Connected components in Erdős-Rényi random graphs

Transcription

1 Coected compoets i Erdős-Réyi radom graphs Kare Guderso 26 November 3 December 203 These otes cotai the material preseted i class o the compoet structure of G,p. The particular focus is whe is G,p likely to have just oe compoet, ad i the case that G,p is ot coected, how big are the largest compoets likely to be? There are may resources o these topics, of varyig degrees of difficulty (e.g. [, 2, 4, 7]). Coectedess The first questio we shall address is, for which values of p is G,p either very likely to be coected, or very likely to be discoected. Oe of the simplest reasos for a graph to be discoected is that it cotais a isolated vertex. I this sectio, it is show that, i fact, isolated vertices are the mai obstructio to coectedess i G,p. For ay k, let X k deote the umber of coected compoets of k vertices i G,p. The, X is the umber of isolated vertices i G,p ad E(X ) = ( p). Set µ = µ (, p) = ( p). As a rough estimate, µ e p = e log p ad so whe p is much larger tha log, oe might expect very few isolated vertices, o average, ad whe p is much smaller tha log, oe might expect may isolated vertices, o average. To make this rough estimate precise, let α() be a fuctio such that α() ad α() log log. Note that, i much of the literature, ω() is used to deote a fuctio growig arbitrarily slowly ad tedig to ifiity, but α is used here to avoid cofusio with a previous use of ω. First cosider log + α() p = p() = = log ( + o()). For these values of p, O the other had, for µ = ( p) e p( ) = e log α() e p e α() = o(). p = p() = log α() the the expected umber of isolated vertices is = log ( + o()), µ = ( p) ( p) e (p+p2 ) = e α() e p2. The fial expressio teds to ifiity sice p 2 = o(). Neither of these estimates for µ is sufficiet, o its ow, to determie whether or ot G,p() is likely to be coected or discoected. I the case that p() < log, the secod momet kare.guderso@bristol.ac.uk

2 Kare Guderso method shall be used to show that with high probability, there is a isolated vertex ad hece G,p is discoected. Coversely, for the case p() > log, it is show below that the expected umber of coected compoets with fewer tha vertices is small ad hece that G,p is likely to be coected. These are each proved separately. Propositio. Let α() 0 be such that α() ad α() log log. Let p() = the G,p() is discoected with high probability. log α(), Proof. Chebyshev s iequality is used to show that P(X = 0) = o(). I order to compute Var(X ), cosider the radom variable X (X ), which is precisely the umber of ordered pairs of distict vertices (v, w) so that both v ad w are isolated. Thus, E(X (X )) = ( )( p) ( p) 2 2 p = µ2 p µ 2 +. (for large eough) Oe ca boud the variace of X by Var(X ) = E(X 2 ) E(X ) 2 By Chebyshev s iequality, ad usig µ, = E(X (X )) + E(X ) E(X ) 2 µ µ µ 2 = µ +. P(X = 0) Var(X ) E(X ) 2 = µ + µ 2 = o(). Therefore, with high probability, there are isolated vertices i G,p ad hece G,p is discoected. To show that for p slightly larger tha log, G,p is coected, compoets of early all sizes are cosidered. I order to compute the expected umber of compoets of size k, oe ca use the fact that every coected compoet cotais a spaig tree ad there are exactly k k 2 differet labelled trees o k vertices. Sice every tree o k vertices has exactly k edges, E(X k ) = P(V coected)( p) k( k) V =k V =k k k 2 p k ( p) k( k) ( ) = k k 2 p k ( p) k( k) k ( e ) k k k 2 p k ( p) k( k) k k e k p k e pk( k). Propositio 2. Let α() 0 be such that α() ad α() log log. Let p() = the G,p() is coected with high probability. log +α(), 2

