arXiv:1302.2446v1 [math.CO] 11 Feb 2013
Degree sequences of random digraphs and bipartite graphs Brendan D. McKay∗ and Fiona Skerman† Research School of Computer Science Australian National University Canberra ACT 0200, Australia
[email protected],
[email protected]
Abstract We investigate the joint distribution of the vertex degrees in three models of random bipartite graphs. Namely, we can choose each edge with a specified probability, choose a specified number of edges, or specify the vertex degrees in one of the two colour classes. This problem can also be described in terms of the row and sum columns of random binary matrix or the in-degrees and out-degrees of a random digraph, in which case we can optionally forbid loops. It can also be cast as a problem in random hypergraphs, or as a classical occupancy, allocation, or coupon collection problem. In each case, provided the two colour classes are not too different in size or the number of edges too low, we define a probability space based on independent binomial variables and show that its probability masses asymptotically equal those of the degrees in the graph model almost everywhere. The accuracy is sufficient to asymptotically determine the expectation of any joint function of the degrees whose maximum is at most polynomially greater than its expectation. Our starting points are theorems of Canfield, Greenhill and McKay (2008–2009) that enumerate bipartite graphs by degree sequence. The resulting theory is analogous to that developed by McKay and Wormald (1997) for general graphs. ∗ †
Research supported by the Australian Research Council. Current address: University of Oxford, Department of Statistics, Oxford OX1 3TG, UK
1
1
Introduction
We prefer to use graph terminology, but will first describe the setting in the matrix and other formulations. Consider a probability space of m × n matrices over {0, 1}. Three probability spaces will be considered. In the first case, which we call Gp , some number p ∈ (0, 1) is specified and each entry of the matrix is independently equal to 1 with probability p and equal to 0 otherwise. In the second case, which we call Gk , some integer k is specified, and all m×n binary matrices with exactly k ones have the same probability. In the third case, which we call Gt , a list of n integers t1 , . . . , tn is specified, and all m × n binary matrices with column sums t1 , . . . , tn , respectively, are equally likely. We can interpret the matrix as a bipartite graph in the standard fashion. Associate distinct vertices U = {u1 , . . . , um } with the rows, and V = {v1 , . . . , vn } with the columns, and place an edge between ui and vj exactly when the matrix entry in position (i, j) equals 1. The row and column sums of the matrix correspond to the degrees of the vertices. These probability models have also appeared in other settings. Given m bins, at each stage j = 1, . . . , n throw tj balls into distinct bins (with all m possible placings equally tj likely). Then the distribution of the number of balls in each bin S = (S1 , . . . , Sm ) can be studied. This model is referred to as allocation by complexes and is precisely our Gt model. If we allow the number of balls thrown to be a random variable Tj , binomially distributed with parameters (m, p), we attain the Gp model.
Similarly, in the coupon collection problem a customer repeatedly buys a random number, T , of distinct coupons from a set of m possible different coupons. This covers both our Gp case when T is binomially distributed with parameters (m, p) and our Gt case where Tj = tj with probability 1. (Here, our vector s describes the number of each coupon collected and t the number of coupons collected at each stage.) Finally, consider a hypergraph on m vertices. At each stage j = 1, . . . , n, choose at random a hyperedge of size tj , allowing multi-edges. Then if we let Si be the number of hyperedges which contain the ith vertex, we obtain the Gt model.
If m = n, we can also associate the matrix with a directed graph. There are n vertices {w1 , . . . , wn }. A matrix entry equal to 1 in position (i, j) corresponds to a directed edge from wi to wj . Note that i = j is possible, so these directed graphs can have loops. The row and column sums of the matrix correspond to the out-degrees and in-degrees, respectively, of the directed graph. We will also treat the case of loop-free digraphs, which correspond to square matrices with zero diagonal. Our methods would also work if some other limited set of matrix entries are required to be zero, but we have not applied them in that case. 2
We now continue using the bipartite graph formulation. For each of the three probability spaces of random bipartite graphs, we seek to examine the (m+n)-dimensional joint distribution of the vertex degrees. If G is a bipartite graph on U ∪ V (respecting the partition into U and V ), then s = s(G) = (s1 , . . . , sm ) is the list of degrees of u1 , . . . , um , and t = t(G) = (t1 , . . . , tn ) is the list of degrees of v1 , . . . , vn . We call the pair (s, t) the degree sequence of G. n Define In = {0, 1, . . . , n} and Im,n = Inm × Im . Also let G(s, t) be the number of bipartite graphs on U ∪ V with degree sequence (s, t). In the case of m = n, we also ~ t) to be the number of digraphs with in-degrees s and out-degrees t. define G(s,
For precision we need to distinguish between random variables (written in uppercase) and the values they may take (written in lowercase). For each probability space of random graphs, as determined by the context, S = (S1 , . . . , Sm ) will denote the random variable given by the degrees in U and T = (T1 , . . . , Tn ) will denote the random variable given by n the the degrees in V . We will take S to have range Inm and T to have range Im . Also define random variables K=
m X
Si =
i=1
n X j=1
Tj ,
Λ=
K . mn
As usual, q is an abbreviation for 1 − p.
