Asymptotic enumeration of symmetric integer matrices with uniform row sums
aa r X i v : . [ m a t h . C O ] J a n Asymptotic enumeration of symmetric integermatrices with uniform row sums
Brendan D. McKay
Research School of Computer ScienceAustralian National UniversityCanberra ACT 0200, Australia [email protected]
Jeanette C. McLeod
Department of Mathematics and StatisticsUniversity of CanterburyChristchurch 8140, New Zealand [email protected]
Abstract
We investigate the number of symmetric matrices of non-negativeintegers with zero diagonal such that each row sum is the same. Equiv-alently, these are zero-diagonal symmetric contingency tables withuniform margins, or loop-free regular multigraphs. We determine theasymptotic value of this number as the size of the matrix tends toinfinity, provided the row sum is large enough. We conjecture thatone form of our answer is valid for all row sums.
Let M ( n, ℓ ) be the number of n × n symmetric matrices over { , , , . . . } with zeros on the main diagonal and each row summing to ℓ . Our interestis in the asymptotic value of M ( n, ℓ ) as n → ∞ with ℓ being a function1 = 9 ℓ = 20 λ = ℓn − = 2 . M (9 ,
20) =1955487489759152410696.of n . Alternative descriptions of the class M ( n, ℓ ) are: adjacency matricesof loop-free regular multigraphs of order n and degree ℓ , and zero-diagonalsymmetric contingency tables of dimension n with uniform margins equalto ℓ . An example appears in Figure 1.Very little seems to be known about this problem. The asymptotic valueof M ( n,
3) was determined by Read in 1958 [12]. According to Bender andCanfield [3], de Bruijn extended this to M ( n, ℓ ) for fixed ℓ but failed topublish it. In any case, [3] generalised the result to bounded but possiblynon-equal row sums. By the method of switchings, Greenhill and McKay [7]found the asymptotic number of matrices with given small row sums over arange that includes M ( n, ℓ ) for ℓ = o ( n / ).In this paper we treat the case of large ℓ and manage to find the asymp-totics whenever ℓ > Cn/ log n for any C > . We will use the multidi-mensional saddle-point method, which was previously applied successfully tothe corresponding { , } problem by McKay and Wormald [11] and to thecorresponding non-symmetric problem by Canfield and McKay [5]. For thenon-symmetric problem with mixed row and column sums, see Barvinok andHartigan [1].Our theorem is as follows. Theorem 1.1.
Let a and b be positive real numbers such that a + b < . Let ℓ = ℓ ( n ) be such that ℓn is even and λ = ℓ/ ( n − satisfies λ ≥ a log n . (1)2 hen as n → ∞ , M ( n, ℓ ) = √ (cid:0) πn (1 + λ ) − ℓ − n +2 λ ℓ +1 (cid:1) − n/ exp (cid:18) λ + 14 λ − λ (1 + λ ) + O ( n − b ) (cid:19) = (cid:18) λ λ (1 + λ ) λ (cid:19) ( n ) (cid:18) n + ℓ − ℓ (cid:19) n √ e / (cid:0) O ( n − b ) (cid:1) . (2)In Section 2, we express M ( n, ℓ ) as an integral in n -dimensional complexspace and divide the domain of integration into three parts, then in Section 3we estimate the integral in two of the parts. In Section 4, we show that thethird part is negligible in comparison provided ℓ is bounded by a polynomialin n . We complete the proof for large ℓ in Section 5 using the theory ofEhrhart quasipolynomials.In Section 6, we show that the form of expression (2) is motivated bya na¨ıve probabilistic model. We also note that (2) agrees with [7], apartfrom the error term, when 1 ≤ ℓ = o ( n / ), and closely matches many exactvalues computed as described in Section 7. This leads us to suspect that (2)is true whenever ℓ >