3 Kare Guderso Proof. Note that, if a graph G is discoected, the there is at least oe coected compoet o at most /2 vertices. ( Therefore, to show that with high probability, G,p is to coected, it /2 ) suffices to show that E k= X k = o(). This sum is broke ito a few parts that are estimated separately. Throughout the proof, it is assumed that is large eough so that p (2 log )/. Note that, as was show previously, E(X ) = µ = o(). Cosider ow the case k log log. Usig p > (log )/ E(X k ) k e k p k e pk( k) k e k ( 2 log Thus, summig over k log log, ) k k log +pk2 e ( ) k 2 log k e k k e (sice pk 2 = o()) ( ) k 2e log = e 2. log k=2 E(X k ) log k=2 ( ) k 2e log 2e log e 2 e 2 2e log = o(). For k > log log, we use the fact that k /2 so that from which, oe ca show that /2 k=log log + E(X k ) k e k p k e pk( k) ( ) k 2 log k e k e pk/2 e k (2 log ) k k/2 log e ( ) k 2e log = e /2, /2 E(X k ) ( ) log log 2e log 2 /2 = o(). /2 Thus, /2 k= E(X k) = o() ad so with high probability, there are o compoets of G,p with fewer tha /2 vertices. Thus, with high probability G,p is coected. Exercise. Recall that a spaig tree of a graph G o vertices is a subgraph of G that is a tree (coected ad acyclic) that cotais all vertices.. For ay, p, compute the expected umber of spaig trees i G,p. (Hit: use Cayley s formula for the umber of trees). 2. Fid a fuctio p() so that with high probability G,p() cotais a isolated vertex ad the expected umber of spaig trees teds to ifiity. 2 Review of brachig processes Brachig processes are a radom model of ifiite trees that ca be used to describe a breadthfirst search i a graph radom. A few basic facts about brachig processes are give here. 3

4 Kare Guderso Defiitio 3. Let ξ be a probability distributio o the o-egative itegers. A probability distributio o the set of rooted ordered trees (fiite ad ifiite) called a brachig process is defied usig ξ as follows: Start with a sigle vertex, called the root. Every vertex has a radom umber of childre each with distributio ξ, idepedetly of all others. Vertices are joied to each of their child vertices by a a edge ad the radom rooted tree is deoted by T ξ. For each 0, let Z deote the umber of vertices i T ξ at distace from the root. The Z 0 = ad for every 0, let ξ, ξ 2,..., ξ Z be idepedet copies of ξ, the Z + ξ + ξ ξ Z. Exercise 2. Show that the sequece (Z ) =0 is a Markov Chai o N = {0,, 2,...}. Oe questio of iterest is whe the tree T ξ is fiite or ifiite. Note that if ξ, the with probability, the tree T ξ is ifiite. O the other had, if P(ξ = 0) > 0, the P(T ξ is fiite) > 0 (Exercise: prove this). I geeral if there exists such that Z = 0, the the process is said to die out or become extict. If, istead, for all 0, Z > 0, the process is said to survive. Defiitio 4. Let for every k 0, set p k = P(ξ = k) ad defie µ = E(ξ) = k=0 kp k [0, ]. Lemma 5. For every 0, E(Z ) = µ. Proof. The proof proceeds by iductio. The base case holds sice E(Z 0 ) =. Assume for some 0 that E(Z ) = µ. E(Z + ) = E(E(Z + Z )) = P(Z = k)e(z + Z = k) = P(Z = k)e(ξ + + ξ k Z = k) = P(Z = k)e(ξ + + ξ k ) = P(Z = k)ke(ξ i ) = E(Z )µ = µ +. Corollary 6. If µ < the P(T ξ survives) = 0. Proof. For every 0, P(T ξ survives) P(Z > 0) = E( {Z>0}) E(Z ) (sice Z Z + ) = µ. Sice was arbitrary ad µ 0, P(T ξ survives) = 0. Oe ca show that if µ, the P(T ξ survives) > 0 directly, but the proof ca also be obtaied by lookig at the aalytic properties of the geeratig fuctio for ξ. Recall that for a radom variable ξ N where for every k, p k = P(ξ = k), the geeratig fuctio for ξ is defied as f(x) = E(x ξ ) = p k x k. k=0 Note that f(x) is a aalytic fuctio for all x C with x <. Example 7. If ξ is give by a Biomial biomial distributio: p k f(x) = ( k) p k ( p) k x k = ( p + px). = ( k) p k ( p) k, the 4