Similar results for the degree sequences of ordinary (not necessarily bipartite) graphs were obtained by McKay and Wormald [27, 28].
1.1
Historical notes
The Gt model has received wide ranging attention, in particular the distribution of the number of isolated vertices. This is also a natural question in the alternative (non-graph) wordings of the model. It corresponds to the number of empty bins in the allocation model [4,11,12,18,29,36,42], the number of uncollected coupons in the collector’s problem [24,41], the number of isolated vertices in the hypergraph model and the number of zero rows in the binary matrix model [13]. More generally, the number of vertices with a particular degree (or range of degrees) in Gt has been studied in allocation [30, 37, 38], graph [1, 22] and matrix models [7]. A different extension on this theme is to study the distribution of the number of draws required to go from i to j non-empty bins [2, 20, 30, 39, 40]. In a similar direction, Khakimullin and Enatskaya studied the distribution of the number of draws to exceed a particular lineup in the bins in the Gt model [17] and in the i.i.d. case which includes the Gp model as well [19]. The monograph by Kolchin gives many results on Gt phrased as the balls and bins model [21]. 3
We are interested in asymptotic results as we take m, n roughly equal as they tend to infinity, but another natural option is to fix m, the number of vertices in one part, and let n, the number of vertices in the other part, tend to infinity. There seems to be a consistent divide in the literature that when considered as a graph the asymptotics of Gt are studied with m, n both tending towards infinity while the balls and bins and coupon collection articles (including those cited above) fix m and take n tending toward infinity. This corresponds to fixing the number of bins and taking the number of balls to infinity or having a fixed number of coupons and letting the number of sampling rounds tend to infinity. In the other two probability models on bipartite graphs Gp and Gk two types of results are known, those on the minimum and maximum degrees [1,5,34] and those on the number of vertices with a given degree [22, 31, 32]. For results in the digraph counterpart G~p see [35] (and below). The model Gp also appears in papers on ball and bin models. Sometimes the numbers of balls thrown at each stage are allowed to be i.i.d. random variables [15]. If we then set these random variables to be binomially distributed with parameters m, p we recover the Gp model. Godbole et. al. [8] make a study of the number of sets of r mutually threatening rooks. This corresponds to the number of vertices with h ≥ r weighted by hr in our Gp and Gk models.
Of the papers cited, we highlight some which concern the minimum and maximum degrees, a fixed number of the smallest and largest degrees and the distribution of the hth largest degree.
Khakimullin determined the asymptotic distribution of the hth largest degree when the average degree increases faster than log m [15]. The model used here allowed the numbers of balls allocated at each step to be independent identically distributed random variables and so includes both our Gp and uniform Gt cases. This extends an earlier result by the same author which gave the asymptotic distribution of the largest degree [16]. Palka and Sperling showed that if we fix p such that np = w(n) log n = o(n), then any fixed number of the smallest and largest degrees are unique in G~p and in the uniform Gt model [35]. A similar result for the G~t model is shown by Palka in [33], where t = (d, d, . . . , d) and d = w(n) log n = o(n). There is also some work on the degrees in random digraphs by Jaworski and Karonski [14] who showed, in the case that t = (d, d, . . . , d) and d = o(n), that the minimum vertex degree in Gt is almost surely the same as that in G~t .
1.2
Asymptotic notation
As we are dealing with asymptotics of functions of many variables, we must be careful to define our asymptotic notations. 4
We will tacitly assume that all variables not declared to be constant are functions of a single underlying index ℓ that takes values 1, 2, . . . , and that all asymptotic statements refer to ℓ → ∞. Thus, the size parameters m, n are in reality functions m(ℓ) and n(ℓ), and a statement like f (m, n) = O(g(m, n)) means that there is a constant A > 0 such that |f (m, n)| ≤ A|g(m, n)| when ℓ is large enough. This should not be cause for alarm, because we will invariably impose conditions implying that m, n → ∞ as ℓ → ∞. c
The expression o˜(1) represents any function of ℓ of magnitude O(e−n ) for some constant c > 0. The constant c might be different for different appearances of the notation. The class o˜(1) is closed under addition, multiplication, taking positive powers, and multiplication by polynomials in n.
1.3
Graph models
We consider three probability spaces of random graphs, and the probability spaces induced on Im,n by the corresponding random variables (S, T). 1. (p-models Gp , G~p , for 0 < p < 1) Generate G by choosing each of the mn possible edges ui vj with probability p, such choices being independent. The probability distribution Gp = Gp (m, n) on Im,n is that of the degree sequence (S, T) of G. If m = n and the edges {ui vi } are forbidden, we obtain the probability distribution G~p instead. We have ProbGp (S = s ∧ T = t) = pk q mn−k G(s, t), 2 ~ t). Prob ~ (S = s ∧ T = t) = pk q n −n−k G(s, Gp
where q = 1 − p and k =
Pm
i=1
si .
