aa r X i v : . [ m a t h . C O ] J un NORMAL BINARY GRAPH MODELS
SETH SULLIVANT
Abstract.
We show that the marginal semigroup of a binary graph model is normalif and only if the graph is free of K minors. The technique, based on the interplay ofnormality and the geometry of the marginal cone, has potential applications to othernormality questions in algebraic statistics. Introduction
The summary of high-dimensional data in a multiway table by lower order marginalsis a staple of the statistical sciences. In the case where the table is a contingency table,that is, a table of counts, the table is a nonnegative integral multiway array, and themarginal summaries are a list of lower order nonnegative integral multiway arrays. Formany applications, e.g. data confidentiality and hypothesis testing [4], we would like todetermine whether a given list of lower order nonnegative integral arrays could actuallybe the list of margins of a high dimensional contingency table.More formally, we have a linear map π ∆ : R r −→ R d which is the linear map that computes (some collection ∆ of) marginals of a r × r × . . . × r n n -way array. The fundamental problem is to characterize the image semigroup S ∆ := π ∆ ( N r ) . Characterizing the semigroup is a problem that typically falls into two pieces. One pieceis to characterize the image cone C ∆ := π ∆ ( R r ≥ ) , by giving its facet defining inequalities. The second piece concerns understanding thediscrepancy between the semigroup S ∆ and its normalization; that is, describing the setof holes: H ∆ := ( C ∆ ∩ Z d ) \ S ∆ . As a very special case of this second problem one is lead to the question:
Question 1.1.
For which ∆ and r is the set of holes H ∆ empty? In other words, forwhich ∆ and r is the semigroup S ∆ normal ?A general answer to question 1.1 seems out of reach with present techniques. Even inthe case of a three cycle ∆ = [12][13][23] it is still open to classify the r which producea normal semigroup (though there remain only a finite number of cases left to check atpresent). In this paper, we focus on the special case where G is a graph, and r = . . . = r n = 2,the so-called binary graph models [3]. In particular, we will show the following: Theorem 1.2.
Let G be a graph with vertex set [ n ] and let r = . . . = r n = 2 . Then themarginal semigroup S G is normal if and only if G is free of K minors. Note, in particular, that Theorem 1.2 implies that marginal semigroups are rarely nor-mal. Section 2 contains the proof of Theorem 1.2, whose main idea is to relate normalityproperties of graphs to normality properties of subgraphs. One of the key take-homemessages of this proof is that these normality problems can often be addressed by takingthe geometry of the cone C ∆ into account, instead of working only with the semigroup.This is the content of Lemma 2.3. Section 3 is devoted to a description of some furtherpossible directions of exploration. 2. The Proof
In this section we formally set up the notion of the marginal cone and the marginalsemigroup. Then we prove some key lemmas for the general normality question for mar-ginal semigroups, and remind the reader of some useful structural results in graph theoryon K minor-free graphs. These ideas come together to provide the proof of Theorem 1.2.Fix a positive integer n . Let [ n ] = { , , . . . , n } . For each i ∈ [ n ] let r i be a positiveinteger. For any set F ⊆ [ n ] let R F = Q i ∈ F [ r i ] be a set of indices. In the special case F = [ n ], let R := R F . For any set A , let R A be the real vector space of dimension { e a : a ∈ A } . In the special case where A = R we say that R R is the spaceof r × r × · · · × r n tables. Elements of R are i = ( i , . . . , i n ). If F ⊆ [ n ], we denote i F = ( i f ) f ∈ F .Let ∆ ⊆ [ n ] be a collection of subsets of n . We define the marginal map π ∆ by theformula: π ∆ : R R −→ M F ∈ ∆ R R F ; e i
