e., MP, = 1 whenever M,, = 1). It is not possible, however, to sign M so that RD(Mo) = D. The problem is that to cover arcs (R, D ) and ( R , E ) we must sign column 2 of M such that M&M&= M,",M,", = - 1. This causes M&M& = 1, so arc ( E , D) cannot be covered. As we shall see from the next concept of minimality, no digraph inverse of this example has fewer than 3 columns. The inverse M of a digraph D is said to minzmal if C ( M ) does not contain a proper subcover of D. Equivalently, M is minimal if we cannot remove a column without violating the defining condition RD( M ) = D. To avoid tedious notation we shall henceforth assume (without loss in generality) that D contains no isolated points. It is not difficult to modify the results, as in the next theorem, to allow isolated points, but we wish to minimize new notation while presenting the main results.

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THEOREM 2 (MINIMALITY THEOREM)

Suppose RD(M) = D. Then, M is a minimal inverse of D if, and only if, there exists a set of arc representatives {(x,, y,)) such that (x,, y,) E XJ * Y, for X, * Y, for I # j. J = 1,. . . , #C(M); and, (x,, y,) If (x,, y,) is an arc representative of X, * Y,, removal of column j in M leaves (x,, y,) uncovered, so no proper subset of C(M) covers D. Conversely, suppose M is a minimal inverse. Then, removal of column j must uncover some arc, which we define as its representative. In words, the Minimality Theorem says that if we have a diclique X, * Y, for which ea& of its arcs is covered by some other diclique-i.e., it has no distinct arc representative-then we can remove column J without changing the topology of

H.J. Greenberg et al. / Extensions zn graph inversion COROLLARY 2.1

The arcset inverse is minimal. COROLLARY 2.2

Every minimal inverse satisfies #C(M) I #A(D). The first corollary is immediate because each arc is covered by precisely one diclique, composed of its endpoints. Corollary 2.2 tells us that the arcset inverse has the greatest number of columns among the set of minimal inverses. This upper bound is particularly useful when combined with the following lower

COROLLARY 2.3

If RD(M) = D, #C(M) 2 diam(D).

The diameter of the digraph, diarn(D), is defined in standard terms as follows. Let R = {{u, u): u + u} (ie, the set of pairs of points for which the second is descendant of the first); and, let L(u, u ) denote the length of the shortest path from u to u for all ( u , u) E R. Then, diam(D) = max[L(u, u): ( u , u) E R]. To prove the corollary let (u,, u, , . . ., up) be a shortest path from u, to up with length p = diam(D). We shall prove that each arc of this path must be in different dicluques for any C(M) for which RD(M) = D. Suppose this is not the case. Let arcs ( u,, u, + ,) and ( u,, uh+ ,) be in X, Y,, and assume h < i. Then, ( u,, u, + ,) is also in X,- Y,, so (u,, 0, ,..., vh, u,+, ,..., up) is a shorter path from u, to up, a contradiction.

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In words, corollary 2.3 says that the arcs'of any shortest path between two points must be in different dicliques. In our example we have diam(D) = 2 (with a max shortest path = (S, R, D)), so #C(M) 2 2. If arc ( E , D ) was not present, this bound would be sharp, but we have more reasoning to do. Corollary 2.3 tells us that arcs ( S , R ) and ( R , D ) must be in different dicliques since they are in the shortest path of S + D. Assign these to dicliques 1 and 2, and look at the remaining digraph. To keep C ( M ) = 2 we must cover arcs (R, E ) and ( E , D ) with the two dicliques already covering (S, R ) and (R, D). Since row E of M is null, (R, E) must be covered by setting ME, = + 1; however, the only way to cover (E, D) is by setting ME, = - 1. Hence, at least one more column must be added to M. We define the core set of models having flow relations described by D as the set M( D) of minimal inverses of D. Corollary 2.1 tells us this set is not empty, and corollary 2.2 tells us it is finite. A form of computer assistance is to characterize M ( D ) to help derive a model having the requisite structure. There is often perceived simplicity in having a minimum number of columns-i.e., inde-


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pendent variables or functionals. Clearly, the problem of finding this is complex, and we do not settle that here. The bounds in corollaries 2.2 and 2.3 are some help, but more important is the further reduction of the core by the modeler's additional specification of structure, beyond flow relations. That is the subject of the next section.