2. (k-models Gk , G~k , for integer k) Generate G by choosing each of the bipartite graphs on U ∪ V having k edges, with equal probability. The probability distribution Gk = Gk (m, n) on Im,n is that of the degree sequence (S, T) of G. If m = n and the edges {ui vi } are forbidden, we obtain the probability distribution G~k instead. We have ProbGk (S = s ∧ T = t) −1 mn G(s, t), if Pm s = Pn t = k; j=1 j i=1 i k = 0, otherwise, 5
ProbG~k (S = s ∧ T = t) −1 2 n − n G(s, ~ t), if Pn si = Pn tj = k; i=1 j=1 k = 0, otherwise, n 3. (t-models Gt , G~t , for t ∈ Im ) Generate G by choosing each of the bipartite graphs on U ∪ V having t(G) = t, with equal probability. Since the random variable T is fixed at the value t for these graphs, we will define our probability space using S only. The probability distribution Gt = Gt (m) on Inm is that of the degree sequence S of G in U. If m = n and the edges {ui vi } are forbidden, we obtain the probability n distribution G~t instead. For a given t ∈ Im , we have
ProbGt (S = s) = ProbG~t (S = s) =
n −1 Y m
j=1 n Y j=1
tj
n−1 tj
G(s, t),
−1
~ t). G(s,
The probability spaces Gp , Gk and Gt are clearly related, by mixing and conditioning. Note that the first relationships on lines (2) and (3) are independent of p and assume 0 < p < 1. n mn X Y X m tj m−tj mn k mn−k Gt , p q p q Gk = Gp = t k j n j=1 t∈Im k=0 n −1 Y m X mn Gt , Gk = Gp K=k = k t Pn j j=1 t: j=1 tj =k Gt = Gp T =t = Gk T =t ,
with similar relations between G~p , G~k and G~t .
(1) (2) (3)
Note that the separate distributions of S and T in Gp and Gk are elementary. In Gp the components of S are independent binomial distributions, while in the Gk model S has a multivariate hypergeometric distribution. The difficulty is in quantifying the dependence between S and T when all m + n components are considered together.
6
1.4
Binomial models
Our aim is to compare the degree sequence distributions defined above to some distributions derived from independent binomials. Our motivating observation is the known marginal distributions of S and T in the models Gp and Gk . ~ p , for 0 < p < 1) Generate m components distributed 1. (Independent models Ip , I Bin(n, p) and n components distributed Bin(m, p), all m + n components being independent. The joint distribution on Im,n is Ip = Ip (m, n). If instead we have m = n and the 2n components are all distributed Bin(n−1, p), the joint distribution ~p = I ~ p (n). We have on In,n is I ProbIp (S = s ∧ T = t) =p
P
i si +
P
j tj 2mn−
q
P
i si −
P
j tj
n m Y Y m n i=1
ProbI~p (S = s ∧ T = t) =p
P
i
si +
P
j tj
q
2n2 −2n−
P
i si −
P
j tj
si
j=1
tj
,
n n Y n−1 Y n−1 . tj si j=1 i=1
2. (Binomial p-models Bp , B~p , for 0 < p < 1) The distribution Bp = Bp (m, n) on Im,n Pn P is the conditional distribution of Ip subject to m j=1 Tj . For m = n, the i=1 Si = ~ p by the same conditioning. We distribution B~p = B~p (n) on In,n is obtained from I have ProbBp (S = s ∧ T = t) Pn Pm p (S = s ∧ T = t) ProbIP , if Pn m j=1 tj ; i=1 si = = ProbIp j=1 Tj i=1 Si = 0, otherwise,
and similarly for B~p .
3. (Binomial k-models Bk , B~k , for integer k) The distribution Bk = Bk (m, n) on Im,n Pn P is the conditional distribution of Ip subject to m j=1 Tj = k. For m = n, i=1 Si = ~ ~ ~ Bk = Bk (n) is derived from Ip in the same way. In both cases, the distribution
7
doesn’t depend on p. We have ProbBk (S = s ∧ T = t) m n −2 Y Pm Pn n Y m mn , if i=1 si = j=1 tj = k; k si j=1 tj = i=1 0, otherwise,
ProbB~k (S = s ∧ T = t) Y −2 Y n n 2 Pn Pn n−1 n−1 n − n , if j=1 tj = k; i=1 si = t s k j i = j=1 i=1 0, otherwise.
n 4. (Binomial t-models Bt , B~t , for t ∈ Im ) The distribution Bt = Bt (m, n) on Inm consists P of the first m components of Bk , where k = nj=1 tj . For m = n, B~t = B~t (n) is ~ p in the same way. For a given t ∈ I n , we have derived from I m
−1 m Y n Pn Pm mn if j=1 tj ; i=1 si = s k i ProbBt (S = s) = i=1 0, otherwise, 2 −1 Y n Pn Pn n−1 n −n if j=1 tj ; i=1 si = si k ProbB~t (S = s) = i=1 0, otherwise, ~ p , for 0 < p < 1) The distribution Vp = Vp (m, n) on Im,n 5. (Integrated p-models Vp , V ~p = V ~ p (n) on is a mixture of Bp′ distributions, while for m = n the distribution V In,n is a mixture of B~p′ distributions. Let ′
Kp (p ) =
mn πpq 1
V (p) =
Z
1/2
mn ′ 2 (p − p) , exp − pq
Kp (p′ ) dp′.