5 Kare Guderso Example 8. Let λ > 0 ad let ξ Po(λ). The geeratig fuctio for ξ is f(x) = k e λ λ k k! x k = e λ( x). Some basic properties of the geeratig fuctio of a distributio ξ are summarized i the followig lemma. Lemma 9.. f(x) ad all derivatives exist ad are o-egative for x [0, ). 2. f(x) is cotiuous, icreasig, covex i [0, ]. 3. f(0) = p 0, f() =. { f () if left derivative f () exists 4. Eξ = otherwise 5. Var(ξ) = { f () + f ()( f ()) if left secod derivative f () exists otherwise Proof. The fuctio f(x) = p k x k is aalytic i x < ad so all derivatives are also aalytic ad obtaied by differetiatig term by term i x <. Sice all coefficiets are positive, the derivatives of f are o-egative for x i [0, ). This proves part. The remaiig parts ca be deduced from part, takig oe-sided limits where ecessary. Usig iteratios, the fuctio f ca also be used to ecode the distributio of the -th level of a brachig process T ξ. For every, defie f Z (x) def = E(x Z ). Lemma 0. Let f(x) be the geeratig fuctio for ξ. For every, f Z (x) = f(f( f(x) )), where this is the -fold iterate of f(x). Proof. The proof is by iductio o. f Z0 (x) = E(x Z0 ) = E(x ) = x. f Z+ (x) = E(x Z+ ) = E(E(x Z+ Z ) = P(Z = k)e(x ξ+ +ξ k ) = P(Z = k)e(x ξ )E(x ξ2 ) E(x ξ k ) = P(Z = k)f(x) k = f Z (f(x)) Defiitio. For ay distributio ξ, deote the extictio probability by p e = p e (ξ) = P(T ξ is fiite). Set ρ = ρ(ξ) = p e, called the survival probability. Theorem 2. Let ξ be a distributio with geeratig fuctio f(x). The extictio probability p e [0, ] is the smallest solutio of the equatio f(x) = x. Proof. Let p be smallest solutio to f(x) = x. To see that p e is a solutio to f(x) = x, ote that Thus, sice the fuctio f is cotiuous, p e = P(T ξ goes extict) = P(Z = 0 for some ) = P( {Z = 0}) = lim P(Z = 0) = lim f () (0) f(p e ) = f( lim f () (0)) = lim f(f () (0)) = lim f (+) (0) = p e To show that p e = p, ote that sice 0 p the f(0) f(p) = p sice f is icreasig. So, f () (0) p. Thus p e = lim f () (0) p ad by miimality of p, p e = p. 5