0, and we conjecture explicit bounds for M ( n, ℓ ) inConjecture 7.Throughout the paper, asymptotic notation like O ( f ( n )) refers to thepassage of n to ∞ . We will also use a modified notation e O ( f ( n )). A function g ( n ) belongs to this class provided that g ( n ) = O ( f ( n ) n aε ) , for some numerical constant a that might be different at each use of thenotation. M ( n, ℓ ) We now express M ( n, ℓ ) as an integral in n -dimensional complex space andoutline a plan for estimating it.We begin with a generating function in n variables x , . . . , x n , Y ≤ j Let ε ′ , ε ′′ , ε ′′′ , ˇ ε be constants such that < ε ′ < ε ′′ < ε ′′′ , and ˇ ε > . The following is true if ε ′′′ is sufficiently small.Let ˆ A = ˆ A ( n ) be a real-valued function such that ˆ A ( n ) = Ω( n − ε ′ ) . Let ˆ B = ˆ B ( n ) , ˆ C = ˆ C ( n ) , ˆ E = ˆ E ( n ) , ˆ F = ˆ F ( n ) , ˆ G = ˆ G ( n ) , ˆ H = ˆ H ( n ) , and ˆ I = ˆ I ( n ) be complex-valued functions of n such that ˆ B, ˆ C, ˆ E, ˆ F , ˆ G, ˆ H, ˆ I = O (1) . Suppose ˆ ε ( n ) satisfies ε ′′ ≤ ε ( n ) ≤ ε ′′′ for all n and define U n = (cid:8) z ⊆ R n : | z j | ≤ n − / ε ( n ) for ≤ j ≤ n (cid:9) . Suppose that for z = ( z , z , . . . , z n ) ∈ U n we have f ( z ) = exp (cid:18) − ˆ An n X j =1 z j + ˆ Bn n X j =1 z j + ˆ C n X j,k =1 z j z k + ˆ Dn − n X j,k,p =1 z j z k z p + ˆ En n X j =1 z j + ˆ F n X j,k =1 z j z k + ˆ Gn / n X j,k =1 z j z k + ˆ Hn − / n X j,k,p =1 z j z k z p + ˆ In − / n X j,k,p,q =1 z j z k z p z q + δ ( z ) (cid:19) , here δ ( z ) is continuous and δ ( n ) = max z ∈ U n | δ ( z ) | = o (1) . Then, providedthe O ( ) term in the following converges to zero, ˆ U n f ( z ) d z = (cid:18) π ˆ An (cid:19) n/ exp (cid:16) Θ + O (cid:0) n − / ε + ( n − / + δ ( n )) ˆ Z (cid:1)(cid:17) , where Θ = 15 ˆ B 16 ˆ A + 3 ˆ B ˆ C A + ˆ C 16 ˆ A + 3 ˆ E A + ˆ F A , and ˆ Z = exp (cid:18) 15 Im( ˆ B ) + 6 Im( ˆ B ) Im( ˆ C ) + Im( ˆ C ) 16 ˆ A (cid:19) . The following lemma defines a linear transformation, adapted from [11]. Lemma 3.2. Define c and z = ( z , z , . . . , z n ) by c = 1 − s n − n − 1) = 1 − − / + O ( n − ) , (7)(1 + λ ) θ j = z j − cn n X k =1 z k (1 ≤ j ≤ n ) . (8) The transformation θ = T ( z ) defined by (8) has determinant (1 − c ) / (1+ λ ) n .For m ≥ , define µ m = P nj =1 z mj . Then we have the following translations. (1 + λ ) n X j =1 θ j = (1 − c ) µ , (1 + λ ) X ≤ j For all real X , (cid:0) − λ ( e iX − (cid:1) − = exp (cid:0) λiX − λ (1 + λ ) X − iλ (1 + λ )(1 + 2 λ ) X + λ (1 + λ )(1 + 6 λ + 6 λ ) X + O (( λ + λ ) X ) (cid:1) . We now present the main result of this section. Theorem 3.4. Under the conditions of Theorem 1.1, there is a region R ′ such that R ⊆ R ′ ⊆ R ⊆ [ − π, π ] n \ R π and I R ′ ( n ) = 1 √ (cid:18) πλ (1 + λ ) n (cid:19) n/ exp (cid:18) λ + 14 λ − λ ( λ + 1) + O ( n − b ) (cid:19) . Proof. Consider the transformation θ = T ( z ) defined by (8) . Define R z = { z : | z j | ≤ n − / ε } and R ′ = T ( R z ) . From (8) we have | θ j | ≤ y for all j = ⇒ | z j | ≤ (1 + λ )(1 − c ) − y for all