0
Then we define ProbVp (S = s ∧ T = t) = V (p) ProbV~ p (S = s ∧ T = t) = V (p)
−1
Z
1
Z
1
Kp (p′ ) ProbBp′ (S = s ∧ T = t) dp′ ,
0
−1
0
8
Kp (p′ ) ProbB~p′ (S = s ∧ T = t) dp′ .
Our main theorems will show that, under certain conditions, Gp is very close to Vp , Gk to Bk , and Gt to Bt . Similar relationships hold for the digraph models. We first record a few elementary properties. Pm Pn Lemma 1. If i=1 si = j=1 tj = k and pqmn → ∞, then ProbBp (S = s ∧ T = t) −1
2k 2mn−2k √
= 2 + O((pqmn) ) p q
n m Y Y m n , πpqmn si j=1 tj i=1
ProbB~p (S = s ∧ T = t) 2
−1
2k 2n2 −2n−2k
= 2 + O((pqn )) ) p q
p
n n Y n−1 Y n−1 . πpqn(n−1) t s j i j=1 i=1
uniformly over s, t. Proof. In Ip , both
Pm
i=1
ProbIp
Si and
X m
Pn
Si =
i=1
j=1 Tj
n X j=1
Tj
have the distribution Bin(mn, p). Therefore =
2 mn X mn k=0
k
p2k q 2mn−2k
(4)
1 1 + O((pqmn)−1 ) , = √ 2 πpqmn where the last line is proved by standard methods. The first claim now follows from the formulas for ProbBp (S = s ∧ T = t) and ProbIp (S = s ∧ T = t). The second claim is proved in the same manner. Lemma 2. If pqmn → ∞, then V (p) = 1 − o e−pqmn .
Proof. Kp (p′ ) is a normal density with mean p and variance pq/(2mn), so we just need to apply standard normal tail bounds to the definition of V (p). The next lemma demonstrates how statistics of variables in Bp can be converted into statistics in Vp . Lemma 3 ([27]). Let X be a random variable on Im,n . Then Z 1 −1 Kp (p′ ) EBp′ (X) dp′, EVp (X) = V (p) Z0 1 VarVp (X) = V (p)−1 Kp (p′ ) VarBp′ (X) + (EVp (X) − EBp′ (X))2 dp′ . 0
9
1.5
Enumerative background
Consider positive integers m, n and real variable x ∈ (0, 1). (As mentioned in Section 1.2, these variables are actually functions of a background index ℓ.) For constants a, ε > 0, we say that (m, n, x) is (a, ε)-acceptable if m, n → ∞ with m = o(n1+ε ), n = o(m1+ε ), and 5m 5n (1 − 2x)2 1+ < a log n. + 4x(1 − x) 6n 6m Note that (5) implies x(1 − x) = Ω (log n)−1 .
(5)
For ε > 0, a vector (x1 , x2 , . . . , xN ) will be called ε-regular if N 1 X xi − xj = O(N 1/2+ε ) N j=1
Pn P uniformly for i = 1, . . . , N. We say that (s, t) is ε-regular if m j=1 tj and s, t i=1 si = are both ε-regular. P P Pn Finally, define λm (t) = (mn)−1 nj=1 tj . If m i=1 si = j=1 tj , the common value of λn (s) and λm (t) will be denoted λ. Note that λ is the value in [0, 1] that gives the density of a bipartite graph with degrees (s, t), relative to Km,n . The bases for our analysis are the following enumerative results of Canfield, Greenhill and McKay [6, 9]. Also see Barvinok and Hartigan [3] for an overlapping result. Theorem 4. Let a, b > 0 be constants such that a + b < 12 . Then there is a constant ε0 = ε0 (a, b) > 0 such that the following is true for any fixed ε with 0 < ε ≤ ε0 . If (s, t) is ε-regular, then −1 Y n m Y m n mn G(s, t) = si j=1 tj λmn i=1 Pn Pm 2 2 (s − λn) i j=1 (tj − λm) 1 −b i=1 × exp − 2 1 − 1− + O(n ) . λ(1 − λ)mn λ(1 − λ)mn Moreover, if m = n, then n −1 Y 2 n n −n n−1 Y n−1 ~ G(s, t) = tj λn2 si j=1 i=1 Pn Pn 2 2 (s − λn) i j=1 (tj − λn) 1 i=1 1− × exp − 2 1 − λ(1 − λ)n2 λ(1 − λ)n2 Pn −b i=1 (si − λn)(ti − λn) + O(n ) . − λ(1 − λ)n2 10
1.6
The main theorems
We now state the theorems that are the main contribution of this paper. Their proofs will be given in Section 3, after some preliminary lemmas are given in Section 2. Theorem 5. Let constants a, b > 0 satisfy a + b < 21 . Then there is a constant ε = ε(a, b) > 0 such that the following holds. Let (D, D ′) be a pair of probability spaces on Im,n in one of the following cases. (a) (m, n, p) is (a, ε)-acceptable and (D, D ′) = (Gp , Vp ),
~ p ), (b) m = n, (n, n, p) is (a, ε)-acceptable and (D, D ′) = (G~p , V (c) (m, n, k/mn) is (a, ε)-acceptable and (D, D ′) = (Gk , Bk ),