4. Common inverses In this section we combine the results of digraph inversion with signed graph inverses. Recall that the signed row graph of M, denoted RG+(M), represents correlation (when it exists). In particular, a positive line between two points vi and u, corresponds to a column of M, say j, for which Mi, = M,,+ 0. If Mi, = M, = - 1, we may think of the levels of the row variables yi and y,, respectively, as input requirements for the variable xi, such as feed stocks. If Mi, = M,, = 1, we may think of y, and y, as levels of outputs for x,, such as finished products. In either case the positive line suggests that if the level of one row variable changes due to a change in x,, the other changes in the same direction. These variables are thus called complements, such as the result of joint production of feedstocks or of finished products. A negative line between two points v, and v, corresponds to a column of M, say j, for which Mi,= -Mkj+ 0. If Mi,=-1 and Mk,= + I , we may think of x, as having yi as an input requirement and y, as an output, such as the transfer of a material from one region to another. In this case the negative line suggests that a change in x, causes y, and y, to change in opposite directions (i.e., an increase in yi is accompanied by a decrease in y,). These variables are thus called substitutes, such as regions competing for a common resource. The modeler may know some such correlations in addition to flow relations, so we may address the issue of a common inverse-i.e., a matrix M that satisfies RD(M) = D and RG+(M ) = Gt. Recall that RG+( M) need not exist even if RG(M) = G. This occurs when two columns of M give opposite correlations. It is therefore appropriate to begin with conditions for which an inverse of D is

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THEOREM 3 (SIGNABILITY THEOREM)

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Suppose RD(M) = D = [V, A ] . Then, RG+(M) exists if, and only if, the following signabilityproperty holds for C ( M ) = { Xj Y,}. For each (u, u) E A : u E X, implies v @ Y,.

Suppose RD(M) = D and RG+(M) exists, and let (u,, u,) E A . Then, some column of M, say j, must be such that Mi, = -1 and M,, = 1. The line (ui, u,) must belong to RG(M), and in RG+(M) has s((ui, v,)) = -1. By definition,

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MI, = - 1 for all v, E XJ and MkJ= 1 for all u, E Y,. Thus, if u, and u, are both in XJ or both in Y, for some j, we would have s((u,, u,)) = $1 in RG+(M), a contradiction. Hence, M must possess the signability property if RG+(M) exists. Now suppose the cover of D corresponding to M has the signability property and that (v,, v,) is a line in RG(M) for which there exist columns j and k of M giving opposite signs: MI, M,, = - 1 and MI, M,, = + 1. The first case implies D contains the arc(u,, v,), and the second implies both u, and u, are in X, or both are in Y,,contradicting the signability property. Clearly, the arcset inverse is signable, so M ( D ) always contains a common inverse, A(D), when G+= RG+(A(D)). More generally, suppose R D ( M ) = D and RG+(M) exists. The next theorem answers the question, "When is M a common inverse of D and G+?" THEOREM 4 (COMMON INVERSE EXISTENCE THEOREM) There exists a common inverse of D = [V, A ] and G+ = [V, E , s] if, and only

if, the sign function s satisfies the property: s(u,, u,) (u,, uh) E A or (oh, u,) €A.

= -1

if, and only if,

the negative lines of Gf and generates no positive lines, while R covers its positive lines and generates no negative ones; therefore, RG+(M) = G+. Conversely, let M be a common inverse. If s ( ~ ,0,) , = - 1, there must exist a column of M, say j, for which M,,M,, = - 1. This implies that D contains either the arc ( u,, oh) or the arc ( u,, u,) ; and, conversely, if s(u,, u,) = + 1, there cannot exist j for which MIJMh,= - 1, so D cannot contain the arc (u,, v,). Thus, the existence of a common inverse implies the sign function property in the theorem.

