A note on log-concave random graphs

Transcription

1 A ote o log-cocave radom graphs Ala Frieze ad Tomasz Tocz Departmet of Mathematical Scieces, Caregie Mello Uiversity, Pittsburgh PA53, USA Jue, 08 Abstract We establish a threshold for the coectivity of certai radom graphs whose (depedet) edges are determied by the uiform distributios o geeralized Orlicz balls, crucially usig their egative correlatio properties We also show the existece of a uique giat compoet for such radom graphs Itroductio Probabilistic combiatorics is today a thrivig field bridgig the classical area of probability with moder developmets i combiatorics The theory of radom graphs, pioeered by Erdős-Réyi [], [3] has give us umerous isights, surprises ad techiques ad has bee used to cout, to establish structural properties ad to aalyze algorithms There are by ow several texts [], [6], [4] that deal exclusively with the subject The most heavily studied models beig G,m ad G,p Both have vertex set [] ad i the first we choose m radom edges ad i the secod we iclude each possible edge idepedetly with probability p Let X be a radom vector i [0, ) ( ) with a log-cocave dow-mootoe desity f, that is (i) log f is cocave ad (ii) f(x) f(y) if x y (coordiate-wise) For 0 < p <, let G X,p be a radom graph with vertices,, ad edges determied by X: for i < j, {i, j} is a edge if ad oly if X {i,j} p Such log-cocave radom graphs were itroduced by Frieze, Vempala ad Vera i [5] For istace, whe X is uiform o [0, ] ( ), GX,p is the radom graph G,p The paper [5] itroduced a surprisig coectio betwee radom graphs ad covex geometry Research supported i part by NSF Grat DMS66063 the electroic joural of combiatorics (05), #P00

2 It studied, amog other thigs, the coectivity of G X,p ad foud a logarithmic gap for the threshold There is o gap whe G X,p is defied by uiform samplig from a wellbehaved regular simplex ad we exted this case to Geeralized Orlicz Balls GOBs: that is sets of the form {x R d : d i= f i( x i ) } for some odecreasig lower semicotiuous covex fuctios f,, f d : [0, ) [0, ] with f i (0) = 0, which are ot idetically 0 or + o (0, ) The ey property of Orlicz balls is egative correlatio We say that a radom vector X i R d has egatively correlated coordiates if for ay disjoit subsets I, J of {,, d} ad oegative umbers s i, t j, we have P( i I X i > s i, j J X j > t j ) P( i I X i > s i )P( j J X j > t j ) It was show i [7] that this property holds for radom vectors uiformly distributed o GOBs (see also [8] for a first such result treatig two coordiates ad [9] for a simpler proof of the geeral result) Notatio: Throughout the paper we will let ad σ max be defied by σ mi = mi i<j EX i,j ad σmax = max i<j EX i,j Our result cocerig coectivity is the followig theorem Theorem Let X = (X i,j ) i<j be a log-cocave radom vector i [0, ) ( ) with a dow-mootoe desity ad egatively correlated coordiates (a) For every δ (0, ), there are costats c ad c depedet oly o δ such that for log p < c, we have P(G X,p has isolated vertices) > c δ (b) For every δ (0, ), there are costats C ad C depedet oly o δ such that for p > C σ max log, we have P(G X,p is coected ) > C δ We will also discuss the existece of a giat compoet for smaller values of p Notatio: Let M = max sup max T (i,j)/ T E(X i,j X T = y), () y [0, ) T where the first maximum is over all oempty subsets T of the idex set {(i, j), i < j } ad we deote X T = (X i,j ) (i,j) T For our theorem o the existece of a giat compoet we eed to have M = O() For a } GOB, {x R ( ) : i<j f i,j( x i,j ), this is justified by the followig assumptio: we let a i,j = sup {t > 0 : f i,j (t) } A regular simplex { x R d : a x } for some a 0 if a i /a j K for some ot too large K the electroic joural of combiatorics (05), #P00