~k ), (d) m = n, (n, n, k/n2 ) is (a, ε)-acceptable and (D, D ′) = (G~k , B Then there is an event B = B(D) ⊆ Im,n such that ProbD (B) = o˜(1), and uniformly for (s, t) ∈ Im,n \ B, ProbD (S = s ∧ T = t) = 1 + O(n−b) ProbD′ (S = s ∧ T = t). Moreover, let X : Im,n → R be a random variable and let E ⊆ Im,n be an event. Then, ProbD (E) = 1 + O(n−b ) ProbD′ (E) + o˜(1),
ED (X) = ED′ (X) + O(n−b) ED′ (|X|) + o˜(1) max |X|, (s,t)∈Im,n VarD (X) = 1 + O(n−b ) VarD′ (X) + o˜(1) max X 2 . (s,t)∈Im,n
Corollary 6. Let E ⊆ Im,n be an event. Then, under the conditions of Theorem 5, if ProbBp (E) → 0 then p ProbGp (E) = o˜(1) + o(1) ProbBp (E) ,
so in particular ProbGp (E) → 0. Similarly, if m = n and ProbB~p (E) → 0 then p ProbG~p (E) = o˜(1) + o(1) ProbB~p (E) , so in particular ProbG~p (E) → 0.
Theorem 7. Let constants a, b > 0 satisfy a + b < 21 . Then there is a constant ε = ε(a, b) > 0 such that the following holds whenever (m, n, λm (t)) is (a, ε)-acceptable and t is ε-regular. Let (D, D ′) be a pair of probability spaces on Inm in one of the following cases. (a) (D, D ′) = (Gt , Bt ), 11
(b) m = n and (D, D ′) = (G~t , B~t ). Then there is an event B = B(D) ⊆ Inm such that ProbD (B) = o˜(1), and uniformly for s ∈ Inm \ B, ProbD (S = s) = 1 + O(n−b) ProbD′ (S = s). Moreover, let X : Inm → R be a random variable and let E ⊆ Inm be an event. Then, ProbD (E) = 1 + O(n−b) ProbD′ (E) + o˜(1), |X|, ED (X) = ED′ (X) + O(n−b) ED′ (|X|) + o˜(1) max m s∈In X 2. VarD (X) = 1 + O(n−b) VarD′ (X) + o˜(1) max m s∈In
2
Properties of likely degree sequences
Our first task will be to investigate the bulk behaviour of our various probability spaces, in order to identify some behaviour that has probability o˜(1). We will apply a few concentration inequalities, which we now give. Theorem 8 ([25]). Let X = (X1 , X2 , . . . , XN ) be a family of independent random variables, with Xi taking values in a set Ai for each i. Suppose that for each j the function QN Q ′ ′ f : N i=1 Ai differ only in the i=1 Ai → R satisfies |f (x) − f (x )| ≤ cj whenever x, x ∈ j-th component. Then, for any z, P 2 Prob f (X) − E(f (X)) ≥ z ≤ 2 exp −2z 2 / N c i=1 i . Corollary 9. Let X = (X1 , X2 , . . . , XN ) be a family of independent real random variables P such that |Xi − E(Xi )| ≤ ci for each i. Define X = N i=1 Xi . Then, for any z, P 2 Prob |X − E(X)| ≥ z ≤ 2 exp − 21 z 2 / N i=1 ci Another consequence of Theorem 8 is the following.
Theorem 10. Let A1 , . . . , AN be finite sets, and let a1 , . . . , aN be integers such that 0 ≤ ai ≤ |Ai | for each i. Let Aaii denote the uniform probability space of ai -element subsets Q Ai → R satisfies |f (x) − f (x′ )| ≤ cj of Ai . Suppose that for each j the function f : N i=1 ai Q Ai whenever x, x′ ∈ N are the same except that their j-th components xj , x′j have i=1 ai |xj ∩ x′j | = aj − 1 (i.e., the aj -element subsets xj , x′j are minimally different). If X = (X1 , . . . , XN ) is a family of independent set-valued random variables with distributions A1 , . . . , AaNN , then for any z, a1 −2z 2 . Prob f (X) − E(f (X)) ≥ z ≤ 2 exp PN 2 i=1 ci min{ai , |Ai | − ai } 12
Proof. We start by reminding the reader of a classical algorithm called “reservoir sam(i) (i) pling”, attributed by Knuth to Alan G. Waterman [23, p. 144]. Let Yai +1 , . . . , Y|Ai| (i) be independent random variables, where Yj has the discrete uniform distribution on {1, 2, . . . , j}. Now suppose Ai = {w1 , . . . , w|Ai| }. Execute the following algorithm: For j = 1, . . . , ai set xj := wj ; (i) For j = ai + 1, . . . , |Ai |, if Yj ≤ ai then set xY (i) := wj . j
(i)
(i)
Define Xi = Xi (Yai +1 , . . . , Y|Ai| ) to be the value of {x1 , . . . , xai } when the algorithm finishes. The raison d’ˆetre of the algorithm, which is easy to check, is that Xi has distribution Ai ; i.e., it is uniform. It is also easy to check that the maximum change to Xi resulting ai (i)
from a change in a single Yj
is that one element is replaced by another.