j, | z j | ≤ y for all j = ⇒ | θ j | ≤ (1 + λ ) − (1 + c ) y for all j. These imply, for n ≥ , that T − R ⊆ R z and R ⊆ R ′ ⊆ R . From Lemma 3.3 we have, for θ ∈ R ′ , F ( θ ) = exp (cid:18) − A X ≤ j In the previous section we proved that the contribution to I ( n ) from the box R ′ is I R ′ ( n ) = (cid:18) πA n (cid:19) n exp (cid:0) O (1 + λ − ) (cid:1) . We now consider the contribution to I ( n ) from the region R c (defined in (5))and show, provided λ is not too large, that it is negligible compared to I R ′ ( n ).First we import from [5] some useful lemmas.9 emma 4.1. The absolute value of the integrand F ( θ ) of (4) is | F ( θ ) | = Y ≤ j Define t = (1 + λ ) − and g ( x ) = − A x + ( A + 9 A ) x .Then, uniformly for λ > and K ≥ , ˆ t − t exp (cid:0) Kg ( x ) (cid:1) dx ≤ p π/ ( A K ) exp (cid:0) O ( K − + ( A K ) − ) (cid:1) . Theorem 4.3. Suppose that the conditions of Theorem 1.1 hold, and inaddition that λ = n O (1) . Then ˆ R c | F ( θ ) | d θ = O ( n − ) I R ′ ( n ) . Proof. The proof follows a similar pattern to that of [11, Theorem 1]. Define t and g ( z ) as in Lemma 4.2.Define n , n , n , n , functions of θ , to be the number of indices j suchthat θ j lies in [ − t, t ] , ( t, π − t ) , [ π − t, π + t ] , and ( − π + t, − t ) , respectively. Let R ′′ be the set of all θ such that max { n n , (cid:0) n (cid:1) , (cid:0) n (cid:1) } ≥ n ε . Any θ ∈ R ′′ has the property that f ( θ j + θ k ) ≤ f (2 t ) for at least n ε pairs j, k . Since f ( z ) ≤ for all z , and the volume of R ′′ is less than (2 π ) n , we have ˆ R ′′ | F ( θ ) | d θ ≤ (2 π ) n f (2 t ) n ε . Applying (9) and the assumption that λ = O ( n O (1) ) , we find that ˆ R ′′ | F ( θ ) | d θ = O ( e − c n ε/ ) I R ′ ( n ) (10) for some c > .For θ ∈ R c \ R ′′ we must have n , n = O ( n / ε ) and either n = O ( n / ε ) or n = O ( n / ε ) . The latter two cases are equivalent, so we willassume that n = O ( n / ε ) , which implies that n = n − O ( n / ε ) . efine S , S , S , functions of θ , as follows. S = { j : | θ j | ≤ t } ,S = { j : t < | θ j | ≤ t } ,S = { j : | θ j | > t } . Define s i = | S i | for each i . Since s = n , we know that s + s = O ( n / ε ) .Now we bound | F ( θ ) | in R c \ R ′′ using f ( θ j + θ k ) ≤ f ( t ) ≤ exp (cid:18) − λ λ ) (cid:19) if j ∈ S , k ∈ S , exp (cid:0) − A ( θ j + θ k ) + ( A + A )( θ j + θ k ) (cid:1) if j, k ∈ S , otherwise . Let I ( s ) be the contribution to I ( n ) from those θ ∈ R c \ R ′′ with the givenvalue of s , and let θ ′ denote the vector ( θ j ) j ∈ S . The set S can be chosenin at most n s ways. Applying the bounds above, and allowing (2 π ) s + s forintegration over θ j ∈ S ∪ S , we find I ( s ) ≤ n s (2 π ) s + s exp (cid:18) − s s λ λ ) (cid:19) I ′ ( s ) , (11) where I ′ ( s ) = ˆ t − t · · · ˆ t − t Y j,k ∈ S ,j 2) exp (cid:0) O (1 + λ − ) n − (cid:1)(cid:19) s ≤ (cid:18) πA n (cid:19) n/ exp (cid:0) O ( n / ε ) (cid:1) . The third line of the above follows from the bounds X ≤ j 11 + λ (cid:19) ( n ) (cid:18) λ λ (cid:19) nℓ/ . (Proof: Apply (21) to each entry in the upper triangle and use the assumedindependence of the entries there. The result is independent of the actualmatrix entries.) Therefore, M ( n, ℓ ) = Prob( E all ) P . Now make a na¨ıve assumption that the events E j are