7→ ⊕ F ∈ ∆ e i F on unit vectors and extending the map linearly to arbitrary elements of R R . Note thatthe “marginal of a marginal is a marginal”, so that will assume that if F ∈ ∆ and S ⊆ F ,then S ∈ ∆ as well. Thus, we will refer to ∆ as a simplicial complex. The coordinates on L F ∈ ∆ R R F are denoted p Fi F , where F ∈ ∆ and i F ∈ R F .The marginal cone C ∆ is the image of the nonnegative orthant under the marginal andthe marginal semigroup S ∆ is the image of the lattice points in the nonnegative orthantunder the marginal map C ∆ := π ∆ ( R R≥ ) , S ∆ := π ∆ ( N R ) . Clearly both C ∆ and S ∆ depend on both ∆ and R , but we suppress the dependence on R in the notation.First, we consider how two general operations on a simplicial complex ∆ relate to thegeometry of the marginal cone and the marginal semigroup. The first operation is edgecontraction. Suppose the L ⊆ [ n ] are a set of vertices L ∈ ∆. Define the edge contractionby ∆ /L , on ([ n ] ∪ { v } ) \ L by∆ /L := { S ∈ ∆ : S ∩ L = ∅} ∪ { S ∪ { v } : S ∩ L = ∅} . ORMAL BINARY GRAPH MODELS 3
When we contract the edge, we set r v = min f ∈ F r f . The second operation is vertexdeletion. If v ∈ [ n ] define the vertex deletion by∆ \ v = { S ∈ ∆ : v / ∈ S } . The following lemma seems to be known in the literature on marginal polytopes, thoughit is difficult to find a precise reference.
Lemma 2.1.
Suppose that Γ is obtained from ∆ by either (1) deleting a vertex, or (2) contracting an edge.Then C Γ is (isomorphic to) a face of C ∆ and S Γ is isomorphic to S ∆ ∩ C Γ . In particular,if S ∆ is normal then so is S Γ .Proof. Faces of a cone (or semigroup) are obtained by taking the intersection of the cone(or semigroup) with a hyperplane of the form c T p = 0, where c T p ≥ v , consider the hyperplane given by c T p = r v X j =2 p { v } j . Clearly c T p ≥ C ∆ , so C ∆ ∩ { p : c T p = 0 } is a face of C ∆ .Furthermore, both it, and the corresponding semigroup are generated by π ∆ ( e i ,...,i n ) suchthat i v = 1. So this is the same as the marginal cone (or semigroup) with the same ∆and r v = 1. However, in this case, if v ⊂ F , then the F marginal of a table in this face isthe same as the F \ { v } marginal. Hence, this facial cone (or semigroup) is isomorphic to C ∆ \ v (or S ∆ \ v .For the case of contracting an edge L , we can take the hyperplane c T p = X i L ∈R L \ D p Li L where D is the diagonal D = { i L ∈ R F : i l = i l · · · } . Clearly c T p ≥ C ∆ , so C ∆ ∩ { p : c T p = 0 } is a face of C ∆ . Furthermore, both it, and thecorresponding semigroup are generated by π ∆ ( e i ,...,i n ) such that i l = i l = · · · . Then if F ∩ L is nonempty, then the F marginal of a table on this face, can be recovered fromthe marginal of F \ L ∪ l where l is any element of L . Since we can take the same l l for all such F , we deduce that this facial cone (or semigroup) is isomorphic to C ∆ /L (or C ∆ /L ). (cid:3) The holy grail for studying the geometry of the marginal cone would be a result aboutremoving an element of ∆. It is unlikely that there is a very general result that removingelements from ∆ preserves normality. Our next crucial result, Lemma 2.3 concerns aspecial case of when normality is preserved on removing a face.
SETH SULLIVANT
To explain Lemma 2.3 we first need to introduce a slightly modified version of ourcoordinate system for speaking of marginal cones a semigroups, which allows us to workwith full dimensional cones and semigroups. To reduce the dimensionality, we show thatwe only need to consider those i F that do not contain r f , for all f ∈ F . Note howeverthat we always include a coordinate p ∅ , which gives the sample size of a table u .To this end let R F = Q f ∈ F [ r f − Proposition 2.2.