One implication of the Common Inverse Existence Theorem is that a flow relation (i.e., an arc) is a strong correlation. Whereas every flow relation imparts negative correlation, a negative correlation may be viewed as some flow relatlon whose direction is unknown. In our example this result applies to identify quickly a correlation that is inconsistent with the flow relations-i.e., violates the condition in the Common Inverse Existence Theorem. For example, this is the case if the modeler specifies: Supply of crude o i l i s p o s i t i v e l y c o r r e l a t e d w i t h the Level o f r e f i n i n g .


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o f c r u d e o i l i s p o s i t i v e l y c o r r e l a t e d w i t h demand

find a common inverse in M ( D ) or whether we must find an augmented inverse-i.e., with columns 0 * Y or X* 0 , each weakly of the same sign. To begin to illustrate these iSsues before resolving them, suppose the modeler gives the global statement: Every p a i r o f v a r i a b l e s a r e c o r r e l a t e d .

In our example, the second inverse is not signable, so only the arcset inverse remains consistent. It must be augmented since it covers no positive lines. The above specification insists that all nonadjacent points of D must have positive lines in G+; that is, we must have (S, D), (S, E) E E+. We obtain the folldwing augmented common inverse, which is unique up to minimality and sign reversals of the augmented columns.

and G+. The following corollary embellishes the Common Inverse Existence Theorem and tells us when a digraph inverse is a common inverse without

COROLLARY 4.1 Suppose M E M( D) and RG+(M ) = G+. Then, G+ is consistent and E+ = { ( u , v): u and v are in some clique of Gf, all of whose points comprise X or Y

for some X

Y E C( M ) }.

H.J. Greenberg et al. / Extensions in graph znuersion It may at first seem that is difficult to validate relations, as the modeler specifies correlations after flow relations. The next result tells us that we need only check one particular inverse (the arcset) to validate the existence of a common inverse. THEOREM 5 (COMMON INVERSE CONSISTENCY THEOREM)

There exists a common inverse of D and Gf if, and only if, there exists a consistent augmentation of the arcset inverse. Definitionally, if there exists a consistent augmentation of the arcset inverse, there is a common inverse (namely, the augmented inverse). Conversely, suppose of the same sign (hence covering some of the positively signed lines), and C(Q) is a diclique cover of D. If Q is not the arcset inverse, there must exist XJ* Y, for which either #XI> 1 or #Y, > 1. Without loss in generality, suppose the former. Then, partition XJ= XI,U XJ2for which #XI,= 1 and #X,, = #XJ- 1 and replace XJ Y, with XJ,=, Y, and XJ2 * Y,. Let M ' be this new matrix with column j replaced by the two columns, and observe RD(M') = RD(M) = D. If any positively signed line associated with the clique XJ becomes uncovered, augment a cover to R to maintain RG+( M') = RG+(M ) = G+ ( t h s does not affect RD(M') = D). Repeat this until Q is the arcset inverse. The result is an augmented arcset inverse, thus completing the proof.

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We could say the augmented arcset inverse because there is really only one of interest: a column for every positive line. Later, when one seeks to produce a common inverse, alternatives must be considered. The Common Inverse Consistency Theorem lets us simply perform a test of validity (and it provides one candidate model-that is, the inverse-at the end of the specifications). There is a special case where every inverse of D is signable, thus being a common inverse for some (consistent) G+. THEOREM 6 (TRANSITIVITY THEOREM)

RG+(M) exists for all M E M ( D ) if, and only if, D is transitively irreducible. Suppose D = [V, A] is transitively irreducible-i.e., if ( up, u,), ( u,, v,) E A, there does not exist arc (up, u,) E A. Then, suppose M E M( D), and ( up, u,) is a line in RG(M). Since RD(M) = D, M must contain a column j for which

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(up, uq), (uq, u,) and (up, u,.) are all arcs of A (i.e., D is not transitively irreducible). Then, define M as follows. Let Q be the arcset inverse of D with the three arcs removed, and let M be Q augmented by the two dicliques { { up ) ( u,), { up, ug) { u,)). Clearly, M E M(D). Further, M contains the forbidden submatrix that makes RG(M) unsignable.

