Hypercycles in a random hypergraph

We introduce the notion of a hypercycle in a hypergraph which is defined by a Τ χ Ν (0, l)-matrix A. The maximum number of independent hypercycles s(A) is connected with the rank r (A) of the matrix A by the equality r(A) + s(A) T. We prove that for a regular random hypergraph with N vertices and T edges, whose each edge contains not more than r vertices, the total number of hypercycles S(A) = 2*() 1 has the following threshold property as Ν, Τ -» oo, Ν/Τ -* α: there exists a constant ar such that MS(A) —> 0 for α < ar, and MS(A) -» oo for a > ar. I. MAIN RESULTS We consider a hypergraph G A which is defined by a Τ χ Ν matrix A = \\atj\\ in GF(2). The set of vertices of the hypergraph GA is the set {1,..., TV} of the numbers of columns of the matrix, and the set of enumerated hyperedges is the set {61,..., &r}, where bt = {j: atj = 1}, t = Ι,.,.,Τ. Thus, there exists a correspondence between a row at = (ati, · , CUN) and the hyperedge bt, t = 1,..., T. Note that the empty edge corresponds to a row consisting of zeros. The multiplicity of a vertex j in the set of hyperedges B = {6^,...,b tm} is the number of hyperedges in B which contain this vertex. A set of hyperedges B = {btl,..., btm} is a hypercycle if every vertex of the hypergraph G A has an even multiplicity in B, in other words, if the sum by coordinates of rows atl + ... + atm in GF(2) is equal to the zero vector. If every row of the matrix A consists of exactly two units, then the hypergraph G A is an ordinary graph (parallel edges may be encountered), and a hypercycle is an ordinary cycle or a union of cycles. The set of numbers of hyperedges which form a hypercycle is called in [1] the critical set of rows of the matrix A. The union of two sets of hyperedges BI and B2 is the set BI Δ Β2 containing those, and only those, hyperedges which are contained in one of the sets BI and B2 and are not contained in the both sets. Let ει , . . . ,ε s take values 0 and 1. Hypercycles £?i,..., £s are independent, if ειΒιΔε2Β2Δ...Δε3Β3 = 0 iff ει = ... = εβ = 0. We denote by s(A) the maximum number of independent hypercycles in the hypergraph GA, and by r(A), the rank of the matrix A. In Section 2 it will be proved that r(A) + s(A) = T. (1) *UDC 519.2. Originally published in Diskretnoya Malematika (1991) 3, No. 3, 102-108 (in Russian). Translated by V. F. Kolchin. 564 G.V. Balaton, K F. Kolchin and V. L Khokhlov This relation allows us to apply hypercycles of G A to the investigations of the rank of the matrix A in GF(2). In the present paper, a random regular hypergraph G> is considered, whose matrix A = Ar = ||α*/|| has the following structure. The elements of the matrix atj, t = 1,..., Τ, j = 1,..., Ν, are random variables; the rows of the matrix are independent In each row r units are consecutively allocated; every unit, independently of one another, falls into each of Ν positions with the probability l/N, and the element atj is equal to one if an odd number of units is fallen into position j. Therefore, there are not more than r units in each row. Similar models of random matrices with a fixed number of units in each row were considered in [2, 3]. For such regular hypergraphs the following threshold property holds: if N,T —> oo in such a way that T/N —» a, then an abrupt change in the behaviour of the rank of the matrix Ar occurs while the parameter α is passing a critical value ar. We denote by S(Ar) the total number of hypercycles in GA, which is equal to 2-1. Theorem 1. Let r > 3 be fixed, T, N -> oo in such a way that Τ/Ν -» a. Then there exists a constant ar such that MS(Ar) —> Ofor a < ar and MS(Ar) -» oo for a > ar. The constant ar is the first component of the vector which is the unique solution of the system of equations e-*coshA(——V = 1, \ar-xj χ (ar-x\l λ V χ λ tanh λ = χ with respect to the variables α, χ, λ. The numerical solving of the system of equations gives us the following values of the critical constants: a3 = 0.8894..., a4 = 0.9671..., a5 = 0.9891..., a6 = 0.9969..., a? = 0.9986..., a8 = 0.9995... These values were first obtained by Balakin under slightly different assumptions about the matrix Ar. Expanding the solution of the system into powers of e ~, one can obtain the approximate formula r r Γ to 2 In2 which gives values close to the precise ones for r > 4. 