6 Kare Guderso Corollary 3. If Eξ, ad ξ is ot idetically, the P(extictio) =. If Eξ >, the P(extictio) <. Proof. If Eξ > the f () > ad so there exists h > 0 such that f() f( h) h >. That is, f( h) < h. Apply the itermediate value theorem to f(x) x to obtai x [0, h] such that f(x) = x. So, p e h <. Coversely, if p e <, the sice f(p e ) = p e ad f() =, the covexity of f implies that for every x [p e, ], f(x) x. Thus for h > 0 (small eough), f() f( h) h ad so µ = f (). Two cases: either x 0 (p e, ) such that f(x 0 ) < x 0 ad the for x [x 0, ], ( ) f(x0 ) f(x) ( x) x 0 for x [x 0, ]. Therefore, f() f( h) h f(x0) x 0 > as h 0. Thus f () >. Otherwise f(x) = x for all x (p e, ). The f(x) = x everywhere ad the ξ =. 3 The giat compoet Oe of the, perhaps, most startlig results give by Erdős ad Réyi i their 959 ad 960 papers o radom graphs [5, 6] was that for p = λ/, there is a sudde phase trasitio i the size of the largest coected compoet of G,p, as λ varies. Whe λ <, all compoets are relatively small, O(log ), ad o the other had, whe λ >, there is compoet o Θ() vertices, with the ext largest compoet havig size O(log ). I this sectio, we shall use brachig processes to explore the compoet structure of a radom graph ad deduce a form of Erdős ad Réyi s result. Oe of the useful facts that shall be eeded from probability is that Bi(, λ/) p Po(λ), meaig that as teds to ifiity, biomial radom variables with parameters ad λ/ coverge, i probability to a Poisso radom variable with mea λ. Ideed, for ay fixed c > 0, Bi( c, λ/) p Po(λ). Thus, for small values of k, a breadth-first search i G,λ/ i a compoet o k vertices looks like a brachig process with offsprig distributio Po(λ) that has exactly k vertices. For ay vertex v, let C v deote the coected compoet of v, i G,λ/. Oe ca explore the compoet of C v vertex by vertex startig with R 0 = B 0 = {v 0 }. For k 0, R k is the set of vertices reached at time k ad B k is the set of boudary vertices at time k (those that have bee reached, but have uexplored eighbourhoods). For each k 0, pick w B k ad let R k+ = N(w) V \(R 0 R k ): the ewly discovered vertices reached whe the eighbourhood of w is explored. Defie B k+ = B k \ {w} R k+. This exploratio stops whe B k = 0 ad at this time, we will have costructed a spaig tree of the compoet of v. Note that at each time, coditioed o the values of R 0, R,..., R k, R k+ Bi ( R 0 R R k, p). Sice the distributios of umber of ewly reached vertices is ot always the same, this exploratio is ot a brachig process. However, it ca be closely approximated by a appropriately chose brachig process. Use T λ for a brachig process with offsprig distributio Po(λ). For every k, set ρ k (λ) = P( T λ = k). 6

7 Kare Guderso Lemma 4. Let λ, k be fixed costats. For every vertex v of G,λ/, P( C v = k) ρ k (λ). Proof. Fix a rooted, ordered tree T with k vertices ad the P(spaig-tree rooted at v T ) P(T λ = T ). The proof is completed by summig over fiitely may trees o k vertices. For every k, let N k (G) deoted the umber of vertices i a graph G i compoets of order k. Recall that X k (G) was used for the umber of compoets with k vertices so that N k (G) = kx k (G). Use N k for N k (G,λ/ ). Corollary 5. For every fixed k, λ, Proof. lim E(N k) = ρ k (λ). E(N k ) = E(#v s.t. C v = k) = v V P( C v = k) = P( C v = k). With a little more careful coutig, oe ca show the followig. Lemma 6. For λ, k fixed, lim 2 E(N 2 k ) = ρ k (λ) 2. For a radom variables (X ) ad x p R, the otatio X x shall be used to mea that (X ) coverges i probability to x. That is, for every ε > 0, there is a N ε so that for N ε, P( X x > ε) < ε. Usig Chebyshev s iequality, Lemmas 5 ad 6 show that for every k, Defie the followig quatities: Corollary 7. For k, λ fixed. N p k ρ k (λ), ad 2. N p >k ρ k (λ). N k p ρ k (λ). N k (G) = N (G) + + N k (G), N >k (G) = # vertices i compoets with > k vertices, ρ k (λ) = P( T λ k). Proof. For the first, ote that k is a fixed fiite umber, so For the other, ote that N k = k i= N i p ρ i (λ) = ρ k (λ). N >k = N p k ρ k (λ). i 7