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We may topologically sort D (and find cycles if the relations do not form a partial ordering). With some modifications, this gives more information to the modeler, especially about possible correlations. In practice, the arcs may represent precedence relations, in which case flow is passage of time. In such cases topological sorting must be successful (or the modeler has an inconsistency just in the flow relations because then it is not a partial ordering), so the Transitivity Theorem can be used to deduce correlations implied by each minimal inverse of D. In other cases, as in commodity flows, cycles are not unusual. In those cases the Transitivity Theorem can be used to reduce the set of candidate models-i.e., the set of common inverses that are formed by augmentations of M E M(D). We may want to derive an inverse that has no augmented columns, and where D contains transitive arcs. We may still have some members of M ( D ) consistent with G+. The next result shows when this is the case. Define points u and v as common successors if there exists a point x for which u, v E S(x") Similarly, define u and u as common predecessors if there exists x for which u, v E P(x). THEOREM 7 (UNAUGMENTED COMMON INVERSE EXISTENCE THEOREM)

Suppose D and Gf are consistent (i.e., satisfy the condition in the Common Inverse Existence Theorem). Then, M ( D) contains a common inverse if, and only if, the endpoints of every line in E+ are either common predecessors or common successors. The positively signed lines of R G ( M ) are precisely those (u, v) for which u, v E XJ or u, u E Y/ for some j. In the former case, u and v are common predecessors of the points in X,. Moreover, since M is minimal, X,, # 0 and Y, # Rr for each j. C] The example digraph violates this condition for any G+ that has a positive line. Recall that the only signable, minimal inverse is the arcset inverse, which covers no positive lines. This generalizes to the following. COROLLARY 7.1

The (unaugmented) arcset inverse cannot be a common inverse of D and G+ if G+ contains any positive lines. The foregoing results apply to enable aids for modeling by quickly identifying inconsistent structural specifications; and if consistent, by providing matrices

H.J. Greenberg et al. / Extensions in graph inversion that serve as candidate models that satisfy flow and correlation specifications between pairs of variables. The modeler may also have a structural class in mind and want the computer to see if the specifications are consistent with that class. That is considered in the next section. 5. Leontief inverses

In this section we consider a class of linear models that arises in economics: ~eontief input-output matrices. Although numerical properties are part of the definition of a Leontief matrix, we consider only its structural property: it is a square matrix with positive diagonal elements and nonpositive off-diagonal elements. THEOREM 8 (LEONTIEF INVERSE THEOREM)

For a digraph D there is a unique inverse L ( D ) that is Leontief. The proof of the Leontief Inverse Theorem is partly constructive. That is, L ( D ) is obtained by { P(v) * { v)) for each v in V, called the inclaw inverse. The uniqueness comes from the fact that if RD( M ) = D and M satisfies the Leontief property (for a fixed ordering of the points), then M = L ( D ) because each column of M is precisely P(v) { v). If D contains a source, its Leontief inverse cannot be minimal because source columns are singletons and could be dropped. We define the reduced Leontief inverse L*(D) to be the removal of the singleton columns of L ( D ).

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COROLLARY 8.1

L*(D) E M(D). In our example we have:

Note that L*(D) is equivalent to the second inverse, which recall is not signable. The critical property is that G+ must have ( x , y ) E E+ if, and only if, P ( x ) n P ( y ) # 0 . That is, we have the following. COROLLARY 8.2

L ( D ) is a common inverse of D and G.? if, and only if, G+ is consistent and E+ is the set of common predecessors.


141

More study is needed to deal with the 2-graph inversion, not treated here. There is also need to study more special classes of models; the Leontief matrices comprise one important class, but there are many others. Indeed, the main advances must come from a richer representation of model structures, amenable to natural specification, so the computer can do more to help build a model from partial information about relations among its variables and parameters. There is also need for algorithmic developments. Obtaining clique covers is difficult, but some special cases may be tractable (Pullman [28]). All of this is to support an artificially intelligent environment for modeling and analysis.

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