On the number of cycles in a random non-equiprobable graph

We consider a non-equiprobable random graph Gn,# with n labelled vertices and N edges which is obtained by N independent trials. In each trial one edge is drawn: it connects vertices i and j with the probability 2ρ,·ρ;·, or it forms a loop at vertex i with the probability p? ; i, j = 1,. , n, pi,. , pn > 0, Pi + . + Pn = 1. The main result of the paper is the following assertion. Assume that p,· = α,·/η where a,· = α,·(η), 0 < ε < a,· < Ε < oo, i = 1,. , η, ε and Ε are constants, and the limit 1 n 2 l · · * · V~^ 2 a = hm -2_X „_>oon^ exists. Then, if n and N tend to infinity so that 2N/n -» λ and λα < 1, the distribution of the number of cycles in the graph converges to the Poisson distribution with parameter Λ = —| ln(l λα). 1. THE MAIN STATEMENTS Among a number of papers devoted to random graphs (surveys of the results can be found in [1-3]) only a few publications deal with random graphs which take their values with unequal probabilities [4-6]. However, the stability of the results obtained under the condition of strict equiprobability must be studied without doubts. In a number of cases it is useful to know how the results are modified if the condition of equiprobability is omitted and what level of deviations of the probabilities can be considered as a negligible one which preserves the results valid for the equiprobable case. In the present paper an important characteristic of a random graph, namely the number of cycles, is investigated. For a number of models of equiprobable graphs under the conditions on parameters such that with the probability tending to one the graph has no giant component, the number of cycles has in limit the Poisson distribution. In [7] it is pointed out that for a random equiprobable graph with n vertices and TV edges one can show by the method of moments that the distribution of the number of cycles an,jv converges, as η, Ν -> oo, 2N/n -> λ, Ο < λ < 1, to the Poisson distribution with parameter -|ln(l λ) λ/2 λ/4. The analogous result for a random graph with n vertices, in which any edge exists independently of the others with the probability ρ = λ/η, Ο < λ < 1, is proved in [8]. In the present paper we consider a non-equiprobable graph Gn^ with n vertices labelled with the numbers 1,2,. , n and with TV edges obtained by the following process. One carries out N independent trials in each of which one edge is drawn. The edge connects two different vertices or forms a loop; the vertices with labels i and j are connected with the probability 2pipj, and the loop at vertex i is formed with the *UDC 519.12. Originally published in Diskretnaya Matematika (1990) 2, No. 3, 137-145 (in Russian). Translated by A. V. Kolchin. 110 K F. Kolchin and K /. Khokhlov probability p; ij = l,. ,n, pi,. ,pn > 0, pi + . + pn = 1. Thus, after TV trials we have a realization of the random graph Gn,N, which, generally speaking, has loops and multiple edges. The main result of the paper is the following assertion. Theorem 1. Assume that pt = α,/π where a, = a,(n), 0 < ε < a,· < jE < oo, i = 1,. , η, ε and Ε are constants, and the limit a 2 = lim-Y> ~*°° n i^i exists. Then, if η and Ν tend to infinity so that 2N/n —> λ, λα < I , the distribution of the number of cycles an^ in the graph GH,N converges to the Poisson distribution with parameter Λ = — |ln(l λα). While proving the theorem, the limit distribution of the random variable ar, which is equal to the number of the cycles of length r, and the joint limit distribution of an,. , otTs are obtained. Theorem 2. Under the conditions of Theorem 1, without the requirement λα < 1, the distribution of the random variable ar for any fixed r tends to the Poisson distribution with parameter \T = Aa/(2r). Theorem 3. Under the conditions of Theorem 1, without the requirement λα < 1, the joint distribution of ari,. , ar, for any fixed 1 < r\ < . < rs converges to the distribution of s independent random variables which have the Poisson distributions with parameters A r i , . , \Te, respectively. The proof will be performed by the method of moments. As one realizes this laborconsuming approach, the natural desire to omit bulky calculations arises. These circumstances, probably, explain the fact that a complete ρτοοί earned out by the method of moments has never been published even for the equiprobable case. Therefore we give detailed proofs in the next section. 2. PROOFS OF THE MAIN STATEMENTS Denote by ar the number of cycles without self-intersections of length r, r > 3, in the random graph Gn,w· Let us give a more precise definition of the variable. For r distinct vertices n,. ,ir assume &! ,. ,,·,. = 1 if in GH,N there exists a cycle composed of these r vertices and containing exactly r edges of the graph GU,N (such cycles are called the cycles without self-intersections); in other cases assume &,,. ,,,. = 0. Then «r = Σ &I. -. V, (1) «l,. M*r where the summation is taken over all CN distinct unordered sets of r distinct indices. In the complete graph with vertices zi , . ,z r there exist (r 1)1/2 distinct cycles containing exactly r edges. We label these cycles in an arbitrary order with the numbers On the number of cycles in a random &aph 111 j = 1, . , (r — l)! /2 and represent the random variable £,lt.,tr as the sum of indicators: (r-l)! /2 &,-,V = Σ #L,V> (2) j=i where £t^. Mt= 1 if the jth (under the labelling chosen) cycle exists in <3n,yv, and f£L«v = 0 otherwise. In the paper the random variable is investigated, where the variables ar are defined by (1) for r > 3, a\ is the number of loops, and a2 is the number of pairs of parallel edges in Gn^. Each cycle in the graph GUiN may be considered as the set of edges which compose this cycle; therefore, the following assertion is needed for evaluating such probabilities as P{£t^. |t-r = 1}. Let Vr = {(ii,ji),. ,(ir,jr)} be the set of r distinct pairs of vertices in the graph Gn,N where u ^ j k , k l,. ,r. Denote by P(Vr) the probability of the event that all the edges from Vr exist in Gn,N· Lemma 1. If n,N — > oo, 2N/n — > λ, 0 < λ < oo, 0 < ε < α, < Ε < οο, i l, . , η, then for arbitrary fixed ε, Ε, r P(VT) = ̂ α,. ,α;, . a,vair (l + O Q) (3) uniformly with respect to αϊ, . , αη ΖΑΖ f/ze mentioned bounds and to all sets Vr . Moreover, for any δ > 0 there exists a constant c such that for all r and n