8 Kare Guderso Usig some geeral properties of sequeces of radom variables, this result exteds to the followig. Theorem 8. Let λ be costat. There is a fuctio α() slowly eough so that. N α()(g,λ/ ) p ρ(λ), ad 2. N >α()(g,λ/ ) p ρ(λ). Proof. Use the fact that k ρ k(λ) = P(T λ is fiite) = ρ(λ). From this theorem, we kow that a ρ(λ) fractio of vertices i G,λ/ are i big compoets (of size at least α()) ad the rest are all i small compoets (of size at most α()). What remais is to show that there are ot may big compoets - there is oly a sigle giat compoet. As otatio, let L (G) deote the umber of vertices i the largest compoet of a graph G ad L 2 (G) deote the umber of vertices i the secod largest compoet. Theorem 9 (Erdős, Réyi [5, 6]). Let λ be costat, the. L (G,λ/ ) = ρ(λ) + o p (), ad 2. L 2 (G,λ/ ) = o p (). Proof. First we shall show that there is a sigle giat compoet that covers most of the vertices i big compoets. Assume that λ > so that ρ(λ) > 0 (otherwise there is othig to prove) ad fix ε (0, ρ(λ)). Sice the fuctio ρ is cotiuous, let λ be such that ρ(λ ) ρ(λ) ε/3. Usig a techique sometimes kow as spriklig, the radom graph G,λ/ is costructed as a uio of two idepedet graphs: G (, λ /) ad G 2 (, δ), i which case δ λ λ. Exercise 3. Check that G,λ/ ca be defied as G (, λ /) G 2 (, δ). As before, N >α()(g ) p ρ(λ ). Defie a set of vertices i big compoets, B = {v compoets of v i G with > α() vxs}. For sufficietly large, with probability at least ε, B ρ(λ ) ε/3 ρ(λ) 2ε/3. The goal is to show that B is coected. So far, the edges arisig from G 2 have bee igored. They are ow used to show that, eve if B is discoected i G, the subgraph i B becomes coected whe the G 2 -vertices are icluded. Note that if L (G) (ρ(λ) ε), the L (G) (ρ(λ) ε) B ε/3. Call a partitio of B = B B 2 a bad partitio if B, B 2 ε/6, there are o G -edges betwee B ad B 2, ad there are o G 2 -edges betwee the two sets. If L (G) B ε/3, the there exists a partitio satisfyig the first two coditios ad, give (B, B 2 ), for some costat c > 0, P(o G 2 edges b/w B, B 2 ) = ( δ) B B2 exp( δ 2 ε 2 /36) = exp( c). Thus, sice there are at most /α() differet compoets i B, P(L (G,λ/ ) B ε/3) P( a bad partitio) 2 /α() e c = o(). 8

9 Kare Guderso The fial iequality uses the fact that α(). Thus, for every ε > 0, P(L (ρ(λ) ε)) ε. It remais to show that the size of the secod largest compoet is sigificatly smaller (at least i the case λ > ). Let be large eough so that P(N >α() (ρ(λ) + ε/3)) ε ad α() ε/3. Determiistically (ot just with high probability), L max{α(), N >α() } α() + N >α() ad L + L 2 N >α() + 2α(). Thus, with probability at least ε, L α() + N >α() (ρ(λ) + ε), L + L 2 N >α() + 2α() (ρ(λ) + ε). From the previous, with probability at least ε, L (ρ(λ) ε) ad so with probability at least 2ε, L 2 2ε. As ε > 0 was arbitrary, L 2 = o p (). 3. A more careful look at the sub-critical case I the case where λ <, oe ca use a slightly differet compoet exploratio, itroduced by Karp, to get a more precise boud o the size of the largest compoet i G,λ/. The exploratio is defied as follows. Give v 0 V, set R 0 = B 0 = {v 0 }. The sets R are the reached vertices, ad the sets B are the boudary vertices - those with uexplored eighbourhoods. At each step 0, choose a vertex v B ad set X + = N V \R (v) ad update B + = B \ {v} N V \R (v) R + = R N V \R (v). Cosider the distributio of the radom variables (X ) : X Bi(, p) ad for every 2, coditioed o the values of X,..., X, X + Bi( (X + X X ), p). With iequalities i terms of stochastic domiatio, for every, X Bi(, p). Furthermore, oe ca check that for every time t, R t = t + ad ( t t ) B t = + (X i ) = X i (t ). i= The above process stops whe there are o boudary vertices. At this poit, we have foud all vertices i the compoet of v 0 : C v0. Thus, C v0 = + mi{t i= t X i t }. The stochastic domiatio of the variables X i ad the characterizatio of C v0 i terms of these variables ca be used to give bouds o the size of the largest compoet. i= 9

10 Kare Guderso Propositio 20. Let λ < be fixed, the with high probability L (G,λ/ ) = O(log ). Proof. Usig the vertex exploratio described above, for ay t, P( C v0 t + 2) P( P( t X i t) i= t Bi(, λ/) t) i= P(Bi(t, λ/) t). Usig a iequality about tails for Biomial radom variables due to Jaso, sice λt < t, ( ( λ) 2 t 2 ) P(Bi(t, λ/) t) exp 2(λt + ( λ)t/3) ( exp ( ) λ)2 t. 2 Settig t = 3 log ( λ), we see that 2 P( C v0 t + 2) exp ( 32 ) log = 3/2 ad hece Thus, with high probability, P( cmpt with t + 2 vxs) 3/2 = /2 = o(). L (G,λ/ ) 3 log + 2 = O(log ). ( λ) 2 4 Fixed degree distributios 4. The cofiguratio model The cofiguratio model was itroduced by Bollobás to study radom regular graphs. I geeral, give a sequece d, d 2,..., d with the coditio that i= d i is eve, a radom (multi-)graph with degree sequece d, d 2,..., d ca be defied as follows. For each i, let S i = {(i, j) j d i }. A radom graph is geerated by choosig, uiformly at radom, a matchig of the set S i ad the idetifyig edges: xy E iff there are j x, j y so that {(x, j x ), (y, j y )} is i the matchig. The set S i ca be thought of as the edge stubs emaatig from vertex i ad a edge stubs joied uiformly at radom. Note that i geeral, such a procedure ca produce loops ad multiple edges. Oe ca either work with the resultig multigraph, or coditio o the graph beig simple. Exercise 4. Show that, coditioed o the graph beig simple, the cofiguratio model produces a graph with degree sequece (d, d 2,... d ) uiformly at radom. I the case of r-regular graphs, oe ca geerate a radom r-regular multigraph usig the cofiguratio model with d = d 2 = = d = r (o the coditio that their sum is eve). Bollobás showed that i this radom model, the radom multi-graph G r satisfies P(G r is simple) > 0. This allows oe to move betwee a radom multi-graph ad the radom r-regular graphs: if a property holds with high probability i the radom multi-graph, the property also holds with high probability i the radom simple graph. 0

11 Kare Guderso 4.2 Exploratio by brachig process It is kow that for r 3, with high probability, a radom r-regular graph is coected (see, e.g. []). I fact, Robiso ad Wormald [0] showed that a radom regular graph will have, with high probability, a Hamilto cycle! Molloy ad Reed [8, 9] looked at sequeces of radom graphs with fixed degree distributio ad foud a critical poit for the appearace of a giat compoet i well-behaved sequeces of graphs. Defiitio 2. A sequece of iteger valued fuctios D = {d i } i 0 is called a asymptotic degree sequece iff. for all i, d i () = 0, ad 2. for all, i 0 d i() =. A asymptotic degree sequece is called feasible iff for every, there exists at least oe graph, G, o vertices so that for every i, G cotais d i () vertices of degree i. Such a sequece is called smooth iff there exist a sequece of costace {λ i } so that lim d i()/ = λ i. Give such a (feasible, smooth) sequece D, oe ca cosider a sequece of radom graphs, G,D, give by the cofiguratio model ad the degree distributio give by D. That is, for every, a radom graph is chose, accordig to the cofiguratio model so that for every i, there are d i () vertices of degree i. Whe is large, such a graph will have, for every i, approximately λ i vertices of degree i. Assumig some other techical coditios, Molloy ad Reed [8, 9] showed that if 0 < i i(i 2)λ i <, the there exists c, c 2 > 0, depedig oly o D so that with high probability, G,D has exactly oe compoet with at least c vertices ad all other compoets have order at most c 2 log. O the other had, if i i(i 2)λ i < 0, the with high probability, every compoet is of order o(). How does this compare to the results regardig brachig processes from the previous sectio? Set µ = i iλ i, the (asymptotic) average degree of a vertex i the sequece of radom graphs. The, i i(i 2)λ i > 0 iff i(i )λ i >. µ i 0 Cosider a compoet exploratio from a fixed vertex v, as was doe i the Erdős-Réyi radom graph. Whe is large ad v is chose at radom, the umber of eighbours of v is goig to be k with probability approximately λ k. From there, oe ca explore the u-matched edge stubs of v. Sice it is edge stubs that are matched up iitially (ad ot vertices), a vertex w of degree i has probability of havig oe of its edge stubs matched to a edge stub of v proportioal to approximately iλ i. If such a vertex of degree i is chose, it will have oly i eighbours i future exploratio of the graph. Thus, a brachig process that is slightly differet from what was see previously ca be used to describe the compoet exploratio. For the first level (Z ) of the brachig process, the offsprig distributio is ξ 0 give by P(ξ 0 = k) = λ k ad for all 2, the -th level is give from the ( )-st with a offsprig distributio ξ so that P(ξ = i ) = iλ i µ. survives with positive prob- Note that, as we have see previously, the brachig process T ξ ability if E(ξ ) = i(i )λ i >. µ i

12 Kare Guderso Oe ca check that, if T ξ survives with positive probability, the so does a radom tree with offsprig distributio ξ 0 for the first geeratio ad distributio ξ for all further geeratios. There are a lot of details to fill i for a proof that this brachig process is, ideed, a good model for the compoet exploratio i the radom graphs with fixed degree distributio. I additio, the spriklig argumet used for G,p, does ot traslate i a straightforward way to graphs with a fixed degree distributio. Some details ca be foud, for example, i Durrett [4]. Refereces [] Béla Bollobás, Radom graphs, Secod editio, Cambridge Studies i Advaced Mathematics, 73, Cambridge Uiversity Press, Cambridge, 200. [2] Béla Bollobás, Oliver Riorda, Radom graphs ad brachig processes, i Hadbook of large-scale radom etworks, pp 5 5, Bolyai Soc. Math. Stud., 8, Spriger, Berli, [3] Reihard Diestel, Graph theory, Fourth editio, Graduate Texts i Mathematics, 73, Spriger, Heidelberg, 200. [4] Rick Durrett, Radom graph dyamics, Cambridge Series i Statistical ad Probabilistic Mathematics, Cambridge Uiversity Press, Cambridge, 200. [5] P. Erdős, A. Réyi, O radom graphs. I, Publ. Math. Debrece 6 (959), [6] P. Erdős, A. Réyi, O the evolutio of radom graphs, Magyar Tud. Akad. Mat. Kutató It. Közl. 5 (960), 7 6. [7] Svate Jaso, Tomasz Luczak, Adrezej Ruciski, Radom graphs, Wiley-Itersciece Series i Discrete Mathematics ad Optimizatio. Wiley-Itersciece, New York, [8] Michael Molloy, Bruce Reed, A critical poit for radom graphs with a give degree sequece, i Proceedigs of the Sixth Iteratioal Semiar o Radom Graphs ad Probabilistic Methods i Combiatorics ad Computer Sciece, Radom Graphs 93 (Pozań, 993), Radom Structures Algorithms 6 (995), o. 2-3, [9] Michael Molloy, Bruce Reed, The size of the giat compoet of a radom graph with a give degree sequece, Combi. Probab. Comput. 7 (998), o. 3, [0] R. W. Robiso, N. C. Wormald, Almost all regular graphs are Hamiltoia, Radom Structures Algorithms 5 (994), o. 2,