independent.By symmetry, Prob( E j ) is independent of j , so we get a na¨ıve estimate of M ( n, ℓ ): M naive ( n, ℓ ) = Prob( E ) n P . (22)Now consider Prob( E ). The number of possible first rows is (cid:18) n + ℓ − ℓ (cid:19) . (This is the number of ways of writing ℓ as the sum of n − S , each such first row has probability (cid:18) 11 + λ (cid:19) n − (cid:18) λ λ (cid:19) ℓ . E ) = (cid:18) n + ℓ − ℓ (cid:19)(cid:18) 11 + λ (cid:19) n − (cid:18) λ λ (cid:19) ℓ . Substituting this value into (22), we get M naive ( n, ℓ ) = (cid:18) λ λ (1 + λ ) λ (cid:19) ( n ) (cid:18) n + ℓ − ℓ (cid:19) n Therefore, formula (2) in Theorem 1.1 can be written M ( n, ℓ ) = M naive ( n, ℓ ) √ (cid:0) + O ( n − b ) (cid:1) . Note that √ e / ≈ . As noted in Section 5, M ( n, ℓ ) is the number of integer points in ℓ P n , where P n is the polytope defined in that section. Lattice point enumeration tech-niques such as the algorithm in [6] therefore allow the exact computation of M ( n, ℓ ) for small n . In practice this is feasible for n ≤ n ≤ 10, almost irrespective of ℓ .By interpolating the computed values, we obtain the Ehrhart quasi-polynomial for small n . Recall that M ( n, ℓ ) is a polynomial M e ( n, ℓ ) foreven ℓ and a polynomial M o ( n, ℓ ) for odd ℓ . We have M o ( n, ℓ ) = 0 if n isodd, and the following. M e (3 , ℓ ) = 1 M e (4 , ℓ ) = M o (4 , ℓ ) = ℓ + ℓ + 1 M e (5 , ℓ ) = ℓ + ℓ + ℓ + ℓ + ℓ + 1 M e (6 , ℓ ) = ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + 1 M o (6 , ℓ ) = M e (6 , ℓ ) − e (7 , ℓ ) = ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + 1 M e (8 , ℓ ) = ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + 1 M o (8 , ℓ ) = M e (8 , ℓ ) − ℓ − ℓ − ℓ − ℓ − ℓ − M e (9 , ℓ ) = ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + ℓ ℓ + ℓ + ℓ + ℓ + ℓ + ℓ + 1The same method would yield M (10 , ℓ ) with a plausible but large amountof computation. For completeness, we also give the Ehrhart series L n ( x ) = P ℓ ≥ M ( n, ℓ ) x ℓ for n ≤ − x ) L ( x ) = 1(1 − x ) L ( x ) = 1(1 − x ) L ( x ) = ( x + 1) + 16 ( x + x ) + 41 x (1 − x ) (1 + x ) L ( x ) = ( x + 1) + 6 ( x + x ) + 30 ( x + x ) + 40 x (1 − x ) L ( x ) = ( x + 1) + 807 ( x + x ) + 81483 ( x + x )+ 1906342 ( x + x ) + 15277449 ( x + x )+ 50349627 ( x + x ) + 74301542 x (1 − x ) (1 + x ) L ( x ) = ( x + 1) + 90 ( x + x ) + 4726 ( x + x )+ 107050 ( x + x ) + 1261121 ( x + x )+ 8761248 ( x + x ) + 39187016 ( x + x )+ 119662536 ( x + x ) + 259344246 ( x + x )+ 408811676 ( x + x ) + 475095180 x (1 − x ) L ( x ) = ( x + 1) + 52524 ( x + x )+ 169345602 ( x + x )+ 78276428212 ( x + x )+ 10217460516057 ( x + x )+ 527531262668208 ( x + x )+ 13016462628712186 ( x + x )+ 172410423955058664 ( x + x )18 1322251960254170931 ( x + x )+ 6176715510750440488 ( x + x )+ 18182086106689738044 ( x + x )+ 34470475812807166836 ( x + x )+ 42606701216240491693 x For larger n , P n has too many vertices for this method to be useful, butwe can use the technique of [8] and [5]. Define f ( z ) = 1 + z + z + · · · + z ℓ .Then M ( n, ℓ ) is the coefficient of x ℓ x ℓ · · · x ℓn y nℓ/ in Q ≤ j 10) = 613329062511931789477677176839174642138032757885191693120 , which is about 2% higher than the estimate of Theorem 1.1.Machine-readable versions of these exact formulas, along with many otherexact