Application to Random Graphs

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Johnathan Wade
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1 A Applicatio to Radom Graphs Brachig processes have a umber of iterestig ad importat applicatios. We shall cosider oe of the most famous of them, the Erdős-Réyi radom graph theory. 1 Defiitio A.1. Let V be a collectio of vertices, ad let E be the collectio of all possible ( 2) edges betwee them. For a fixed p [0, 1], a radom graph G(, p) o vertices is a subgraph of G = (V, E) obtaied by idepedetly deletig each edge e E with probability 1 p (equivaletly, idepedetly drawig each edge e E with probability p). Exercise A.2. Describe the disrtibutio of the total umber of edges i G(, p). Edges e preset i G(, p) are called occupied; other edges are kow as uoccupied or vacat. Two vertices v, v V are coected i G(, p), if there is a occupied path (sequece of occupied edges) coectig v ad v. For a vertex v V, its coected compoet (or cluster) C v is defied as the maximal coected subgraph i G(, p) cotaiig v. The size C v of the cluster C v i G(, p) is the umber of vertices i C v. For every p (0, 1) the radom graph G(, p) decomposes ito certai umber of coected compoets ad isolated vertices (ie., clusters of size oe). It is ituitively clear that for p > 0 very small, the typical radom graph G(, p) cotais may isolated vertices, whereas for p < 1 close to oe, all vertices i V are joied ito a sigle giat compoet. A atural questio is what happes for itermediate values of p, ad how does the chage from a sea of small compoets to a sigle giat compoet occur. The pheomeo i the followig theorem is ofte referred to as the phase trasitio for radom graphs: Theorem A.3. Let p = c, where c > 0 is a costat. There exist positive costats K 1 ad K 2 such that for the radom graph G(, p) the followig holds: a) If 0 < c < 1, the with probability approachig oe as, every coected cluster C v i G(, p) cotais o more tha K 1 log vertices. b) Let c > 1 ad let β = β c (0, 1) solve the equatio β + e cβ = 1. The with probability approachig oe as, the graph G(, p) cotais a sigle compoet of size of order β, whereas every other cluster C v i G(, p) cotais o more tha K 2 log vertices. Remark A.3.1. I fact, β = 1 ρ with ρ = ρ c beig the extictio probability for the brachig process with Poi(c) offsprig distributio. 1 The origial paper by Erdős ad Réyi (1959) geerated a lot of iterest i applyig probabilistic methods to combiatorics ad other areas of (pure) mathematics. A1

2 A.1 Coectio to brachig processes We ow describe a geometric procedure, which explais the itrisic relatio betwee G(, p) ad a certai brachig processes. Namely, choose a vertex v 0 ad fid all its eighbours (v 1, v 2, v 3,ad v 4 i the picture below). Of course, the umber of eighbours of v 0 is a radom variable with distributio Bi( 1, p). v1 v5 v0 v2 v6 v7 v4 v3 Figure A.1: Iteratig process of exposig the cluster of v 0 i G(, p). Saturated vertices ad their cotributig edges share the same colour. We mark v 0 as saturated or exposed, ad the fid all eighbours of v 1 i V \ {v 0, v 1, v 2, v 3, v 4 (vertices v 5 ad v 6 i the picture) ad mark v 1 saturated. The procedure is repeated util all vertices i the cluster of v are saturated. Notice that the uber X 0 of eighbours of v 0 is a Bi( 1, p) radom variable, the umber X 1 of eighbours of v 1 is a Bi( 1 X 0, p) radom variable, the umber of X 2 of eighbours of v 2 is a Bi( 1 X 0 X 1, p) radom variable etc. For p < 1, the average umber of ewly added vertices is bouded above by p < 1, so that the process is likely to termiate after a fiite umber of steps; moreover, each of these distributios is close to Bi(, p). If C v0 cotais exactly m vertices, the the process termiates oce each of the m 1 ew added vertices becomes saturated, amely whe X 0 + X X m 1 = m 1. (A.1) Observe that such m is the smallest possible value for which this equality holds. Exercise A.4. For m > 0 ad the variables X k as described above, deote S m = X X m 1. Show that the geeratig fuctio of S m satisfies G m (z) def = E ( z S m ) = ( 1 + g m (z 1) ) 1 e ( 1)g m (z 1), (A.2) where g m = 1 (1 p) m = 1 ( 1 c ) m. [Hit: Use iductio, coditioig ad the relatio g m+1 = p + (1 p)g m.] The rest of this sectio sketches the mai ideas of the argumet; it will ot be examied, ad might be skipped, if you are ot iterested. A2