3 Now our assumptios o the f i,j imply that the a i,j s are fiite Furthermore, M max i,j a i,j ad so our assumptio here is that max i,j a i,j is bouded by a absolute costat Theorem Let X = (X i,j ) i<j be a log-cocave radom vector i [0, ) ( ) with a dow-mootoe desity Assume that M = O() There are costats c ad c such that for every β >, we have (i) If p < c, the ( (ii) If p > c M log M ad P(G X,p has a compoet of order β log ) < β ), the P(G X,p has a compoet of order [β log, /]) < β P(G X,p has a uique giat compoet of order > /) > Note that we have dropped the assumptio of egative correlatio 5β log β Coectivity: Proof of Theorem Part (b) is part of Theorem of [5] For (a), we adapt the stadard secod momet argumet used for the Erdös-Réyi model For i, let Y i be equal to if the vertex i is isolated ad 0 otherwise Let Y = Y + + Y be the umber of isolated vertices We have, (EY ) P(G X,p has isolated vertices) = P(Y > 0) EY Thus, if we show that EY ( + ε)(ey ), the P(Y > 0) ε Clearly, EY = EY + EY Y l = l EY + l P(Y = = Y l ) = EY + l P(Y = = Y l ) ad our goal is to show that EY ε (EY ) ad P(Y = = Y l ) l ( + ε ) (EY ) the electroic joural of combiatorics (05), #P00 3

4 From the egative correlatio of coordiates of X as well as a elemetary iequality P(A) P(A B) + P(B), we get P(Y = = Y l ) = P( i X i > p, X il > p, X l > p) P( i X i > p)p( i, l X il > p) P( i X i > p) [ P( i l X il > p) + P(X l > p) ] = P(Y = ) [ P(Y l = ) + P(X l p) ] p By Lemma 35 from [5], P(X l p) (recall that by the Préopa-Leidler iequality, margials of log-cocave vectors are log-cocave; clearly, margials of dow-mootoe desities are dow-mootoe) Therefore, P(Y = ) p P(Y = = Y l ) P(Y = )P(Y l = ) + l l l ( ) P(Y = ) + p P(Y = ) ( + p ) ( (EY ) < + c log EY EY ) (EY ), so it suffices to tae ε such that ε EY ad ε c log EY By Lemma 3 from [5], P(Y = ) e ap/, for some uiversal costat a (the assumptio p < σ log 4 mi of that lemma is clearly satisfied if p < c ), so EY = P(Y = ) e ap/ > ac Thus, ε = c ac log will suffice 3 Giat Compoet: Proof of Theorem Lemma 3 Let X = (X i,j ) i<j be a log-cocave radom vector i [0, ) ( ) with a dow-mootoe desity There are uiversal costats a ad b such that for S, T {(i, j), i < j } ad p > 0, we have ( ) T bp P( s S X s > p, t T X t p) e ap S /M the electroic joural of combiatorics (05), #P00 4

5 Proof Fix disjoit sets S, T {(i, j), i < j } (if they are ot disjoit, the probability i questio is 0) ad y [0, ) T Let f be the desity of (X S, X T ) The coditioal desity of the vector X S give X T = y, f XS X T (x y) = f(x, y) f(x, y)dx is dow-mootoe ad log-cocave Therefore, by Lemma 3 from [5], P( s S X s > p X T = y) e ap S /M We deote the desity of X T by f XT ad get P( s S X s > p, t T X t p) = P( s S X s > p X T = y)f XT (y)dy [0,p] T e ap S /M f XT (y)dy [0,p] T = e ap S /M P( t T X t p) ( ) T bp e ap S /M, where the fial iequality follows directly from Lemma 3 of [5] With this lemma i had, we ca prove Theorem Proof Let Z be the umber of compoets of order (that is, o vertices) i G X,p As for the Erdös-Réyi model, looig at a spaig tree for each compoet ad boudig the correspodig i-out edge probabilities usig Lemma 3 yields If p = M a ( ) ( bp EZ e ap( )/M ( e ) ( bp e ap( )/M = σ [ ] mi eb pe ap bp M ( ) ) ) c, with c beig a costat (chose soo), this becomes where we put A = eb a M EZ e A c [ Ace c e c/], the electroic joural of combiatorics (05), #P00 5

6 Case If c is a small costat, say c (equivaletly, p ), the we boud ea e b e c e c/ crudely by ad get that Thus, EZ e A c (Ac) e(ac) e e E ( β log Z ) e β log By the first momet method, this gives (i) e e e β log e3 = e e < β β Case Let c be a large costat, say such that Ace c/ e ad Ac e, which holds whe, say c 4 log A, provided that A is large eough, which leads to the assumptio o p i (ii) The for /, we have Thus, E β log / EZ e Ac (Ace c/ ) e Z e β log e e < β β By the first momet method, this gives the first part of (ii) To go about the secod part ad show that there is a giat compoet, we shall simply cout the umber of vertices o the small compoets ad show that with high probability, there are strictly less such vertices The uiqueess of a giat compoet plaily follows from the fact that it has more tha / vertices, so there caot be more tha oe such compoets Fix β log ad set t = e For ay positive iteger l et +, we have P(Z et) P(Z (Z ) (Z l + ) et(et ) (et l + )) EZ (Z ) (Z l + ) et(et ) (et l + ) EZ (Z ) (Z l + ) (et l + ) l As for the upper boud for EZ, looig at spaig trees for each l-tuple of distict compoets of order ad boudig the correspodig i-out edge probabilities usig the electroic joural of combiatorics (05), #P00 6

7 Lemma 3 yields EZ (Z ) (Z l + ) ( )( ) ( ) ( (m ) ( ) l e ap bp M l( l) ( e ) ( ) l ( )l ( ) l e ap bp M l( l) ( e [ = Ace c e cl/] ) l A c ) ( )l Provided that l /, uder our assumptio c 4 log A, this is further upper bouded by (t/ ) l, which gives P(Z e ) = P(Z et) ( ) l t l et l + For log, we choose l = ad get P(Z e ) e ( log, log ) For < log, we have t = e > e, so choosig, say l = e yields ( ) ( l P(Z e t ) et e = e e t ) l ( e ( e ) l ) e, < log Combiig the last two estimates, the uio boud gives that the probability of the evet E = { β log, Z e } is at most 4 (β /) log + + log e (log ) (e ) < 5β e log (we chec that log < ad simply boud 4 (β /) log + 4 (e ) e log e (log ) e β+ e ) To fiish, it log remais to chec that o E c, there are few vertices o the small compoets O E c, we have Z e < e e = (e ) < 093 β log β log = the electroic joural of combiatorics (05), #P00 7

8 Remar 3 It was show i [9] that the egative correlatio property holds i fact for radom vectors with desities of the form h( f i (x i )), where h : [0, ) [0, ) is a oicreasig log-cocave fuctio (h = [0,] givig uiform desities o GOBs) For such desities, M is fiite ad ca be bouded as for GOBs i terms of certai parameters depedig o the fuctios f i ad h 4 Coclusio ad Ope Questios We have successfully geeralised the results o the regular simplex i [5] to GOBs The followig questios seem most apposite Q What we prove i Theorem does ot rule out the possibility that i some rage of p there is more tha oe giat compoet Ca the proof be tighteed to rule this out? Q What is the coectivity or giat compoet threshold for the itersectio of two well-behaved regular simplices? Q3 What is the coectivity or giat compoet threshold for the itersectio of a few regular simplices with idepedet radomly chose coefficiets? Refereces [] B Bollobás: Radom Graphs, Academic Press, 985 [] P Erdős ad A Réyi, O radom graphs I, Publ Math Debrece 6 (959) [3] P Erdős ad A Réyi: O the evolutio of radom graphs, Publ Math Ist Hugar Acad Sci 5 (960) 7-6 [4] AM Frieze ad M Karońsi, A itroductio to radom graphs, Cambridge Uiversity Press, 05 [5] AM Frieze, S Vempala ad J Vera, Logcocave radom graphs Electro J Combi 7 (00), o, Research Paper 08, 3 pp [6] S Jaso, T Lucza ad A Rucisi: Radom Graphs, Wiley-Itersciece, 000 [7] M Pilipczu, JO Wojtaszczy, The egative associatio property for the absolute values of radom variables equidistributed o a geeralized Orlicz ball Positivity (008), o 3, [8] JO Wojtaszczy, The square egative correlatio property for geeralized Orlicz balls Geometric aspects of fuctioal aalysis, , Lecture Notes i Math, 90, Spriger, Berli, 007 [9] JO Wojtaszczy, A simpler proof of the egative associatio property for absolute values of measures tied to geeralized Orlicz balls Bull Pol Acad Sci Math 57 (009), o, 4 56 the electroic joural of combiatorics (05), #P00 8