values of M ( n, ℓ ), can be found at [10].After observing a large number of exact values, we have noted that (2)appears to have an accuracy much wider than we can prove. We can evenguess extra terms. We express our observations in the following conjecture.19 onjecture. For even nℓ , define ∆ ( n, ℓ ) by M ( n, ℓ ) = M naive ( n, ℓ ) √ (cid:18) 34 + 3 ℓ + 112 ℓ ( n − 1) + ∆ ( n, ℓ ) n ( n − (cid:19) . Then | ∆ ( n, ℓ ) | < for n ≥ , ℓ ≥ . In this section we note a simple corollary of Theorem 1.1. Choose X uni-formly at random from the set M ( n, ℓ ) of zero-diagonal symmetric non-negative integer matrices of order n and row sums ℓ . Let X min be the leastoff-diagonal entry of X . If X min ≥ k for integer k ≥ 0, we can subtract k from each entry to make a matrix of row sums ℓ − ( n − k . This elementaryobservation shows thatProb( X min ≥ k ) = M ( n, ℓ − ( n − k ) M ( n, ℓ ) . Theorem 1.1 can thus be used to estimate this probability whenever it appliesto the quantities on the right. We can provide some information even in othercases; note that (1) is not required for the following. Theorem 8.1. Let k = k ( n ) ≥ and ℓ = ℓ ( n ) ≥ with nℓ even. Define a = kn /ℓ . Then, as n → ∞ , Prob( X min ≥ k ) ( → if a → ∞∼ e − a/ if a = O (1) . Proof. We begin with a case incompletely covered by Theorem 1.1, namely ℓ = o ( n ) . Define M , M to be the sets of those matrices in M ( n, ℓ ) withno off-diagonal zeros, and exactly two or four off-diagonal zeros, respectively.Given X ∈ M , choose distinct q, r, s, t and replace a qr , a rs , a st , a tq (and a rq , a sr , a ts , a qt consistently) by a qr − δ, a rs + δ, a st − δ, a tq + δ , where δ =min { a qr , a st } . This can be done in Θ ( n ) ways and creates an element of M .Alternatively, if X ∈ M , choose distinct q, r, s, t such that either a qr or a st or both are 0. Then replace a qr , a rs , a st , a tq (and a rq , a sr , a ts , a qt consistently)by a qr + δ, a rs − δ, a st + δ, a tq − δ , where ≤ δ ≤ min { a rs , a tq } − . If his produces an element of M , it is the inverse of the previous operation.Given a choice of a qr = 0 , s and δ can be chosen in at most ℓ ways since P s a rs = ℓ , then t can be chosen in at most n ways. Similarly for q st = 0 .Therefore, this operation can be done in at most O ( ℓn ) ways. It followsthat either |M | = 0 or |M | = o ( |M | ) , which completes this case since Prob( X min ≥ k ) ≤ Prob( X min ≥ for k ≥ .In case ℓ = Θ ( n ) , define k ′ = min { k, ⌊ ℓ/ (2 n ) ⌋} and estimate the valueof Prob( X min ≥ k ′ ) using (16) . This gives the desired result when k = k ′ .For k > k ′ the value obtained tends to 0, so again the desired result followsby monotonicity with respect to k . In this paper we have begun the asymptotic enumeration of dense symmetricnon-negative integer matrices with given row sums, by considering the specialcase of uniform row sums and zero diagonal. Further cases, which can beapproached by the same method, are to allow the row sums to vary, and toallow diagonals other than zero. The structure of random matrices in theclass can also be investigated by specifying some forced matrix entries. Wehope to return to these problems in the future. References [1] A. Barvinok and J. A. 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