Therefore, we can apply Theorem 8 if we consider f (X) as a function of all the in(i) dependent variables {Yj }. If ai < |Ai |/2, we can represent Xi by its complement; this justifies the term min{ai , |Ai | − ai } in the theorem statement. We next apply these concentration inequalities to show that certain events are very likely in our probability spaces. Theorem 11. The following are true for sufficiently small ε > 0. (a) Suppose that (m, n, p) and (m, n, k/mn) are (a, ε)-acceptable. Then ProbD (S, T) is ε-regular = 1 − o˜(1) for D being any of Gp , Gk , Ip , Bp , Bk , or Vp . The same is true for m = n when D ~ p , B~p , B~k , or V ~ p. is any of G~p , G~k , I n (b) If t ∈ Im is ε-regular, and (m, n, λm (t)) is (a, ε)-acceptable, then
ProbD S is ε-regular = 1 − o˜(1).
for D being Gt or Bt . The same is true for m = n when D is either of G~t or B~t . Proof. By symmetry, we need only show that S is almost always ε-regular. In the case that D is Gp or Ip , each Si has the binomial distribution Bin(n, p), and K has the distribution Bin(nm, p). Therefore, by Corollary 9, ProbD |Si − pn| ≥ n1/2+ε/2 = o˜(1), ProbD |Λ − p| ≥ n−1+2ε = o˜(1), 13
i = 1, . . . , m, (6)
from which it follows that ProbD S is ε-regular = 1 − o˜(1).
The cases that D is Gk , Bp , or Bk follow, since these are the same as slices of Gp or Ip of size n−O(1) , using p = k/mn. Also, the distribution of S in Bt is the same as in Bk for P k = nj=1 tj , so that case follows too.
For D = Gt , note that each Si is the sum of independent variables X1 , . . . , Xn , where Xj is a Bernoulli random variable with mean tj /m. The theorem thus follows using the same argument as we used for Gp . Finally consider D = Vp . Taking X to be the indicator of the event that S is not ε-regular, Lemmas 2–3 give Z 1 EVp (X) = O(1) Kp (p′ ) EBp′ (X) dp′ 0 Z p−n−1+ε Z p+n−1+ε Z 1 = O(1) + + Kp (p′ ) EBp′ (X) dp′. 0
p−n−1+ε
p+n−1+ε
The first and third integrals are o˜(1) since the tails of Kp (p′ ) are small (recall that it is a normal density with mean p and variance O((mn)−1 )), while the second integral is o˜(1) by the present theorem in the case D = Bp′ . (Note that if (m, n, p) is (a, ε)-acceptable, then all p′ ∈ [p − n−1+ε , p + n−1+ε ] are (a′ , ε) for slightly different a′ .) For the digraph models, the proofs are essentially the same.
Theorem 12. The following are true for sufficiently small ε > 0. (a) Suppose that (m, n, p) and (m, n, k/mn) are (a, ε)-acceptable. Then m X 2 −1/2+2ε ProbD (Si − nΛ) = 1 + O(n ) Λ(1 − Λ)mn = 1 − o˜(1), ProbD
i=1 n X j=1
(Tj − mΛ)2 = 1 + O(m−1/2+2ε ) Λ(1 − Λ)mn = 1 − o˜(1),
(7) (8)
when D is Gp or Gk . When m = n, the same bounds hold when D is G~p or G~k . n (b) If t ∈ Im is ε-regular, and (m, n, λm (t)) is (a, ε)-acceptable, then (7) holds when D is Gt , and when m = n and D is G~t .