Consider the map from L : L F ∈ ∆ R R F → L F ∈ ∆ R R F that deletesall the p Fi F such that there is an f ∈ F with i f = r f . Then there is a linear map M : L F ∈ ∆ R R F → L F ∈ ∆ R R F such that M ◦ L ( C ∆ ) = C ∆ . Furthermore, L ( C ∆ ) is a fulldimensional polyhedral cone.Proof. The full-dimensionality will follow from the existence of the inverse map M , be-cause the dimension of the space L F ∈ ∆ R R F equals the dimension of the marginal coneby Corollary 2.7 in [7].To prove the existence of the inverse map, we use a familiar M¨obius inversion styleformula. In particular, we need to show that we can recover p Fi F where i F contains somezeros from all the p Fi F where these last contain no zeroes. It suffices to show this in thecase where i F is the vector of all elements equal to r f , we we denote r F . For each S ∈ ∆let q F = P i F ∈ R F p Fi F . Then we have p Fr F = X S ⊆ F ( − S q S . (cid:3) Lemma 2.3.
Suppose that ∆ and r , . . . , r n are such that S ∆ is normal. Let F be amaximal face of ∆ . Let A, B be integral matrices such that C ∆ = (cid:26) ( x, y ) = ( p Si S : S ∈ ∆ \ F, p Fi F ) : ( A B ) (cid:18) xy (cid:19) ≥ (cid:27) . If the matrix B satisfies the property that the system By ≥ b has an integral solution forall b such that the system has a real solution, then S ∆ \ F is normal.Proof. Note that this lemma is merely a general property of projections of cones (
A B ) (cid:0) xy (cid:1) ≥ C ∆ \ F , and let x be an integralpoint in it. If we can find an integral y such that ( x, y ) ∈ C ∆ , we will be done because C ∆ is normal since ( x, y ) ∈ S ∆ implies x ∈ S ∆ \ F . The condition on B guarantees that anintegral y always exists, since the set of such y is the solution to the system By ≤ − Ax where − Ax is integral and real feasible. (cid:3) One more tool we will need for proving normality, is that normality is preserved whengluing two simplicial complexes together according to a reducible decomposition. A sim-plicial complex ∆ has a reducible decomposition (∆ , S, ∆ ) is ∆ = ∆ ∪ ∆ , ∆ ∩ ∆ = 2 S (where 2 S is the power set of S ), and either ∆ nor ∆ = 2 S . A simplicial complex with areducible decomposition is called reducible . A simplicial complex is decomposable if it iseither a simplex (of the form 2 K ) or it is reducible and both ∆ and ∆ are decomposable. ORMAL BINARY GRAPH MODELS 5
Lemma 2.4.
Let ∆ be a reducible simplicial complex with decomposition (∆ , S, ∆ ) . If S ∆ and S ∆ are normal then so is S ∆ .Proof. Let x = ( x ∆ , x ∆ ) ∈ C ∆ . (Without changing the issues of normality where canrepeat all the elements of x Ti such that T ⊆ S ). We must show that x ∈ S ∆ . Now, both S ∆ and S ∆ are normal, which implies that x ∆ ∈ S ∆ and x ∆ ∈ S ∆ . Let | ∆ i | = ∪ T ∈ ∆ i T be the ground set of ∆ i . Let Γ be the simplicial complex with facets | ∆ | and | ∆ | .For each i = 1 ,
2, by normality of S ∆ i , there exist integral vectors y i ∈ S | ∆ i | suchthat π ∆ i ( y i ) = x i . Furthermore, since π S ( x ) = π S ( x ) we have that the pair ( y , y )satisfies the natural equality relations to lie in the cone C Γ . However, Γ is an exampleof decomposable simplicial complex, so its only facet defining inequalities come frompositivity [6], so ( y , y ) lies in C R . It is also known that decomposable models are normal[7], so ( y , y ) ∈ S R . This implies that ( π ∆ ( y ) , π ∆ ( y )) = ( x ∆ , x ∆ ) ∈ S ∆ , so S ∆ isnormal. (cid:3) To prove Theorem 1.2 we need some important folklore decompositions for K minor-free graphs. Recall that a graph G is chordal if every cycle of length ≥ G is a chordal graph H such that G ⊆ H .The tree width of G , denoted τ ( G ) is one less than the minimal clique number over allchordal triangulations of G . Note that chordal graphs are closely related to decomposablesimplicial complexes: a simplicial complex is decomposable if and only if it is the complexof cliques of a chordal graph. A folklore result relates tree width and K minor free graphs. Theorem 2.5.