2. CONNECTION WITH THE RANK OF MATRIX Let us prove equality (1). In GF(2) we consider the homogeneous system of equations A'Y = 0 (2) with the transposed matrix. The following one-to-one correspondence between solutions of system (2) and hypercycles of the hypergraph GA exists: the solution Hypercycles in a random hypergraph 565 ^«i,~.,<m = {2/ι>···>3/τ} whose components yt,,...,ytm are equal to one and the other components are equal to zero corresponds to the hypercycle B = {6tl,... ,6tm}. Linear independence of solutions corresponds to independence of hypercycles. Therefore, the maximum number of independent hypercycles is equal to the maximum number of linearly independent solutions of system (2), which, evidently, is equal to Τ — r(A). Thus, equality (1) is proved. 3. THE AVERAGE NUMBER OF HYPERCYCLES AND CONNECTION WITH AN ALLOCATION PROBLEM The total number of hypercycles S(Ar) in the hypergraph GT with the matrix Ar can be represented in the form of a sum of indicators. Put 6lf...,tm = 1 if the hypercycle Β = {δ<ι,.·. Am} in GT exists, and 6i,...,«m = 0 otherwise. It is clear that P{6lv..,<m = 1} does not depend on the indices *i,...,<m. Indeed, from the definition of the random hypergraph GT it follows that &i,...,*m = 1 if and only if the number of units in each column of the submatrix consisting of the rows with the numbers <i, . . . , tm is even. The number of units in TV columns of any m rows before modulo 2 reducing the number of units fallen has the polynomial distribution with rra trials and N outcomes. Denote by 771 (n, N), . . . , ηΝ(η, Ν) the occupancies of cells in the equiprobable scheme of allocating n particles into Ν cells. In these notations, the number of units in columns of any m rows before reducing modulo 2 has the distribution which coincides with the distribution of the variables η\ (rra, TV), . . . , 7/^(rra, TV). Therefore Ptfti ..... tm = 1} = P{r7i(rm,7V) <E £, . . . , ̂ (rm, N) G E}, where E is the set of even numbers, and the average number of hypercycles in GT can be written out in the following form: MS(A r )= £C?Ps(rra,7V), (3) m = l where ?E(rrn, N) = P{7n(rra, N) G E, . . . , ηΝ(ττη, Ν) (Ξ Ε}. Thus, for estimating MS(Ar) we ought to know the asymptotic behaviour of PE(rra,7V). 4. A LOCAL THEOREM We consider a more general case and obtain the asymptotic behaviour of the probabilities Pfl(n,7V) = Ρ{ηι(η,Ν) G Κ,...,ηΝ(η,Ν) G R}, where R is a set of non-negative integers (see [4]). The joint distribution of random variables 771 (n, TV), . . . , 77^(71, N) can be expressed as a conditional distribution of independent random variables £1 , . . . , £//, identically distributed by the Poisson law with an arbitrary parameter λ, in the following way (see, for example, [5]): for any non-negative integers ni, . . . , nN, n\ + . . . + nN = n, P{77i(n,7V) = ni,...,777v(n,7V) = nN] = ?{ξι = ni,...,£v = nN \ & + ... + £N = n}. 566 G. V. Baialan, V. F. Kolchin and V. I. Khokhlov Therefore JV) = Ρ{ηι(η,Ν)€Η,...,ηΝ(η,Ν)£Ε} " s " 6fl} Let us introduce independent identically distributed random variables fi,.·., with the distribution = k} = P{6 = k 1 6 G }, k = 0,1,... It is not difficult to see that P{6 + ... + 6v = n |6 e A,..., and therefore ΡΛ(», ΛΟ = (P{6 e Α } Γ ρ + · · Γ . (4) "{ξι + ... + ςΛτ = ") We introduce the notation χ = n/./V and choose the parameter λ of the Poisson distribution in such a way that