(c) If m = n, (n, n, p), (n, n, k/n2 ) and (n, n, λn (t)) are (a, ε)-acceptable, and t ∈ Inn is ε-regular, then n X ProbD (Si − nΛ)(ti − mΛ) = O(n−1/2+2ε )Λ(1 − Λ)n2 = 1 − o˜(1) i=1
14
when D is G~p , G~k or G~t . P 2 Proof. Write R = m i=1 (Si − nΛ) . For i = 1, . . . , m and j = 1, . . . , n, let Xij be the indicator for an edge from ui to vj . Then some rearrangement of terms yields R=
m n 1 X X ∆ii′ jj ′ , 2m ′ ′
(9)
i,i =1 j,j =1
where ∆ii′ ii′ = (Xij − Xi′ j )(Xij ′ − Xi′ j ′ ). When D is either Gp or Gt , Xij is independent of Xi′ j ′ if j 6= j ′ , and ED (Xij ) is independent of i. This shows that ED (∆ii′ jj ′ ) = 0 for j 6= j ′ , leaving us with ED (R) =
m n 1 XX ProbD (Xij 6= Xi′ j ). 2m ′ j=1 i,i =1
This gives EGp (R) = pq(m − 1)n, n 1 X tj (m − tj ). EGt (R) = m j=1 P 2 1+2ε Now define R∗ = m }. If Sj is changed by 1 for some j, which i=1 min{(Si − nΛ) , m 2 1+2ε changes Λ by 1/mn, then min{(Si − nΛ) , m } changes by O(m1/2+ε ) for i = j and by O(m−1/2+ε ) for i 6= j. Consequently, R∗ changes by O(m1/2+ε ). Applying Theorem 8, we find that ProbD |R∗ − ED (R∗ )| ≥ 21 m1+ε n1/2+ε/2 = o˜(1)
for D = Gp . It also holds for D = Gt , using Theorem 10 in the same way.
Now Theorem 11 shows that ProbD (R 6= R∗ ) = o˜(1), which implies that ED (R∗ ) = ED (R) + o˜(1). Therefore we can argue ProbD |R− ED (R)| ≥ m1+ε n1/2+ε/2 ≤ ProbD (R 6= R∗ ) + ProbD |R∗ − ED (R)| ≥ m1+ε n1/2+ε/2 ≤ o˜(1) + ProbD |R∗ − ED (R∗ )| ≥ m1+ε n1/2+ε/2 + o˜(1) = o˜(1).
P We also have that Λ is fixed at the value λm (t) = (mn)−1 nj=1 tj in Gt and that ProbGp |Λ − p| ≥ n−1+2ε = o˜(1),
by (6). From these bounds, inequality (7) follows for Gp and Gt , and (8) follows for Gp by symmetry. By choosing p = k/mn and noting that Gk is a slice of size n−O(1) of Gp , the theorem is proved for Gk too. 15
For D = G~p , G~k , G~t , the proofs of (7) and (8) follow the same pattern. Since (9) still holds, we can note that EG~p (∆ii′ jj ′ ) = EGp (∆ii′ jj ′ ) and EG~t (∆ii′ jj ′ ) = EGt (∆ii′ jj ′ ) unless {j, j ′ } ⊆ {i, i′ }, to infer that EG~p (R) = EGp (R) + O(n) and EG~t (R) = EGt (R) + O(n). This is enough to ensure that the rest of the proof continues in the same way. (For the record, EG~p (R) = pq(n − 1)2 .) We now prove part (c). Take D = G~t first, with t being ε-regular and (n, n, λn (t)) being (a, ε)-acceptable. We have
EG~t (Si ) =
X j6=i
tj λn2 ti = − , n−1 n−1 n−1
from which it follows that Pn n 2 X j=1 (tj − λn) = O(n1+2ε ). EG~t (Si − λn)(ti − λn) = − n−1 i=1 Now we can apply Theorem 10 to conclude that (c) holds. In the case of D = G~p , Theorem 11 says that T is ε-regular with probability 1 − o˜(1), so (c) holds in that case too. Finally, G~k is a substantial slice of G~p if p = k/n2 , so (c) holds for G~k as well.
3
Proofs of the main theorems
In this section we will prove the theorems and corollaries in Section 1.6. We first consider Gp . Suppose that a, b > 0 are constants with a + b < 21 , and that (m, n, p) is (a, ε)-acceptable. According to Theorems 4, 11 and 12, and (6), there is an / B, event B ⊆ Im,n such that ProbGp (B) = o˜(1) and, for (s, t) ∈ |K − pmn| ≤ mn2ε ,
(10) −1 Y n m mn n Y m k nm−k −b , exp O(n ) ProbGp (S = s ∧ T = t) = p q si j=1 tj k i=1 n m Y Y p m n 2k 2mn−2k =p q 2πpqmn si j=1 tj i=1 (k − pmn)2 −b × exp + O(n ) (11) 2pqmn P P for i si = j tj = k, where the last step follows by Stirling’s formula and, as always, we are assuming that ε is sufficiently small. 16
We wish to show that (11) closely matches the probability in Vp . Define P (p, s, t) = ProbBp (S = s ∧ T = t). By the definition of Vp , we have ProbVp (S = s ∧ T = t) = V (p)
Z
−1
1
Kp (p′ )P (p′, s, t) dp′.
0
By Section 1.4 item 2, we have 2k Pn Pm 2mn−2k T S = ProbIp 1 − p′ p′ P (p′, s, t) j i j=1 i=1 Pm Pn = . P (p, s, t) p 1−p ProbIp′ i=1 Si = j=1 Tj
(12)
We will divide the integral into three parts. Define Jp = [p − n−1+3ε , p + n−1+3ε ]. By Lemma 1 and (10), for p′ ∈ Jp and (s, t) ∈ / B, we have P (p′, s, t) mn ′ 2(k − pmn) ′ 2 −1/2 = exp (p − p) − (p − p) + O(n ) , (13) P (p, s, t) pq pq which gives Z
′
′
′
Kp (p )P (p , s, t) dp = 2
−1/2
Jp
(k − pmn)2 −1/2 P (p, s, t) exp + O(n ) . 2pqmn
2mn−2k To bound the integral outside Jp , note that (p′ /p)2k (1 − p′ )/(1 − p) is increasing ′ −1+3ε −1+3ε for p ≤ p − n and decreasing for p ≥ p + n . Also, since the mean square of a set of numbers is at least as large as the square of their mean, we can infer from (4) that Pn Pm −1 ProbIp′ for all p′ . Since mn o˜(1) = o˜(1), we obtain j=1 Tj ≥ (mn + 1) i=1 Si = from (12) that Z Kp (p′ )P (p′ , s, t) dp′ = o˜(1)P (p, s, t).
[0,1]\Jp
Recalling Lemma 2, we conclude that Z 1 −1 V (p) Kp (p′ )P (p′ , s, t) dp′ 0
=2
−1/2
(k − pmn)2 −1/2 + O(n ) , P (p, s, t) exp 2pqmn
which matches (11) when the value of P (p, s, t) given by Lemma 1 is substituted. This completes the proof of the first claim of Theorem 5(a). The next two claims follow on summing the first claim over all (s, t). For the variance, we can apply the formula for the
17
expectation to argue VarGp (X) = min EGp (X − µ)2 µ∈R
= min o˜(1) max (X − µ)2 + (1 + O(n−b )) EVp (X − µ)2 µ∈R (s,t) = min o˜(1) max X 2 + (1 + O(n−b )) EVp (X − µ)2 µ∈R
(s,t)
= o˜(1) max X 2 + (1 + O(n−b )) min EVp (X − µ)2 µ∈R
(s,t)
2
−b
= o˜(1) max X + (1 + O(n )) VarVp (X). (s,t)
For the third line we have used the obvious fact that the minimum in the first line occurs somewhere in the interval [min X, max X]. The proof of Theorem 5(b) is the same. To prove Theorem 5(c), note that according to Theorems 4, 11 and 12, there is an event B ⊆ Im,n such that ProbGk (B) = o˜(1) and, for (s, t) ∈ / B, m Y n mn −2 Y m n −b , ProbGk (S = s ∧ T = t) = exp O(n ) t s k j i j=1 i=1 which matches ProbBk (S = s ∧ T = t) up to the error term. Similarly for Theorem 5(d). Theorem 7 follows from a similar argument, on noting that the ε-regularity of t implies n X j=1
(Tj − λm)2 ≤ n2+2ε ≤ m4ε λ(1 − λ)mn.
Finally, we prove Corollary 6 for D = Gp , which is representative of the two cases. In view of Theorem 5, it will suffice to prove that p ProbVp (E) ≤ o˜(1) + o(1) ProbBp (E) (14) if ProbBp (E) → 0. Define o n p ε 1 y = max n , − log(ProbBp (E)) − 2 log log(ProbBp (E)) and
√ Eˆ = (s, t) ∈ E : |K − pmn| ≤ y pqmn .
By a suitable normal approximation of the binomial distribution, such as [26, Thm. 3], ˆ = o˜(1) + O(e−y2 /2 /y). ˆ = O(e−y2 /2 /y), so by Theorem 5, ProbVp (E \ E) ProbGp (E \ E) Also note that Z 2 Kp (p′ ) dp′ = O(e−y /2 /y). √ |p′ −p|>y
pq/2mn
18
Therefore, since V (p)−1 = 1 + o˜(1) by Lemma 2, Z −y 2 /2 /y) + ProbVp (E) = o˜(1) + O(e
|p′ −p|≤y
√
pq/2mn
ˆ dp. Kp (p′ ) ProbBp′ (E)
p √ Now note that, by (13), for |p′ − p| ≤ y pq/2mn and |k − pmn| ≤ y pqmn we have ProbBp′ (S = s ∧ T = t)
ProbBp (S = s ∧ T = t) mn ′ 2(k − pmn) ′ 2 −1/2 (p − p) − (p − p) + O(n ) ≤ exp pq pq ≤ (1 + O(n−1/2 ))ey
2 /2
and so ˆ ˆ ≤ (1 + O(n−1/2 ))ey2 /2 ProbBp (E). ProbBp′ (E) Since
R
Kp (p′ ) dp < 1, we have proved that ProbVp (E) ≤ o˜(1) + O(e−y
2 /2
/y) + (1 + o(1))ey
2 /2
ProbBp (E),
which gives (14) when the value of y is substituted.
4
Concluding remarks
A theorem similar to Theorem 4 holds also in the sparse domain. This was shown by P Greenhill, McKay and Wang in the case that (maxi si )(maxj tj ) = o ( i si )2/3 [10]. That theorem can probably be used to develop a similar theory of degree sequences in that domain. However the lack of a precise enumeration in the gap between the sparse domain and the dense domain of Theorem 4 currently thwarts a theory which spans both the sparse and dense domains.
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