A graph G is free of K minors if and only if τ ( G ) ≤ . The following lemma, which can be handled computationally using the program Nor-maliz [2].
Lemma 2.6.
Let r i = 2 for all i . Then the semigroup S K l for the complete graph K l isnormal if and only if l = 1 , , .Proof. For l = 1 , C K l is a unimodular simplicial cone and normality follows. For l = 3,the semigroup S K is 7 dimensional with 8 generators and normality is verified withNormaliz. For l = 4, a computation with Normaliz shows that the set ( C K ∩ L F ∈ K Z R F ) \S K consists of a single point (see also, [9]). For l ≥ S K l is not normal by Lemma 2.1. (cid:3) Now for our last piece of the argument, we need to recall a result about the polyhedralstructure of cone C G in the case the G is K minor free. We work with the alternatecoordinate system introduced in Proposition 2.2. Since we are working will that case that r i = 2, each coordinate p Fi F will have i F = (1 , , . . . ), a vector of all ones. To simplifynotation, we merely use p F to denote this coordinate. Theorem 2.7. [1]
Let r i = 2 for all i . If the graph G is free of K minors, the cone C G is the solution to the following system of inequalities: p jk ≥ , p j − p jk ≥ , p k − p jk ≥ , p ∅ − p j − p k + p jk ≥ for all jk ∈ E SETH SULLIVANT X jk ∈ O p jk − X jk ∈ C \ O p jk − X j ∈ V ( O ) p j + X j ∈ V ( C \ O ) p j + O − p ∅ ≥ for all cycles C ∈ G and odd subsets O ⊆ C where V ( O ) and V ( C \ O ) denotes the set of vertices that appear in O and C \ O , respec-tively. We now have all the tools in hand to prove our main results on normality.
Proof of Theorem 1.2.
First of all, we will show that if a graph G has a K minor, then S G could not be normal. If a graph has a K minor, then that minor can be realized byvertex deletions and edge contractions alone. Since Lemma 2.6 implies that S K is notnormal, Lemma 2.1 implies that S G is not normal.Now suppose that G is free of K minors. By Theorem 2.5, G has tree width ≤
2. If G is a chordal graph with τ ( G ) ≤ G , these reducible decompositions preservenormality. Lemma 2.6 implies that S K l is normal if l = 1 , , or 3. Thus, S G is normal if G is a chordal graph. Since every K minor free graph is the edge subgraph of a chordal K minor free graph, it suffices to show that normality is preserved when deleting an edgefrom a K minor free graph.To prove this last claim, let G be a K minor free graph such that S G is normal, andlet e be an arbitrary edge in G , and let H = G \ e , the edge deletion of G . According toTheorem 2.7, the cone C G is the solution to the system of equations that have the form( A, B )( x, p e ) T ≥
0, where all the entries of B are either ± ,
0. In particular, because thereis only one coordinate p e , B is a column of 0 , ±
1. Thus, the system By ≥ b , where b is anintegral vector such that By ≥ b has a real solution is equivalent to a system c ≤ y ≤ c ,where c and c are integers. Every such system which has a real solution has an integralsolution. Lemma 2.3 implies that S H is normal. (cid:3) Further Directions
This paper has solved the normality question for binary graph models, however, thisis still very far from a complete solution to the normality problem for arbitrary sets ofmarginals for arbitrary sized tables. In this section, we outline some possible directionsfor further research.First of all, the technique used in this paper (gluing along facets and then removing afacet preserving normality, together with knowledge of the defining inequalities of struc-ture of marginal cone) can be used to prove normality in many more situations besidesjust for all r i = 2. Example 3.1.
Consider the three cycle ∆ = [12][13][23], with r = 2 , r = 4 , and r = 3.A direct computation with Normaliz shows that S ∆ is normal in this case, and the facetdescription computed there has that, for the edge 13, the corresponding B matrix is a110 × , , ± (0 , , ± (1 , ± (1 , . ORMAL BINARY GRAPH MODELS 7
It is easy to see that such a B satisfies the conditions of Lemma 2.3, so that we can deletethe edge [13] to preserve normality. Of course, this is not very interesting as this producesa decomposable complex [12][23], which are always normal.Now suppose that we have the complex ∆ = [12][13][14][23][34], with r = 2 , r =4 , r = 3 , and r = 4. This is reducible into two models of the above type, along the edge[13], so S ∆ is normal. Since ∆ is reducible, the polyhedral structure of C ∆ is obtainedby taking the union of the two constraint sets for the two halves. In particular, if we tryto delete the edge [13], we get a B matrix of size 220 ×
2, where each row is one of thevectors (0 , , ± (0 , , ± (1 , ± (1 , . Hence the resulting four cycle ∆ ′ = [12][14][23][34] is normal with r = 2 , r = 4 , r = 3 , and r = 4. (cid:3) The preceding example illustrates that the techniques can be applied to more generalgraphs. It seems natural to hope that once the (non)normality of all the three cycle models[12][13][23] have been determined for all r , r , and r , that a combination of techniquesfrom the preceding Section could be used to decide normality for arbitrary graphs.A second situation where these techniques are likely to apply fruitfully is for arbitrarysimplicial complexes but with all r i = 2. This is because the condition of Lemma 2.3 isespecially easy to verify in this case, because B is always a column vector. In fact, inexamples we have investigated, the conditions of Lemma 2.3 seem to be necessary andsufficient for guaranteeing normality on removing a maximal face. Example 3.2.
Let ∆ = [12][134][234], and r = r = r = r = 2. A direct computationin Normaliz shows that S ∆ is normal. The facet defining inequalities of C ∆ have 0 , ± , ± p . Removing facet [134] produces the complex Γ =[12][13][14][234], and Normaliz verifies that S Γ is not normal. (cid:3) Question 3.3.
Let r i = 2 for all i . Suppose that ∆ is such that S ∆ is normal. Let F bea maximal face of ∆. Let A, B be integral matrices such that C ∆ = (cid:26) ( x, y ) = ( p Si S : S ∈ ∆ \ F, p Fi F ) : ( A B ) (cid:18) xy (cid:19) ≥ (cid:27) , with irredundant and minimal description. If the matrix B has an entry of absolute value > S ∆ \ F is not normal?Note, however, that Lemma 2.3, while adequate for the examples we have encounteredhere, is probably not the best possible result along these lines. This is because the set of b on which the condition “real solution implies integral solution” needs to be valid is, infact, very small. Problem 3.4.
Find a stronger version of Lemma 2.3.Lastly we would like to address what we think is the main take-away message of thispaper. This is that the polyhedral structure of the cone C ∆ seems crucial to studying thenormality of S ∆ . Unfortunately, the only class where a nice polyhedral description of themarginal cone C ∆ is known is the case of K -minor free graphs with all r i = 2. SETH SULLIVANT
Problem 3.5.
Find new classes of ∆ where there is a elegant uniform description of thepolyhedral cone S ∆ .A related situation where there is an elegant polyhedral description concerns the cutcones [5]. We hope that these properties could also be used to resolve the normalityquestions for cut cones from [8]. Acknowledgments
The author received support from the U.S. National Science Foundation under grantDMS-0840795.
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