3 A.2 Some useful iequalities The followig simple iequalities are very useful i our argumet. Exercise A.5. a) Show that for every real y 0, we have 1 + y < e y ad 1 y < e y. b) Show that for every y (0, 1) log(1 y) + y y2 2. (A.3) [Hit: Ivestigate mootoicity of the fuctio f(y) = log(1 y) + y + y2 2.] c) Show that for all real y > 0, d) Let ε (0, 1) be fixed. Show that for every y (0, ε], 1 e y > y y2 2. (A.4) log(1 y) < y < (1 ε) log(1 y). (A.5) [Hit: The fuctio g(y) = y + (1 ε) log(1 y) is strictly mootoe i (0, ε].] A.3 Subcritical case c (0, 1) The first claim of Theorem A.3 follows directly from Exercise A.4 ad the followig observatios: Exercise A.6. Let p = c with c (0, 1). Use the Markov iequality to estimate P(S m+1 = m) P(S m+1 m) E ( z S m+1 m ) = G m+1 (z) z m, for all z 1. Next, check the simple iequality g m < mp ad use it to show that ( ( ) ) log G m+1 1/c c m (m + 1)(1 c) + m log c. Fially, use the iequality i (A.3) to deduce that for all m > 0. log P ( C v m + 1 ) log P(S m+1 m) 1 c (1 c)2 m 2 It follows from the previous exercise that with c (0, 1) ad K > 0 large eough, the RHS of the iequality P ( v s.t. C v > K log ) P ( C v0 > K log ) goes to zero sufficietly fast as. I particular, every costat K 1 satisfyig the coditio K 1 (1 c) 2 > 2 implies the first statemet of Theorem A.3. A3

4 A.4 Supercritical case c > 1: prelimiaries We start by cosiderig two importat examples. Example A.7. [Poisso brachig] Cosider the brachig process with the offsprig distributio Z 1 Poi(c). The for c > 1 the extictio probability ρ = ρ c is the oly solutio i (0, 1) to the equatio ρ = e c(ρ 1). Writig ρ c = 1 β c, we deduce that β = β c (0, 1) solves the equatio β + e cβ = 1. Example A.8. [Biomial brachig] Let the offsprig distributio Z 1 be Biomial Bi(, p) with p c > 1 as. The for all large eough the extictio probability ρ = ρ(, p) of the correspodig brachig process is smaller tha oe; moreover, ρ(, p) ρ c as, where ρ c is the extictio probability of the brachig process with Poi(c) offsprig distributio. Solutio. Ideed, the claim follows immediately from the poit-wise covergece of geeratig fuctios: G Z1 (s) = ( 1 + p(s 1) ) exp{c(s 1), as, for every fixed s [0, 1]. Usig (A.5), oe ca also compare the correspodig extictio probabilities: Lemma A.9. Let c > 1 ad ε (0, 1) be fixed. The for all large eough 2 ad all m satisfyig 1 m ε, we have, uiformly i s [0, 1) ( 1 + c ) ( (s 1) < e c(s 1) < 1 + c ) m (s 1). As a result, the extictio probabilities ρ,m of Bi( m, c ) brachig processes ad that ρ c of Poi(c) brachig process satisfy, for all such ad m, the followig iequality: ρ,0 < ρ c < ρ,m. A.5 Supercritical case c > 1: sketch of the proof The argumet ow splits ito the followig three steps. Fix c > 1 ad α (0, β), where β = β c is the oly solutio i (0, 1) to the equatio β + e cβ = 1. First, we show that with probability approachig oe as, the radom graph G(, c ) cotais o clusters of size betwee K 2 log ad α, provided K 2 > 0 is large eough. Next, we show that with probability approachig oe as, every two clusters of size at least α must be coected; this implies that the limitig desity of vertices of G(, c ) belogig to the sigle large compoet (of size at least α, if exists) is bouded below by β c. We fially show that that with probability approachig oe as, the limitig desity of vertices of G(, c ) belogig to small compoets (of size at most K 2 log ) equals exactly ρ c = 1 β c. Step I: absece of itermediate clusters Exercise A.10. For fixed c > 1, let β = β c (0, 1) solve the equatio β + e cβ = 1. a) Fix ay α (0, β), deote δ = 1 α e cα > 0, ad show that for all y [0, α] we have 1 y e cy δ α y. b) Let S m ad g m be as described i Exercise A.4. Show that ES m = ( 1)g m is bouded below by ( 1 + δ α ) (m 1). 2 e.g., the claim of the lemma holds for all c/ε. A4

5 The followig result is of geeral importace: Lemma A.11. Let X Bi(, p). The for all y > 0, P ( X EX y ) exp { y2. (A.6) 2 EX Proof. Usig (A.4) ad the expoetial Markov iequality P ( X a ) E ( e tx+ta) with t > 0, we deduce P ( X a ) (1 + p(e t 1)) e ta ( ) e ta p(1 e t ) exp{ ta p t t2 ; 2 it remais to otice that for a = EX y = p y the miimum of the expressio i the last expoetial is attaied for t = y y2 (ad equals, ie., (A.6) holds). p 2p By combiig Exercise A.10 ad Lemma A.11, we ca estimate the probability of the evet { S m = m 1 i (A.1), that a fiite cluster cotais exactly m vertices: Exercise A.12. Show that for all m α ad some A δ2, we have 2(α+δ)α P ( S m = m 1 ) P (S m ES m δ ) α (m 1) exp { A(m 1). (A.7) This allows to boud the probability that G(, c ) cotais itermediate clusters: Exercise A.13. Use (A.7) to deduce that for K 2 > 0 large eough, the probability that a cluster of a fixed vertex v is of size betwee K 2 log ad α is small α m=k 2 log Deduce that for AK 2 > 1, P ( S m = m 1 ) P ( for some v, K 2 log C v α ) α m=k 2 log e A 1 e A e AK 2 log. P ( S m = m 1 ) 0 as. From ow o we ca ad shall assume that the radom graph G(, c ) uder cosideratio cotais o clusters of itermediate size. We also otice that the argumet i the proof of Lemma A.11 ca be used to derive the so-called Cheroff bouds for biomial distributios: Exercise A.14. Let X Bi(, p) ad let y > 0 be fixed. a) Optimize the expoetial Markov iequality P ( X a ) E ( e tx+ta) w.r.t. t > 0 ad deduce the estimate P ( X E(X) y ) exp { y2. (A.8) 2 EX b) Optimize the expoetial Markov iequality P ( X a ) E ( e tx ta) w.r.t. t > 0 ad deduce the estimate P ( X E(X) + y ) { exp y 2 2(EX + y/3). (A.9) A5

6 Step II: uiqueess of the giat compoet We ow show that wit probability approachig oe as, all clusters of size at least α (for ay α (0, β c )) are coected, ie., form a sigle giat compoet. Exercise A.15. Use Lemma A.11 to show that the evet { S α ( 1 + δ 2α ) α that after α steps of exposure process as described i Sect. A.1 above the cluster uder cosideratio cotais less tha δm/2 usaturated vertices has small probability, ({ ( P S α 1 + δ ) ) α exp { δ2 2α 8(α + δ). Suppose that vertices v ad v both belog to differet compoets of size larger tha α. By the previous exercise, after the first α steps of the exposure process, with probability approachig oe as, each of these compoets cotais at least δ/2 usaturated vertices. O this evet, the probability that oe of the possible edges coectig these usaturated vertices is ope, is ot larger tha (1 p) δ2 2 /4 exp { cδ2 4 0, as. I other words, with probability approachig oe as, all clusters of size at least α are merged ito a sigle giat compoet. As the argumet above holds for every α (0, β), the giat compoet (if exists) cotais o less tha ( β + o(1) ) vertices, ie., has asymptotic desity at least β c. Step III: small cluster desity It remais to estimate the umber N s of vertices v for which C v < K 2 log. By liearity of the expectatio, 1 EN s = 1 ( ) E 1 Cv <K 2 log = P ( process at v dies out ) ρ c, v where the covergece follows from Lemma A.9. To fiish the argumet, oe verifies that Var(N s )/(EN s ) 2 0 as, so that selectig t = t > 0 such that Var(N s )/t 2 0 but t/en s 0 as, oe applies the Chebyshev iequality, P ( N s EN s > t ) Var(N s )/t 2 0 as, ad deduces that the limitig desity 1 N s of small vertices coverges to ρ c as. Sice ρ c = 1 β c, this fiishes the proof of Theorem A.3. A6

7.1 Convergence of sequences of random variables

7.1 Convergence of sequences of random variables Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite