Sqediasmìc kai Anˆlush AlgorÐjmwn se TuqaÐa Graf ... - Nemertes

1 TUQAŸIA GRAFŸHMATA TOMŸHS ETIKETŸWN 4tètoia montèla. Analutikìtera, sthn enìthta 2 parajètoume merikˆ apotelèsmata twn [10] kai[21] pou aforoÔn to montèlo G n,m,p kai ta sugkrÐnoume me apotelèsmata gia dh upˆrqonta tuqaÐamontèla grafhmˆtwn. H enìthta 3 aforˆ th sqèsh pou èqei to omoiìmorfo montèlo tuqaÐwngrafhmˆtwn tom c me to montèlo tuqaÐwn grafhmˆtwn G n,p tou Erdös. 'Opwc ja doÔme, gia orismènectimèc twn n, m kai p, oi q¸roi pou parˆgontai apì ta duo montèla eÐnai isodÔnamoi. Tèloc,sthn enìthta 4 antimetwpÐzoume to prìblhma thc eÔreshc megˆlwn anexˆrthtwn sunìlwn koruf¸n.Sugkekrimèna, upologÐzoume th mèsh tim kai th diasporˆ tou arijmoÔ twn anexˆrthtwn sunìlwnkorf¸n megèjouc k gia thn perÐptwsh tou genikoÔ motèlou tuqaÐwn grafhmˆtwn tom c kai sthsunèqeia parousiˆzoume treÐc apodotikoÔc pijanotikoÔc algìrijmouc gia thn kataskeu megˆlwnanexˆrthtwn sunìlwn koruf¸n se tètoia graf mata.1.1 Gnwstˆ montèla tuqaÐwn grafhmˆtwnH jewrÐa tuqaÐwn grˆfwn jemeli¸jhke kai anaptÔqjhke arketˆ apì touc P. Erdös kai A. Rényi thdekaetÐa tou '50 (blèpe [7, 8]). To megalÔtero mèroc thc douleiˆc touc basÐsthke sto montèlo G n,p ,to opoÐo apoteleÐ èna apì ta pr¸ta montèla pou ìrisan. Se autì to montèlo tuqaÐwn grafhmˆtwn,o arijmìc n eÐnai o arijmìc twn koruf¸n kai p = p(n) eÐnai h pijanìthta na upˆrqei mia ˆkmmetaxÔ duo koruf¸n. H pijanìthta Ôparxhc miac akm c eÐnai anexˆrthth apì thn Ôparxh th mhÔparxh twn upìloipwn ( n2)− 1 pijan¸n akm¸n tou grˆfou. Sunep¸c, h pijanìthta na pˆroume ènasugkekrimèno stigmiìtupo autoÔ tou montèlou kajorÐzetai pl rwc apì ton arijmì twn pleur¸ntou. H anexarthsÐa Ôparxhc mh Ôparxhc twn akm¸n se autì to montèlo dieukolÔnei katˆ polÔton upologismì pijanot twn orismènwn gegonìtwn.'Ena ˆllo exÐsou shmantikì montèlo tuqaÐwn grafhmˆtwn eÐnai to G n,M . Se autì to montèloèna grˆfhma epilègetai isopÐjana apì ìla ta graf mata me n korufèc kai M pleurèc. Genikˆ, oipijanìthta na pˆroume èna sugkekrimèno grˆfhma se autì to montèlo eÐnai diaforetik apì thnpijanìthta na pˆroume to Ðdio grˆfhma sto montèlo G n,p . 'Omwc, o T. Luczak sto [11] apèdeixeìti gia mia arketˆ megˆlh perioq tim¸n twn paramètrwn n, M kai p ta duo autˆ montèla eÐnaiisodÔnama ìson aforˆ mia megˆlh perioq idiot twn.Ektìc bèbaia apì ta parapˆnw duo polÔ shmantikˆ montèla, upˆrqou kai ˆlla montèla tuqaÐwngrˆfwn, merikˆ apì ta opoÐa èqoun exetasjeÐ se megˆlo bˆjoc. 'Ena tètoio montèlo eÐnai kai tomontèlo tuqaÐwn kanonik¸n grafhmˆtwn R n,k , sto opoÐo epilègoume isopÐjana èna k-kanonikìgrˆfhma n korufèc, dhlad èna grˆfhma sto opoÐo kˆje koruf èqei bajmì akrib¸c k. Miashmantik paralag autoÔ tou montèlou eÐnai kai to montèlo tuqaÐwn kanonik¸n grafhmˆtwn melˆjh stic akmèc pou orÐsthke kai analÔjhke apì touc sugrafeÐc sto [16].1.2 Omoiìmorfo montèlo tuqaÐwn grafhmˆtwn tom c etiket¸nAc jewr soume èna sÔmpan M = {1, 2, . . . , m} apì stoiqeÐa pou ja onomˆzoume etikètec kaièna sÔnolo V = {v 1 , v 2 , . . . , v n } apì korufèc. O akrib c majhmatikìc orismìc tou omoiìmorfoumontèlou tuqaÐwn grafhmˆtwn tom c faÐnetai parakˆtw.

1 TUQAŸIA GRAFŸHMATA TOMŸHS ETIKETŸWN 5Orismìc 2 (Omoiìmorfo Montèlo TuqaÐwn Grafhmˆtwn Tom c [10, 21]) 'Estw ìti se kˆjekoruf v ∈ V antistoiqoÔme èna sÔnolo S v ⊆ M pou diamorf¸netai epilègontac kˆje etikètai ∈ M me pijanìthta p, anexˆrthta apì thn epilog twn ˆllwn etiket¸n kai anexˆrthta apì ticepilogèc pou kˆnoume gia ˆllec korufèc. An jewr soume to grˆfhma G me sÔnolo koruf¸n Vkai sÔnolo akm¸n to e(G) = {(u, v) : u, v ∈ V, S u ∩ S v ≠ ∅}, tìte to G eÐnai èna stigmiìtupo touomoiìmorfou montèlou tuqaÐwn grafhmˆtwn tom c (uniform random intersection graphs model),to opoÐo sumbolÐzoume me G n,m,p .Shmei¸noume ìti, lìgw thc anexarthsÐac epilog c twn etiket¸n apì tic korufèc, h pijanìthtagia mia sugkekrimènh koruf na epilèxei èna sugkekrimèno sÔnolo etiket¸n S eÐnai p s (1 − p) m−s ,ìpou s = |S|.1.2.1 Perigraf tou montèlou me pÐnakecEÐnai profanèc ìti ta sÔnola S v , v ∈ V pou perigrˆfoun monos manta èna stigmiìtupo tou montèlouG n,m,p mporoÔn na anaparastajoÔn me èna n × m pÐnaka apì 0 kai 1, ston opoÐo ja anaferìmastewc pÐnaka anaparˆstashc R n,m,p . Kˆje gramm tou pÐnaka anaparˆstashc antistoiqeÐ semia koruf kai kˆje st lh se mia etikèta. Ta stoiqeÐa tou pÐnaka orÐzontai gia kˆje v ∈ V kaii ∈ M wc ex c{ 1 an i ∈ SR vi =v0 an i /∈ S v .Apì ton orismì tou montèlou G n,m,p blèpoume ìti kˆje stoiqeÐo tou pÐnaka anaparˆstashc eÐnai1 anexˆrthta me pijanìthta p. 'Eqontac ton pÐnaka anaparˆstashc, mporoÔme na kataskeuˆsoumeto grˆfhma sto opoÐo antistoiqeÐ bˆzontac mia pleurˆ metaxÔ duo koruf¸n u, v ∈ V an kai mìnoneˆn upˆrqei toulˆqiston mia st lh i tou R n,m,p tètoia ¸ste R vi = R ui = 1. Merikèc forèceÐnai eukolìtero na fantastoÔme ìti to grˆfhma pou antistoiqeÐ se ènan pÐnaka anaparˆstashcparˆgetai apì èna sÔnol klik¸n pou orÐzontai apì tic st lec tou pÐnaka. Autì sumbaÐnei giatÐ anjewr soume mia sugkekrimènh st lh, ìlec oi korufèc pou èqoun 1 se aut eÐnai geitonikèc kai ˆrasqhmatÐzoun mia klÐka.EÐnai bèbaia profanèc ìti èna grˆfhma mporeÐ na paraqjeÐ apì perissìterouc apì ènan pÐnakecanaparˆstashc. Prˆgmati, kˆje grˆfhma mporeÐ na antistoiqeÐ se perissìtera apì èna graf -mata tom c kai kˆje grˆfhma tom c èqei akrib¸c ènan pÐnaka anaparˆstashc. Gia parˆdeigma, oiparakˆtw pÐnakec anaparˆstashc parˆgoun kai oi duo èna trÐgwno.⎛⎝1 0 0 01 0 1 01 0 1 1⎞⎠kai⎛⎝1 0 0 11 0 1 00 0 1 1Sunep¸c, an kai lìgw thc anexarthsÐac h pijanìthta na pˆroume ènan pÐnaka anaparˆstashcr me x ˆssouc eÐnai akrib¸c P R (r) = p x (1 − p) mn−x , h akrib c tim thc pijanìthtac na pˆroume⎞⎠

1 TUQAŸIA GRAFŸHMATA TOMŸHS ETIKETŸWN 6èna sugkekrimèno grˆfhma èqei elafr¸c pio polÔplokh èkfrash. Sugkekrimèna, gia èna grˆfhmaG èqoumeP (G) =∑r∈R n,m,p:to r parˆgei to GP R (r).1.2.2 Akmèc sto montèlo G n,m,pEÐnai eÔkolo na doÔme ìti h pijanìthta Ôparxhc miac sugkekrimènhc akm c sto G n,m,p montèlo eÐnaiP (∃(u, v)) = 1 − P (S v ∩ S u = ∅) = 1 − (1 − p 2 ) m .To shmantikì stoiqeÐo ìmwc tou montèlou G n,m,p pou to diaforopoieÐ apì to montèlo G n,p eÐnaiìti oi akmèc den eÐnai anexˆrthtec metaxÔ touc. 'Opwc faÐnetai kai sto Je¸rhma 1 parakˆtw, hpragmatopoÐhsh enìc gegonìtoc akm¸n 1 mporeÐ na ephrreˆsei jetikˆ thn pijanìthta ikanopoÐhshcenìc ˆllou gegonìtoc akm¸n sto G n,m,p . AxÐzei na parathr soume ìti h apìdeixh tou Jewr matocmporeÐ na genikeujeÐ ¸ste na sumperilˆbei opoiesd pote aÔxousec grafojewrhtikèc idiìthtec stomontèlo G n,m,p .Je¸rhma 1 ([21]) Upˆrqei mia mh arnhtik susqètish metaxÔ duo opoiwnd pote gegonìtwnakm¸n B 1 := {e i1 , e i2 , . . . , e ik } kai B 2 := {e j1 , e j2 , . . . , e jl }, dhladP (B 1 |B 2 ) ≥ P (B 1 ).Apìdeixh. EÐnai eÔkolo na deÐ kaneÐc ìti tìso to gegonìc B 1 ìso kai to gegonìc B 2 auxˆnontaiìtan to p auxˆnetai. 'Omwc, mia eidik perÐptwsh thc FKG anisìthtac (blèpe [2]) lèei ìti an Akai B eÐnai duo aÔxousec idiìthtec, tìte P (A ∩ B) ≥ P (A)P (B). Epomènwc, efarmìzontac thnanisìthta aut sthn perÐptws mac kai qrhsimopoi¸ntac ton orismì thc desmeumènhc pijanìthtacpaÐrnoume to epijumhtì apotèlesma.□H anisìthta tou jewr matoc 1 isqÔei austhrˆ ìtan to gegonìc akm¸n B 1 perièqei akmèc pouèqoun koinˆ shmeÐa me tic akmèc sto B 2 kai 0 < p < 1. Gia parˆdeigma, sthn perÐptwsh pouB 1 = {∃(u, v)} kai B 2 = {∃(v, w)} èqoumeP (∃(u, v)|∃(v, w))P (∃(u, v))P (∃(u, v), ∃(v, w))=P (∃(u, v))P (∃(v, w))1 − P (∄(u, v)) − P (∄(v, w)) + P (∄(u, v), ∄(v, w))=P (∃(u, v))P (∃(v, w))= 1 − 2(1 − p2 ) m + (1 − p 2 + p 3 ) m> 1(1 − (1 − p 2 ) m ) 21 Me ton ìro gegonìc akm¸n (edge event) ennooÔme èna gegonìc thc morf c {e1 , e 2 , . . . , e k }, ìpou e j eÐnai togegonìc Ôparxhc thc pleurˆc j sto tuqaÐo grˆfhma.

1 TUQAŸIA GRAFŸHMATA TOMŸHS ETIKETŸWN 7gia 0 < p < 1.Tèloc, shmei¸noume ìti mia akìma eidik perÐptwsh thc FKG anisìthtac lèei ìti an A eÐnai miaaÔxousa idiìthta kai B mia fjÐnousa idiìthta, tìte P (A ∩ B) ≤ P (A)P (B). Aut h morf thcFKG mporeÐ na qrhsimopoihjeÐ me trìpo parìmoio me autìn thc apìdeixhc tou jewr matoc 1 giana deÐxoume mh jetik susqètish metaxÔ aÔxousac kai miac fjÐnousac grafojewrhtik c idiìthtactou montèlou G n,m,p .1.2.3 Perioq tim¸n gia {qr sima} graf mataApì th suz thsh se prohgoÔmenh enìthta gÐnetai emfanèc ìti to kenì grˆfhma den prokÔptei mìnoìtan R n,m,p = ∅. OmoÐwc, to pl rec grˆfhma mporeÐ na prokÔyei akìma kai apì araioÔc pÐnakecanaparˆstashc. Diaisjhtikˆ, mporoÔme na poÔme ìti an krat soume stajerˆ ta m kai n, tìteauxˆnontac thn tim tou p paÐrnoume ìlo kai pio puknˆ graf mata. TÐjetai loipìn to er¸thma poiaprèpei na eÐnai h sqèsh metaxÔ twn paramètrwn n, m kai p ètsi ¸ste to montèlo G n,m,p na mhnparˆgei sqedìn bèbaia to kenì to pl rec grˆfhma. Oi parakˆtw Protˆseic apantoÔn akrib¸c seautì to er¸thma. H apìdeix touc eÐnai arketˆ megˆlh gia touc skopoÔc aut c thc diplwmatik ckai gia to lìgo autì paraleÐpetai. Shmei¸noume aplˆ ìti gia tic apodeÐxeic kai twn duo protˆsewnqrhsimopoieÐtai h mèjodoc thc pr¸thc kai thc deÔterhc rop c (blèpe [2, 17]) kai pio sugkekrimèna oianisìthta Markov kai h anisìthta Chebychev. H pr¸th Prìtash upologÐzei to kat¸fli emfˆnishcpleur¸n.Prìtash 1 (kat¸fli emfˆnishc pleur¸n [10, 21]) 'Estw mia sunˆrthsh g(n) → ∞. IsqÔounta ex c(a) An p(n) =1g(n)n √ m, tìte to G n,m,p eÐnai sqedìn bèbaia to kenì grˆfhma.(b) An p(n) = g(n)n √ m, tìte to G n,m,p èqei sqedìn bèbaia toulˆqiston mia akm .Apìdeixh. Blèpe [21].H epìmenh Prìtash aforˆ to kat¸fli pl rouc graf matoc.Prìtash 2 (kat¸fli pl rouc graf matoc [10, 21]) 'Estw mia sunˆrthsh g(n) → ∞. IsqÔounta ex c(a) An p(n) =(b) An p(n) =√√2 log n−g(n), tìte to Gmn,m,p den èqei toulˆqiston mia pleurˆ.2 log n+g(n), tìte to Gmn,m,p eÐnai sqedìn bèbaia to pl rec grˆfhma.□

1 TUQAŸIA GRAFŸHMATA TOMŸHS ETIKETŸWN 9Epiplèon, axÐzei na shmeiwjeÐ ìti sthn perÐptwsh mp ≥ α log n, gia kˆpoia sugkekrimènhstajerˆ α, ta montèla G n,m,p kai G n,m,λ , me λ ∈ (1 ± ɛ)mp, ɛ ∈ (0, 1) eÐnai parìmoia, me thn ènnoiaìti o arijmìc twn etiket¸n pou epilègei mia koruf eÐnai sqedìn bèbaia perÐpou o Ðdioc kai sta duomontèla. Prˆgmati, eÐnai E|S v | = mp, gia kˆje v ∈ V kai ˆra, qrhsimopoi¸ntac thn anisìthta touBoole 2 kai frˆgmata Chernoff paÐrnoume, gia mia stajerˆ ɛ ∈ (0, 1)P (∃v ∈ V : |S v | /∈ (1 ± ɛ)mp) ≤ n2e −ɛ2 mp/3 ≤ 2n exp{log n − ɛ2 α log n3}→ 0gia α > 3 ɛ 2 . H parat rhsh aut mporeÐ Ðswc na qrhsimopoihjeÐ gia na metafrˆsei kaneÐc idiìthtectou enìc montèlou sto ˆllo. Oi akribeÐc idiìthtec isodunamÐac twn duo montèlwn eÐnai ènaendiafèron prìblhma.1.3 Efarmogèc'Opwc èqoume peÐ, kˆje grˆfhma mporeÐ na anaparastajeÐ apì èna grˆfhma tom c (blèpe [12]).Sunep¸c, ta montèla pou perigrˆyame eÐnai arketˆ genikˆ. Akìma, ìpwc ja doÔme sthn enìthta3, gia orismènec timèc twn paramètrwn m, p (m = n α , α > 6), oi q¸roi G n,m,p kai G n,p eÐnaiisodÔnamoi. Sugkekrimèna, oi J.A. Fill, E.R. Sheinerman kai K.B Singer-Cohen apèdeixan sto [9]ìti h sunolik apìluth apìklish (total variation distance) twn duo montèlwn teÐnei sto 0 ìtann → ∞.Prèpei akìma na parathr soume ìti se merikèc efarmogèc pou montelopoioÔntai me graf mata,h anexarthsÐa emfˆnishc twn akm¸n pou proôpojètei to montèlo G n,p den mporeÐ na dikaiologhjeÐapìluta. Antijètwc, to nèo montèlo G n,m,p (sto opoÐo oi akmèc den eÐnai anexˆrthtec metaxÔ touc)mporeÐ na montelopoi sei me arketˆ megalÔterh akrÐbeia tètoiec efarmogèc. Prˆgmati, upˆrqounpeript¸seic sth prˆxh ìpou ènac prˆktorac epikoinwnÐac (ìpwc gia parˆdeigma ènac asÔrmatockìmboc) èqei prìsbash mìno se orismènec (statistikˆ) jÔrec mèsa apì èna sÔnolo jur¸n epikoinwnÐac.'Otan ènac ˆlloc prˆktorac epikoinwnÐac epilègei mia jÔra pou èqei epilèxei kai kˆpoiocˆlloc, tìte dhmiourgeÐtai èmmesa ènac sÔndesmoc epikoinwniac metaxÔ touc. Me autì ton trìpodhmiourgeÐtai èna grˆfhma epikoinwnÐac (me korufèc touc epikoinwniakoÔc prˆktorec) to opoÐomoiˆzei me èna tuqaÐo grˆfhma tom c. Epiplèon, akìma kai epidhmiologikˆ fainìmena (ìpwc giaprˆdeigma h diˆdosh enìc ioÔ) mporoÔn na afairetikopoihjoÔn me pio akrib trìpo apì to montèloG n,m,p (apì ìti me to G n,p ), epeid mporeÐ na anaparistˆ thn katˆ kˆpoio trìpo {suggèneia} duokìmbwn. Tèloc, ˆllec efarmogèc pou mporoÔn na montelopoihjoÔn me èna montèlo tuqaÐwn grafhmˆtwntom c eÐnai h qrhsimpoÐhsh twn pìrwn se katanemhmèna sust mata, oi allhlepidrˆseickinht¸n praktìrwn pou diasqÐzoun to diadÐktuo ktl.2 H anisìthta tou Boole lèei ìti h pijanìthta thc ènwshc gegonìtwn eÐnai mikrìterh Ðsh apì to ˆjroisma twnpijanot twn twn gegonìtwn aut¸n.

2 ORISMŸENES IDIŸOTHTES STO MONTŸELO G N,M,P 102 Orismènec idiìthtec sto montèlo G n,m,pTo montèlo G n,m,p analÔjhke diexodikˆ sth dhmosÐesh [10] kai sth didaktorik diatrib [21].3Sthn enìthta aut parajètoume merikˆ apotelèsmata sqetikˆ me to montèlo G n,m,p , kaj¸c kaisuzhtˆme th sqèsh pou èqoun autˆ me antÐstoiqa apotelèsmata sto montèlo G n,p . Mia sugkentrwmènhanalutik parousÐash twn diˆforwn apotelesmˆtwn gia to montèlo G n,p mporeÐ kaneÐc nabreÐ sto [4].2.1 Sunart seic katwflioÔ Ôparxhc mikr¸n upografhmˆtwnKatarq n, ac orÐsoume kˆpoiec ènnoiec pou ja mac qreiastoÔn se aut thn enìthta. 'Estw G =(V, e(G)) èna grˆfhma. Lème ìti to C = {C 1 , C 2 , . . . , C k } eÐnai èna skèpasma klik¸n (clique cover)tou G an kˆje C i eÐnai mia klÐka tou G kai kˆje pleurˆ e ∈ e(G) an kei se toulˆqiston mia klÐka C i .Gia parˆdeigma, an L j , j ∈ M eÐnai to sÔnolo twn koruf¸n pou èqoun epilèxei thn etikèta j se ènatuqaÐo grˆfhma tom c, tìte eÐnai profanèc ìti to C = {L 1 , L 2 , . . . , L m } eÐnai èna skèpasma klik¸ntou G. Lème akìma ìti èna skèmasma klik¸n C eÐnai ameÐwto (irreducible) an kamÐa apì tic klÐkecC i ∈ C den perièqetai olìklhrh se kˆpoio uposÔnolo tou C pou den perièqei th C i . Gia parˆdeigma,an se èna tuqaÐo grˆfhma tom c èqoume |L i | ≤ 1, tìte to C = {L 1 , L 2 , . . . , L m } den apoteleÐ ameÐwtoskèpasma klik¸n tou G. Tèloc, dedomènou enìc kalÔmatoc klik¸n C kai enìc S ⊆ V , onomˆzoumekˆluma klik¸n periorismèno (restricted) sto S to C[S] = {C i ∩ S : |C i ∩ S| ≥ 1, i = 1, 2, . . . , k}.Afair¸ntac apì to C[S] ìla ta stoiqeÐa pou antistoiqoÔn se klÐkec (sto S) megèjouc 1 paÐrnoumeto sÔnolo C ′ [S] = {C i ∩ S : |C i ∩ S| ≥ 2, i = 1, 2, . . . , k}.Gia thn perÐptwsh mp 2 → 0, to parakˆtw je¸rhma dÐnei to kat¸fli Ôparxhc gia èna opoid potegrˆfhma H me stajerì arijmì koruf¸n sto montèlo G n,m,p .Je¸rhma 2 ([10, 21]) Gia èna opoiod pote grˆfhma H me stajerì arijmì koruf¸nτ H = minCmax∅̸=S⊆V{max{1n |S|/ ∑ |C[S]|m |C[S]|/ ∑ , 1|C[S]|n |S|/ ∑ |C ′ [S]|m |C′ [S]|/ ∑ |C ′ [S]|eÐnai to kat¸fli emfˆnishc tou H san èna epagìmeno upogrˆfhma tou G n,m,p , ìtan mp 2 → 0. Tomin ston tÔpo upologÐzetai anˆmesa se ìla ta ameÐwta kalÔmata klik¸n tou H. Sunep¸c, gia miaopoiad pote sunˆrthsh g(n) → ∞, eÐnai• an p = τ H /g(n), tìte to H emfanÐzetai sqedìn sÐgoura san epagìmeno upogrˆfhma tou G n,m,p• an p = g(n)τ H , tìte eÐnai sqedìn bèbaio ìti to H den emfanÐzetai san epagìmeno upogrˆfhmatou G n,m,p .Apìdeixh. Blèpe [10, 21].3 Diˆforec parallagèc twn tuqaÐwn grafhmˆtwn tom c, ìpwc ta tuqaÐa graf mata diasthmˆtwn (random intervalgraphs) exetˆsthkan stic ergasÐec [19] kai [20].}}□

2 ORISMŸENES IDIŸOTHTES STO MONTŸELO G N,M,P 11PÐnakac 1: Sunart seic katwflioÔ gia thn emfˆnish orismènwn grafhmˆtwn sto G n,m,pGrˆfhmaZ h : kÔkloc h ≥ 4 koruf¸n p =1n 1/2 m 1/2K h : pl rec grˆfhma h ≥ 2 koruf¸n p =K h,k : pl rec dimerèc grˆfhma h ≥ k koruf¸n p =P h : monopˆti h ≥ 2 koruf¸n p =K 1,h : astèri h + 1 ≥ 2 koruf¸n p =Sunˆrthsh KatwflioÔ τ H⎧⎨⎩⎧⎨⎩⎧⎨⎩⎧⎨⎩1nm 1/h1n 1/(h−1) m 1/21n 1/h m1n (h+k)/(2hk) m 1/21n 1/2 m (h−1)/2(h−2)1n h/2(h−1) m 1/21n 1/h m1n (h+1)/(2h) m 1/2, α ≤ 2hh−1, α > 2hh−1, α ≤ h−khk, α > h−khk, α ≤ h−2h−1, α > h−2h−1, α ≤ h−1h, α > h−1h

2 ORISMŸENES IDIŸOTHTES STO MONTŸELO G N,M,P 12Qrhsimopoi¸ntac to Je¸rhma 2 mporoÔme na upologÐsoume th sunˆrthsh katwflioÔ emfˆnishcenìc opoioud pote mikroÔ graf matoc H. O pÐnakac 1 deÐqnei tic sunart seic katwflioÔ gia thnemfˆnish kˆpoiwn sugkekrimènwn grafhmˆtwn sto montèlo G n,m,p , sthn perÐptwsh pou m = n α ,ìpou α > 0 eÐnai mia stajerˆ. Sugkekrimèna, gia p polÔ megalÔtero apì th sunˆrthsh katwflioÔ,to kˆje stigmiìtupo tou montèlou G n,m,p perièqei sqedìn bèbaia to antÐstoiqo grˆfhma. AntÐjeta,gia p polÔ mikrìtero apì th sunˆrthsh katwflioÔ, eÐnai sqedìn bèbaio ìti kˆje stigmiìtupo toumontèlou G n,m,p den perièqei to antÐstoiqo grˆfhma. Gia ton upolismì twn sunart sewn katwflioÔtou pÐnaka (blèpe [21]) jewr jhke ìti to h eÐnai stajerì kai epomènwc milˆme gia graf mata poueÐnai polÔ mikrìtera se mègejoc apì to ìlo grˆfhma. H upìjesh aut ègine gia na aplopoihjoÔnsugkekrimènoi asumptwtikoÐ upologismoÐ stic apodeÐxeic.PrÐn proqwrÐsoume se peript¸seic ìpou to mp 2 den teÐnei sto 0, axÐzei na sugkrÐnoume tic sunartseic katwflioÔ tou pÐnaka 1 me tic sunarthseic katwflioÔ gia thn emfˆnish twn antÐstoiqwngrafhmˆtwn sto montèlo G n,ˆp tou Erdös. H sÔgkrish ja gÐnei me bˆsh to mèso arijmì pleur¸ntou kˆje montèlou sto kat¸fli emfˆnishc twn antÐstoiqwn grafhmˆtwn. 'Opwc eÐnai gnwstì apìta [2, 4], to kat¸fli emfˆnishc enìc opoioud pote graf matoc H me stajerì arijmì koruf¸n wcepagìmeno upogrˆfhma tou montèlou G n,ˆp eÐnai Ðso me n −1/ρ∗ (H) , ìpou ρ ∗ (H) eÐnai mia posìthtapou onomˆzetai mègisth puknìthta (maximum density) kai dÐnetai apì ton tÔpoρ ∗ (H) = maxL≤H|e(L)||V (L)| .To max ston tÔpo autì upologÐzetai anˆmesa se ìla ta upograf mata L tou H.Qrhsimopoi¸ntac thn pijanìthta Ôparxhc miac pleurˆc tou G n,m,p pou upologÐsame sthn enìthta1.2.2, mporoÔme eÔkola na broÔme ìti o mèsoc arijmìc twn pleur¸n tou G n,m,p sthn perÐptwshmp 2 → 0 eÐnaiE[#e(G n,m,p )] =( n2)(1 − (1 − p 2 ) m ) ∼ n22 mp2ìpou qrhsimpoi same to mp 2 → 0 gia na pˆroume (1 − p 2 ) m ∼ 1 − mp 2 . Akìma, o mèsoc arijmìctwn pleur¸n tou G n,ˆp eÐnaiE[#e(G n,ˆp )] =( n2)ˆp ∼ n22 ˆp.Blèpoume loipìn ìti sthn perÐptwsh tou kÔklou Z h , èqoume E[#e(G n,m,τZh )] ∼ n2 m 1 = n 2 nm 2.Epiplèon, epeid ρ ∗ (Z h ) = 1, eÐnai E[#e(G n,1/n )] ∼ n 2. Blèpoume loipìn ìti o mèsoc arijmìcpleur¸n sta duo montèla eÐnai perÐpou o Ðdioc. Akìma ìmwc kai stic peript¸seic twn K h , K h,k , P hkai K 1,h , ìtan to α eÐnai arketˆ megˆlo (tìso ìso na koitˆme sto deÔteo skèloc thc sunˆrthshckatwflioÔ), ta duo montèla èqoun perÐpou ton Ðdio arijmì pleur¸n. Gia parˆdeigma, gia to K h,kèqoume E[#e(G n,m,τKh,k )] ∼ n2 m 1= n2 1. Epiplèon, epeid ρ ∗ (K2 n (h+k)/(hk) m 2 n (h+k)/(hk) h,k ) =h+k, hk eÐnaiE[#e(G n,n −(h+k)/hk)] ∼ n2 1. Autì to fainìmeno den eÐnai se kamÐa perÐptwsh tuqaÐo. Sthn2 nenìthta 3 ja anaferjoÔme (h+k)/(hk) ektenèstera se autì.

2 ORISMŸENES IDIŸOTHTES STO MONTŸELO G N,M,P 13Sthn perÐptwsh pou h posìthta mp 2 teÐnei se mia stajerˆ (dhlad milˆme gia ligo pio {puknˆ}G n,m,p graf mata), tìte faÐnetai ìti opoiod pote grˆfhma H me stajerì arijmì koruf¸n emfanÐzetaisto G n,m,p wc epagìmeno upogrˆfhma sqedìn sÐgoura. To Je¸rhma 3 tonÐzei thn idiaiterìthtatou montèlou G n,m,p se aut thn perioq tim¸n twn m, p.Je¸rhma 3 ([10, 21]) 'Estw H èna grˆfhma me stajerì arijmì koruf¸n kai èstw ɛ ≤ mp 2 ≤1/ɛ, gia kˆpoia jetik stajerˆ ɛ. Tìte to H emfanÐzetai sqedìn sÐgoura san epagìmeno upogrˆfhmatou G n,m,p , ìpou to p an kei sth perioq tim¸n gia {qr sima} graf mata (dhlad mh kenˆ kai mhpl rh).Apìdeixh. Blèpe [10, 21].□Tèloc, to Je¸rhma 4 exetˆzei th perÐptwsh mp 2 → ∞, ìtan dhlad to G n,m,p eÐnai arketˆpuknì. H sunˆrthsh katwflioÔ se aut thn perÐptwsh exartˆtai apì th mègisth puknìthta tou¯H, tou sumplhrwmatikoÔ graf matoc tou H.Je¸rhma 4 ([10, 21]) 'Estw H èna grˆfhma me stajerì arijmì koruf¸n kai èstw mp 2 → ∞.Tìte, gia mia opoiad pote sunˆrthsh g(n) → ∞, eÐnai√• an p = log n−g(n), tìte toρ ∗ ( ¯H)m H emfanÐzetai sqedìn sÐgoura san epagìmeno upogrˆfhma touG n,m,p√log n+g(n), tìte eÐnai sqedìn bèbaio ìti toρ ∗ ( ¯H)mH den emfanÐzetai san epagìmeno upogrˆfhmatou G n,m,p .• an p =Apìdeixh. Blèpe [10, 21].□'Opwc blèpoume apì to Je¸rhma 4, to kat¸fli emfˆnishc tou H eÐnai antistr¸fwc anˆlogo methn tetragwnik rÐza thc mègisthc puknìthtac tou sumplhrwmatikoÔ tou H. Autì eÐnai diaisjhtikˆswstì epeid to G n,m,p eÐnai sqetikˆ puknì ìtan mp 2 → ∞ kai ˆra ìsec ligìterec pleurèc èqeito H, tìso mikrìtero ja prèpei na eÐnai to p.O pÐnakac 2 sunoyÐzei ta apotelèsmata twn Jewrhmˆtwn 2, 3 kai 4.2.2 Sunˆrthsh katwflioÔ sunektikìthtac'Ena apì ta apotelèsmata tou [21] aforˆ kai to kat¸fli sunektikìthtac enìc omoiìmorfou tuqaÐougraf matoc tom c. Sth sunèqeia dÐnoume qwrÐc apìdeixh to antÐstoiqo Je¸rhma.Je¸rhma 5 ([21]) To kat¸fli sunektikìthtac gia to montèlo G n,m,p eÐnai

2 ORISMŸENES IDIŸOTHTES STO MONTŸELO G N,M,P 14PÐnakac 2: Sunart seic katwflioÔ gia thn emfˆnish enìc graf matoc H me stajerì arijmì koruf¸nsto G n,m,pposìthta mp 2Sunˆrthsh KatwflioÔ{mp 2 → 0 min C max ∅̸=S⊆V{maxɛ ≤ mp 2 ≤ 1/ɛ -1n |S|/ ∑ |C[S]| m |C[S]|/ ∑ 1|C[S]| ,n |S|/ ∑ |C ′[S]| m |C′ [S]|/ ∑ |C ′ [S]|}}mp 2 → ∞√log nρ ∗ ( ¯H)m⎧log n⎪⎨ m, α ≤ 1τ c = √⎪⎩ log nnm , α > 1Sugkekrimèna, gia kˆpoia sunˆrthsh g(n) → ∞,• an α ≤ 1 kai p =• an α ≤ 1 kai p =• an α > 1 kai p =• an α > 1 kai p =Apìdeixh. Blèpe [21].log n−g(n), tìte to Gmn,m,p eÐnai sqedìn sÐgoura mh sunektikì, tìte to G n,m,p eÐnai sqedìn sÐgoura sunektikìlog n+g(n)m√√log n−g(n), tìte to Gnmn,m,p eÐnai sqedìn sÐgoura mh sunektikìlog n+g(n), tìte to Gnmn,m,p eÐnai sqedìn sÐgoura sunektikì.Shmei¸noume ìti to kat¸fli sunektikìthtac gia to montèlo G n,ˆp tou Erdös eÐnai ˆp c = log nnˆra o mèsoc arijmìc twn pleur¸n tou G n,ˆp sto kat¸fli sunektikìthtac eÐnai□kai( nE[#e(G n,ˆpc )] = c ∼2)ˆp n22 ˆp c = n log n .2Epiplèon, apì to Je¸rhma 5, o mèsoc arijmìc pleur¸n tou montèlou G n,m,p sto kat¸fli sunektikìthtacìtan mp 2 → 0

2 ORISMŸENES IDIŸOTHTES STO MONTŸELO G N,M,P 15E[#e(G n,m,τc )] =( n2)(1 − (1 − p 2 ) m ) ∼ n22 mp2 =⎧⎨⎩n 2 log 2 n2m, α ≤ 1n log n2, α > 1ParathroÔme loipìn ìti sthn perÐptwsh α > 1 kai mp 2 → 0 o mèsoc arijmìc twn pleur¸n twnduo montèlwn sto kat¸fli sunektikìthtac eÐnai perÐpou o Ðdioc. AntÐjeta, sthn perÐptwsh α ≤ 1(dhlad m ≤ n), to montèlo G n,m,p qreiˆzetai na eÐnai arketˆ pio puknì apì to montèlo tou Erdösgia na gÐnei sunektikì. Apì aut thn parat rhsh kaj¸c kai apì th suz thsh se prohgoÔmenhenìthta eÐnai eÔlogo na anarwthjoÔme an upˆrqei kˆpoio pedÐo tim¸n twn paramètrwn m kai p giato opoÐo oi q¸roi G n,m,p kai G n,p eÐnai isodÔnamoi. Se autì akrib¸c to er¸thma apantˆei h enìthta3.

3 ISODUNAMŸIA TWN MONTŸELWN G N,P KAI G N,M,P ([9]) 163 IsodunamÐa twn montèlwn G n,p kai G n,m,p ([9])Sthn enìthta aut ja asqolhjoÔme me thn isodunamÐa twn montèlwn G n,p kai G n,m,p gia thnperÐptwsh pou m = n α . Gia thn anˆlush jewroÔme ìti h pijanìthta epilog c twn etiket¸n apìtic korufèc eÐnai tètoia ¸ste to grˆfhma G n,m,p na mhn eÐnai kenì kai na mhn eÐnai pl rec, dhlad2 log n−ω(n), ìpou ω(n) eÐnai mia sunˆrthsh pou teÐnei osod pote argˆ sto ˆpeiro.mEpiplèon jewroÔme ìti α > 6, opìte m ≫ n 6 . Me autèc tic upojèseic mporoÔme eÔkola na deÐxoumeìti√ω(n)n √ ≤ p ≤ m• np ≤ √ 2 log nn 2 → 0• mn 3 p 3 ≤ mn ( ) 3 2 log n 3/2m =2 √ 2n 3 (log n) 3/2→ 0.m 1/2Gia thn apìdeixh thc isodunamÐac twn duo montèlwn ja qrhsimopoi soume thn ènnoia thc sunolikc apìluthc apìklishc (total variation distance) thn opoÐa orÐzoume parakˆtw.Orismìc 5 'Estw ìti X, Y eÐnai tuqaÐec metablhtèc pou paÐrnoun timèc se èna koinì sÔnolo S.'Estw akìma L(X), L(Y ) ta mètra pijanot twn sto S twn opoÐwn oi timèc se èna uposÔnolo A touS eÐnai P (X ∈ A) kai P (U ∈ A) antÐstoiqa. H sunolik apìluth apìklish metaxÔ twn L(X) kaiL(Y ) orÐzetai wc ex c∥∥ L(X) − L(Y ) = max |P (X ∈ A) − P (U ∈ A)|.A⊆S'Enac isodÔnamoc trìpoc èkfrashc thc sunolik c apìluthc apìklishc eÐnai o parakˆtw (ˆlloiisodÔnamoi orismoÐ thc Ðdiac posìthtac mporoÔn na brejoÔn sto [1])∥ L(X) − L(Y )∥ ∥ =12∑|P (X = x) − P (U = x)|.x∈SShmei¸noume ìti sthn perÐptws mac oi tuqaÐec metablhtèc X, Y paÐrnoun timèc apì to sÔnolotwn grˆfwn n koruf¸n, dhlad S = G n . Blèpoume loipìn ìti h ènnoia thc sunolik c apìluthcapìklishc mporeÐ na qrhsimopoihjeÐ wc èna mètro sÔgkrishc metaxÔ giaforetik¸n montèlwngrafhmˆtwn.H epìmenh prìtash anagnwrÐzei èna gegonìc pou sumbaÐnei me mhdenik pijanìthta kai sunep¸cmporoÔme na jewr soume ìti de sumbaÐnei.Prìtash 3 'Estw T to gegonìc Ôparxhc toulˆqiston miac st lhc tou pÐnaka R n,m,p me toulˆqistontrÐa 1. Tìte P (T ) = O(mn 3 p 3 ) = o(1).Apìdeixh. 'Estw X o arijmìc twn triˆdwn ˆsswn pou emfanÐzontai stic st lec tou pÐnaka R n,m,p(upojètoume dhlad ìti ènac ˆssoc mporeÐ na katametrhjeÐ pollaplèc forèc se diaforetikèc triˆdec).EÐnai profanèc ìti to gegonìc T pragmatopoieÐtai an kai mìnon eˆn X > 0. Allˆ apì

3 ISODUNAMŸIA TWN MONTŸELWN G N,P KAI G N,M,P ([9]) 17th grammikìthta thc mèshc tim c èqoume ìti E[X] = m ( n3Markov kai th mèjodo thc pr¸thc rop c (blèpe [2, 17])P (T ) = P (X > 0) ≤ E[X] ∼ mn3 p 3)p 3 ∼ mn3 p 36→ 0.6. 'Ara, apì thn anisìthta□H parapˆnw prìtash mac epitrèpei na estiˆsoume thn prosoq mac se pijanìthtec gegonìtwnpou an koun sto T c afoÔP (G ∈ A) = P (G ∈ A|T c )(1 − O(mn 3 p 3 )) + O(mn 3 p 3 ) = P (G ∈ A|T c ) + o(1).Apì ed¸ kai sto ex c ja qrhsimopoioÔme to sumbolismì P T c(A) gia thn pijanìthta pragmatopoÐhshcenìc gegonìtoc A dedomènou ìti èqei sumbeÐ to T c , dhlad dedomènou ìti o pÐnakac R n,m,pden perièqei triˆdec ˆsswn se kamÐa apì tic st lec tou. Qrhsimopoi¸ntac autì to sumbolismìmporoÔme na ekfrˆsoume to apotèlesma thc Prìtashc 3 wc ex c∥ L(Gn,m,p ) − L T c(G n,m,p ) ∥ ∥ → 0.Prèpei ed¸ na parathr soume ìti oi st lec tou pÐnaka R n,m,p paramènoun anexˆrthtec akìma kaidedomènou tou T c . 'Estw loipìn p e h pijanìthta dhmiourgÐac thc pleurˆc e apì mia sugkekrimènhst lh, dedomènou ìti èqei sumbeÐ to T c , dhladTìtep e = P T c(h e parˆgetai apì mia sugkekrimènh st lh).p e ===P (h st lh èqei akrib¸c duo 1 sta ˆkra thc e)P (h st lh den èqei triˆdec ˆsswn)p 2 (1 − p) n−2(1 − p) n + np(1 − p) n−1 + ( )n p2 (1 − p)2n−2p 2(1 − p) 2 + np(1 − p) + ( n2)p2 . (1)Epeid t¸ra eÐnai p → 0 kai np → 0, apì thn (1) èqoume p e ∼ p 2 . Akìma, afoÔ oi m sunolikˆst lec tou R n,m,p eÐnai anexˆrthtec metaxÔ touc (akìma kai metˆ thn pragmatopoÐhsh tou T c ), hpijanìthta emfˆnishc (apì opoiad pote st lh) miac pleurˆc e dedomènou tou T c eÐnaiˆp = P T c(upˆqei h pleurˆ e sto G n,m,p ) = 1 − (1 − p e ) m .Shmei¸noume ìti to ˆp eÐnai Ðdio gia opoiad pote apì tic ( 2) npleurèc jewr soume. EÐmaste t¸raètoimoi na diatup¸soume to parakˆtw apotèlesma.

3 ISODUNAMŸIA TWN MONTŸELWN G N,P KAI G N,M,P ([9]) 18Je¸rhma 6 (IsodunamÐa Montèlwn gia Megˆlo√α) 'Estw m = n α kai α > 6. 'Estw akìmaìti to p = p(n) an kei sthn perioq ω(n)n √ ≤ p ≤ 2 log n−ω(n), ìpou ω(n) eÐnai mia sunˆrthsh poum mteÐnei osod pote argˆ sto ˆpeiro kai èstw ˆp = 1 − (1 − p e ) m . Tìte∥ L(Gn,ˆp ) − L(G n,m,p ) ∥ ∥ → 0 kaj¸c n → ∞.Apìdeixh. Gia th apìdeixh orÐzoume duo tuqaÐec metablhtèc pou ja mac fanoÔn qr simec. GiaeukolÐa jètoume N = ( 2) n. 'Estw loipìn Y mia tuqaÐa metablht pou akoloujeÐ th duwnumikkatanom me paramètrouc N kai ˆp, dhlad'Estw akìmaY ∼ B(N, ˆp).X =N∑i=1ìpou X 1 , . . . , X N eÐnai deÐktriec tuqaÐec metablhtèc gia ta elafr¸c exarthmèna gegonìta sto ex csq ma: Ac jewr soume èna paiqnÐdi sto opoÐo upˆrqoun N + 1 koupìnia c 1 , . . . , c N+1 apì ta opoÐato c N+1 eÐnai kenì kai ìla ta upìloipa kerdÐzoun. O monadikìc paÐqthc tou paiqnidioÔ trabˆei mforèc me epanatopojèthsh èna apì autˆ ta koupìnia sÔmfwna me thn katanomX iP (c i ) = p e , gia i = 1, . . . , NP (c N+1 ) = 1 − Np e .Shmei¸noume ìti afoÔ p e ∼ p 2 , ja eÐnai Np e ∼ (np)2 → 0. Jètoume t¸ra2{ 1 an to cX i =i epilègetai toulˆqiston mia forˆ sta m trab gmata0 alli¸c.'Ara, h tuqaÐa metablht X eÐnai o arijmìc twn diaforetik¸n eid¸n kouponi¸n pou mˆzeye o paÐqthcme m trab gmata. EÐnai eÔkolo na diapist¸sei kaneÐc qrhsimopoi¸ntac th grammikìthta thc mèshctim c ìti E[X i ] = ˆp kai E[X] = N ˆp = E[Y ]. Sunep¸c, h tuqaÐa metablht X antistoiqeÐ stonarijmì twn pleur¸n sto G n,m,p dedomènou ìti èqei sumbeÐ to T c , dhladP T c(|e(G n,m,p | = M) = P (X = M). (2)'Estw t¸ra ìti o pÐnakac R n,m,p den perièqei triˆdec ˆsswn. Tìte eÐnai eÔkolo na diapist¸seikaneÐc ìti kˆje grˆfoc n koruf¸n me M pleurèc èqei thn Ðdia pijanìthta emfˆnhshc. Prˆgmati,an jewr soume ìti èna grˆfhma tom c etiket¸n eÐnai èna grˆfhma me pollaplèc akmèc metaxÔ twnkoruf¸n (mia akm upˆrqei dhlad tìsec forèc ìsec kai oi etikètec pou thn parˆgoun), tìte kˆjetètoia akm katalambˆnei me ˆssouc akrib¸c duo jèseic tou pÐnaka R n,m,p , oi opoÐec mˆlista eÐnai

3 ISODUNAMŸIA TWN MONTŸELWN G N,P KAI G N,M,P ([9]) 19diaforetikèc apì tic jèseic pou katalambˆnoun oi upìloipec akmèc. SumperaÐnoume loipìn ìti hpijanìthta emfˆnhsh opoioud pote uposunìlou tou sunìlou twn grˆfwn me M akmèc eÐnai Ðdiame thn pijanìthta emfˆnhshc tou sugkekrimènou uposunìlou sthn perÐptwsh pou to montèlo pouqrhsimopoioÔme eÐnai to G n,M . 'Ara loipìn katal goume sth parakˆtw sqèsh:P T c(G n,m,p ∈ A ∣ ∣ |e(Gn,m,p )| = M) = P (G n,M ∈ A). (3)Gia na aplopoi soume thn apìdeixh thc sqèshc ∥ ∥ L(Gn,ˆp )−L(G n,m,p ) ∥ ∥ → 0 parajètoume merikècparathr seic gia th sunolik apìluth apìklish.1. H sunolik apìluth apìklish eÐnai mia metrik sto q¸ro ìlwn twn mètrwn pijanot twn stoS.2. An Z eÐnai mia tuqaÐa metablht pou paÐrnei timèc ston Ðdio q¸ro me mia ˆllh tuqaÐa metablhtY , tìte eÐnai∥∥L(X) − L(Y ) ∥ ∥ ≤ ∑ zP (Z = z) ∥ ∥L(X) − L(Y ∣ ∣Z = z) ∥ ∥. (4)Prˆgmati, gia opoid pote A ⊆ S èqoume|P (X ∈ A) − P (Y ∈ A)| = |P (X ∈ A) − ∑ zP (Z = z)P (Y ∈ A ∣ ∣ Z = z)|= | ∑ z≤ ∑ z≤ ∑ zP (Z = z) ( P (X ∈ A) − P (Y ∈ A ∣ ∣ Z = z))|P (Z = z)|P (X ∈ A) − P (Y ∈ A ∣ ∣ Z = z)|P (Z = z) ∥ ∥ L(X) − L(Y∣ ∣Z = z)∥ ∥,ìpou h teleutaÐa anisìthta prokÔptei apì ton pr¸to orismì thc sunolik c apìluthc apìklishc.Tèloc, paÐrnontac to mègisto gia ìla ta A ⊆ S kai efarmìzontac gia ˆllh mia forˆton orismì paÐrnoume thn epijumht sqèsh (4). 43. An upojèsoume ìti upˆrqoun tuqaÐec metablhtèc Z kai Z ′ tètoiec ¸ste,tìteL(X|Z = z) = L(Y |Z ′ = z),gia kˆje z4 H sqèsh aut deÐqnei katˆ kˆpoio trìpo thn kurtìthta (convexity) thc sunolik c apìluthc apìklishc.

3 ISODUNAMŸIA TWN MONTŸELWN G N,P KAI G N,M,P ([9]) 20Prˆgmati, gia kˆje A ⊆ S èqoume∥∥L(X) − L(Y ) ∥ ∥ ≤ 2 ∥ ∥L(Z) − L(Z ′ ) ∥ ∥ . (5)|P (X ∈ A) − P (Y ∈ A)| = | ∑ z= | ∑ z≤ ∑ zP (Z = z)P (X ∈ A ∣ ∑Z = z) − P (Z ′ = z)P (Y ∈ A ∣ Z ′ = z)|(P (Z = z) − P (Z ′ = z)) P (X ∈ A ∣ ∣Z = z)||P (Z = z) − P (Z ′ = z)|P (X ∈ A ∣ ∣Z = z)z≤ ∑ z|P (Z = z) − P (Z ′ = z)|= 2 ∥ ∥ L(Z) − L(Z ′ ) ∥ ∥ (6),ìpou gia thn teleutaÐa isìthta kˆname qr sh tou deÔterou orismoÔ thc sunolik c apìluthcapìklishc. Tèloc, paÐrnontac to mègisto gia ìla ta A ⊆ S kai efarmìzontac ton pr¸toorismì paÐrnoume thn epijumht sqèsh (5).4. An upˆrqei ènac q¸roc pijanot twn ston opoÐo orÐzontai duo metablhtèc X ′ , Y ′ me L(X) =L(X ′ ) kai L(Y ) = L(Y ′ ), tìtePrˆgmati, gia kˆje A ⊆ S èqoume∥ L(X) − L(Y )∥ ∥ ≤ P (X′ ≠ Y ′ ). (7)P (X ∈ A) − P (Y ∈ A) = P (X ′ ∈ A) − P (Y ′ ∈ A)= (P (Y ′ ∈ A, X ′ = Y ′ ) + P (X ′ ∈ A, X ′ ≠ Y ′ )) − P (Y ′ ∈ A)≤ P (Y ′ ∈ A)P (X ′ ≠ Y ′ ) − P (Y ′ ∈ A)= P (X ′ ≠ Y ′ ).Allˆzontac touc rìlouc twn X kai Y sthn parapˆnw sqèsh katal goume sth sqèsh|P (X ∈ A) − P (Y ∈ A)| ≤; P (X ′ ≠ Y ′ )apì thn opoÐa paÐrnoume thn (7) efarmìzontac ton pr¸to trìpo èkfrashc thc sunolik capìluthc apìklishc.

3 ISODUNAMŸIA TWN MONTŸELWN G N,P KAI G N,M,P ([9]) 21Qrhsimopoi¸ntac ta parapˆnw ja prospaj soume na anˆgoume to prìblhma thc apìdeixhc thcsqèshc ∥ ∥ L(Gn,ˆp ) − L(G n,m,p ) ∥ ∥ = o(1) sto prìblhma apìdeixhc thc sqèshc ∥ ∥ L(X) − L(Y )∥ ∥ → 0,ìpou X, Y eÐnai oi tuqaÐec metablhtèc pou orÐsame sthn arq thc apìdeixhc.Katarq n parathroÔme ìti apì thn prìtash 3, thn parat rhsh 1 kai thn trigwnik anisìthta,h sqèsh ∥ ∥ L(Gn,ˆp ) − L(G n,m,p ) ∥ ∥ = o(1) eÐnai isodÔnamh me thn ∥ ∥ L(Gn,ˆp ) − L T c(G n,m,p ) ∥ ∥ = o(1).Allˆ apì thn (3) xèroume ìtikai eÐnai profanèc ìti isqÔeiL T c(G n,m,p∣∣|e(Gn,m,p )| = M) = L(G n,M )L(G n,ˆp∣ ∣|e(Gn,ˆp )| = M) = L(G n,M )afoÔ sto montèlo G n,ˆp ìloi oi grˆfoi me M akmèc èqoun thn Ðdia pijanìthta emfˆnishc.Sunep¸c, apì thn parat rhsh 3 arkeÐ na deÐxoume ìtiAfoÔ ìmwc eÐnai∥ L(|e(Gn,ˆp )|) − L T c(|e(G n,m,p )|) ∥ ∥kai apì th sqèsh (2) eÐnaiL(|e(G n,ˆp )|) = B(N, ˆp) = L(Y )L T c(|e(G n,m,p )|) = L(X)katal goume sto sumpèrasma ìti gia na apodeÐxoume to je¸rhma arkeÐ na deÐxoume ìti ∥ L(X) −L(Y ) ∥ = o(1). Gia na to deÐxoume autì ja qrhsimopoi soume mia trÐth tuqaÐa metablht X(M)gia thn opoÐa ja deÐxoume ìti ∥ ∥ L(X) − L(X(M)) → ∞ kai ∥ ∥ L(X(M)) − L(Y ) → ∞. Qrhsimopoi¸ntacautèc tic sqèseic se sunduasmì me thn trigwnik anisìthta (pou ikanopoieÐ h sunolikapìluth apìklish) katal goume sto epijumhtì apotèlesma.H tuqaÐa metablht X(M) orÐzetai sth sunèqeia. 'Estw M mia tuqaÐa metablht pou akoloujeÐthn katanom Poisson me parˆmetro m. TropopoioÔme to sq ma epilog c kouponi¸n pou d¸samegia ton orismì thc metablht c X ètsi ¸ste o arijmìc twn kouponi¸n pou epilègoume na isoÔtaime thn tuqaÐa metablht M. ParathroÔme bèbaia ìti h mèsh tim tou arijmoÔ autoÔ eÐnai pˆli m.'Oso gia tic tuqaÐec metablhtèc X i , autèc gÐnontai{ 1 an to cX i (M) =i epilègetai toulˆqiston se èna apì ta trab gmata0 alli¸c.kai epiplèonX(M) =N∑X i (M).i=1

3 ISODUNAMŸIA TWN MONTŸELWN G N,P KAI G N,M,P ([9]) 22Se autì to montèlo ac sumbolÐsoume me R i , 1 ≤ i ≤ N, thn tuqaÐa metablht pou antistoiqeÐston arijmì twn trabhgmˆtwn sta opoÐa epilègetai to koupìni i. Gia na broÔme thn katanom autwntwn metablht¸n upologÐzoume thn apì koinoÔ sunˆrthsh puknìthtac pijanìthtac ìpwc faÐnetaisth sunèqeia: 'Estw oi stajeroÐ arijmoÐ k 1 , k 2 , . . . , k N kai s = ∑ Nj=1 k j. Tìte eÐnaiP (R 1 = k 1 , . . . , R N = k N ) =∞∑∣P (M = r)P (R 1 = k 1 , . . . , R N = k N ∣M = r)r=s∞∑()= e m mr rp e (1 − Np e ) r−sr! kr=s1 , k 2 , . . . , k N , r − s= e−m (mp e ) s ∞∑ (m(1 − Np e )) r−sk 1 ! · · · k N ! (r − s)!r=s= e−m (mp e ) s ∞∑ (m(1 − Np e )) rk 1 ! · · · k N ! r!r=0= e−mNpe (mp e ) sk 1 ! · · · k N !N∏ (mp e ) k je −mpe=.k j !j=1Epeid t¸ra epilèxame ta k j aujaÐreta, sumperaÐnoume ìti oi R 1 , R 2 , . . . R N eÐnai anexˆrthtecPoisson tuqaÐec metablhtèc me mèsh tim mp e . 'Ara, gia i = 1, 2, . . . N eÐnaiP (X i (M) = 1) = 1 − P (R i = 0) = 1 − e −mpekai P (X i (M) = 1) = e −mpe . Epiplèon, lìgw thc anexarthsÐac twn R i , oi tuqaÐec metablhtècX i (M), 1 ≤ i ≤ N, ja eÐnai epÐshc anexˆrthtec kai ˆra to ˆjroismˆ touc ja akoloujeÐ th duwnumikkatanom me paramètrouc N kai 1 − e −mpe , dhlad X(M) ∼ B(N, 1 − e −mpe ).Ja prospaj soume t¸ra na zeugar¸soume (couple) tic duwnumikèc tuqaÐec metablhtèc Y kaiX(M). Gia na to petÔqoume autì jewroÔme N anexˆrthtec omoiìmorfa katanemhmènec tuqaÐecmetablhtèc V 1 , V 2 , . . . V N kai jètoumeY i = 1 an kai mìnon eˆn V i ≤ 1 − (1 − p e ) m kaiX i (M) = 1 an kai mìnon eˆn V i ≤ 1 − e −mpe .AfoÔ eÐnai 1 − (1 − p e ) m ≥ 1 − e −mpe , to zeugˆrwma autì exasfalÐzei ìti Y ≥ X(M). 'Ara jaeÐnai

3 ISODUNAMŸIA TWN MONTŸELWN G N,P KAI G N,M,P ([9]) 23P (Y ≠ X(M)) = P { upˆrqei i tètoio ¸ste V i ∈ (1 − e −mpe , 1 − (1 − p e ) m ] }≤ N · P { V i ∈ (1 − e −mpe , 1 − (1 − p e ) m ] }= N ( e −mpe − (1 − p e ) m)ìpou gia th deÔterh sqèsh qrhsimopoi same thn anisìthta Boole. Efarmìzontac t¸ra thn anisìthtaln (1 − x) ≥ −x −x (h opoÐa mporeÐ eÔkola na epalhjeujeÐ), gia x = p2(1−x) 2e kai pollaplasiˆzontacme m, h parapˆnw sqèsh gÐnetaiP (Y ≠ X(M)) ≤ N({e −mpe − exp −mp e − mmp 2 e≤ Ne −mpe2(1 − p e ) 2≤ Nmp 2 e → 0})p 2 e2(1 − p e ) 2gia tic timèc tou p pou upojètei to Je¸rhma. Shmei¸noume ed¸ ìti gia thn deÔterh apì ticparapˆnw anis¸seic qrhsimopoi same th sqèsh 1 − x ≤ e −x .Apì thn parat rhsh 4 gia thn sunolik apìluth apìklish ìmwc èqoume ìti‖L(Y ) − L(X(M))‖ ≤ P (Y ≠ X(M)) → 0.Sth sunèqeia ja sugkrÐnoume tic tuqaÐec metablhtèc X = X(m) kai X(M). Apì th deÔterhparat rhsh sqetikˆ me th sunolik apìluth apìklish èqoume‖L(X(m)) − L(X(M))‖ ≤ ∑ r= ∑ rP (M = r)‖L(X(m)) − L(X(M) ∣ ∣ M = r)‖P (M = r)‖L(X(m)) − L(X(r))‖. (8)Gia na sugkrÐnoume ta L(X(m)) kai L(X(r)) ac jewr soume duo akèraiouc 0 ≤ s ≤ s ′ < ∞.Zeugar¸noume t¸ra tic X(s) kai X(s ′ ) ètsi ¸ste apì ta s ′ trab gmata pou qrhsimopoioÔntai giaton kajorismì thc X(s ′ ), ta pr¸ta s qrhsimopoioÔntai gia ton kajorismì thc X(s). Tìte, apì thnparat rhsh 4 gia th sunolik apìluth apìklish èqoume‖L(X(s ′ )) − L(X(s))‖ ≤ P (X(s) ≠ X(s ′ ))≤ P (X(s − s ′ ) ≤ 1)= 1 − P (X(s − s ′ ) = 0) = 1 − (1 − Np e ) s′ −s .

3 ISODUNAMŸIA TWN MONTŸELWN G N,P KAI G N,M,P ([9]) 24Gia ton upologismì thc pijanìthtac P (X(s − s ′ ) = 0) qrhsimopoi same thn anexarthsÐa metaxÔdiaforetik¸n trabhgmˆtwn kai thn parat rhsh ìti gia na èqoume X(s−s ′ ) = 0 prèpei na epilèxoumekai sta s ′ − s trab gmata to koupìni c N+1 . 'Ara loipìn deÐxame ìti ‖L(X(m)) − L(X(r))‖ ≤1 − (1 − Np e ) |m−r| .Apì thn anisìthta Chebyshev t¸ra èqoume ìtiP (|M − m| < m 1 2 ln m) ≥ 1 −1ln 2 m .H sqèsh (8) mporeÐ t¸ra na grafteÐ wc ex c‖L(X(s ′ )) − L(X(s))‖ ≤∑r:|m−r|≥m 1 2 ln m+∑P (M = r)‖L(X(m)) − L(X(r))‖P (M = r)‖L(X(m)) − L(X(r))‖≤r:|m−r|

3 ISODUNAMŸIA TWN MONTŸELWN G N,P KAI G N,M,P ([9]) 25P (G n,m,p ∈ A) → c an kai mìnon eˆn P (G n,ˆp ∈ A) → ckaj¸c n → ∞, gia kˆje stajerˆ c ∈ [0, 1].Sugkekrimèna, eÐnaikaiP (G n,m,p ∈ A) → 1 an kai mìnon eˆn P (G n,ˆp ∈ A) → 1P (G n,m,p ∈ A) → 0 an kai mìnon eˆn P (G n,ˆp ∈ A) → 0.EÐdame loipìn ìti gia m ≫ n 6 kai p sthn perioq tim¸n gia qr sima graf mata, ta montèlaG n,m,p kai G n,ˆp eÐnai isodÔnama, dhlad eÐnai parìmoia katanemhmèna. O anagn¸sthc mporeÐ nasugkrÐnei to apotèlesma thc enìthtac aut c kai me ta apotelèsmata thc enìthtac 2, ìpou faÐnetaiìti gia orismènec idiìthtec h isodunamÐa twn duo q¸rwn isqÔei kai gia mikrìtera m.

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 264 Anexˆrthta sÔnola koruf¸n: nèa apotelèsmataSthn enìthta aut xefeÔgoume lÐgo apì to klassikì omoiìmorfo montèlo tuqaÐwn grafhmˆtwntom c G n,m,p kai asqoloÔmaste me to genikeumèno montèlo tuqaÐwn grafhmˆtwn tom c G n,m,⃗p ,ìpou ⃗p = [p 1 , p 2 , . . . , p m ]. Sto montèlo autì, upologÐzoume gia pr¸th forˆ sth bibliografÐa thmèsh tim kai th diasporˆ tou arijmoÔ twn anexˆrthtwn sunìlwn koruf¸n megèjouc k (blepe[14]). Gia to skopì autì, qrhsimopoioÔme kˆpoiec endiafèrousec teqnikèc, merikèc apì tic opoÐecqrhsimopoioÔme kai stouc algìrijmouc argìtera 5 .4.1 Sqetikˆ me to mègejoc anexˆrthtwn sunìlwn koruf¸nTo epìmeno Je¸rhma dÐnei ènan akrib tÔpo gia th mèsh tim tou arijmoÔ twn anexˆrthtwn koruf¸nmegèjouc k se èna genikeumèno montèlo tuqaÐwn grafhmˆtwn tom c. Gia thn apìdeixh, blèpoumeto G n,m,⃗p grˆfhma apì thn pleurˆ twn etiket¸n.Je¸rhma 7 ([14]) 'Estw X (k) o arijmìc twn anexˆrthtwn sunìlwn koruf¸n megèjouc k se ènagenikeumèno tuqaÐo grˆfhma tom c G(n, m, ⃗p), ìpou ⃗p = [p 1 , p 2 , . . . , p m ]. EÐnaiE [ X (k)] ( ) n ∏ m(= (1 − pi ) k + kp i (1 − p i ) k−1) .kApìdeixh. 'Estw V ′ èna opoiod pote sÔnolo k koruf¸n kai èstw{ 1 an to VX V ′ =′ eÐnai anexˆrthto sÔnolo koruf¸n0 alli¸c.Profan¸c eÐnaii=1X (k) =∑V ′ ,|V ′ |=kkai ˆra apì th grammikìthta thc mèshc tim c paÐrnoumeE [ X (k)] =( nk)E [X V ′] =X V ′( nk)P {to V ′ eÐnai anexˆrthto sÔnolo koruf¸n}.Gia na upologÐsoume to E [X V ′] arkeÐ na koitˆxoume to grˆfhma G(n, m, ⃗p) apì th meriˆ twnstoiqeÐwn tou M = {1, 2, . . . , m}. Blèpoume tìte ìti to sÔnolo V ′ ja eÐnai anexˆrthto sÔnolokoruf¸n an kai mìnon eˆn kˆje stoiqeÐo tou M epilègetai apì to polÔ mia koruf tou V ′ . AfoÔta stoiqeÐa tou M epilègontai anexˆrthta ja eÐnaiE [X V ′] =m∏i=1P {to stoiqeÐo i ∈ M epilègetai apì to polÔ mia koruf tou V ′ }.5 Ta apotelèsmata thc enìthtac aut c parousiˆsthkan gia pr¸th forˆ sta [14] kai [15].

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 27Akìma ìtan èna stoiqeÐo i ∈ M epilègetai to polÔ mia forˆ apì tic korufèc tou V ′ , tìte eÐteden epilègetai kajìlou, eÐte epilègetai apì akrib¸c mia koruf . 'AraE [X V ′] =m∏i=1(P {kamiˆ koruf den epilègei to i} + P {akrib¸c mia koruf epilègei to i}).Epeid oi epilogèc twn koruf¸n eÐnai anexˆrthtec metaxÔ touc, eÐnai eÔkolo na diapist¸soumeìti h pijanìthta na mhn epilèxei kamÐa koruf tou V ′ to stoiqeÐo i eÐnai (1 − p i ) k , ìpou p i eÐnaih pijanìthta epilog c tou i apì mia opoiad pote koruf . Epiplèon, h pijanìthta na epilegeÐ toi apì akrib¸c mia koruf tou V ′ eÐnai kp i (1 − p i ) k−1 afoÔ upˆrqoun k diaforetikèc korufèc stoV ′ kai h pijanìthta na epilegeÐ to i mìno apì kˆpoia sugkekrimènh apì autèc tic korufèc eÐnaip i (1 − p i ) k−1 .'Ara loipìn apodeÐxame ìtiE [ X (k)] ( ) n ∏ m(= (1 − pi ) k + kp i (1 − p i ) k−1) .ki=1□Sth sunèqeia apodeiknÔoume èna Je¸rhma pou dÐnei ènan akrib tÔpo gia ton upolgismì thcdiasporˆc tou arijmoÔ twn anexˆrthtwn koruf¸n megèjouc k se èna genikeumèno montèlo tuqaÐwngrafhmˆtwn tom c. H apìdeixh qrhsimopoieÐ mia algorijmik teqnik pou sugqwneÔei merikèckorufèc anexˆrthtec metaxÔ touc se mia monadik uperkoruf .Je¸rhma 8 ([14]) 'Estw X (k) o arijmìc twn anexˆrthtwn sunìlwn koruf¸n megèjouc k se ènagenikeumèno tuqaÐo grˆfhma tom c G(n, m, ⃗p), ìpou ⃗p = [p 1 , p 2 , . . . , p m ]. EÐnaiV ar ( X (k)) =k∑( )( ) ( n 2k − sγ(k, s) E [ X (k)] [ ) − E2 X (k)]))2k − s s2s=1ìpou E [ X (k)] eÐnai h mèsh tim tou arijmoÔ twn anexˆrthtwn sunìlwn koruf¸n megèjouc k kaiγ(k, s) =m∏i=1())((1 − p i ) k−s + (k − s)p i (1 − p i ) k−s−1 sp i1 −.1 + (k − 1)p iProof. 'Estw V ′ èna opoiod pote sÔnolo k koruf¸n kai èstw{ 1 an to VX V ′ =′ eÐnai anexˆrthto sÔnolo koruf¸n0 alli¸c.Profan¸c eÐnaiX (k) =∑V ′ ,|V ′ |=kX V ′( nk( nk

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 28kai an V ′1, V ′2 eÐnai duo opoiad pote sÔnola k koruf¸n, tìteAfoÔ eÐnaiV ar ( X (k)) ∑=Cov(X V ′1, X V ′2)V 1 ′,V2 ′,|V1 ′ |=|V 2 ′ |=kk∑ ∑=Cov(X V ′1, X V ′2)s=1 V 1 ′,V2 ′,|V1 ′ |=|V 2 ′ |=k|V 1 ′ ∩V 2 ′ |=sk∑ ∑=E [ ] [ ] [ ]X V ′1X V ′2− E XV ′1E XV ′2s=1 V 1 ′,V2 ′,|V1 ′ |=|V 2 ′ |=k|V 1 ′ ∩V 2 ′ |=sk∑ ∑[=1X V ′2= 1} − E2 X (k)]s=1 V 1 ′,V2 ′,|V1 ′ |=|V 2 ′ |=k|V 1 ′ ∩V ′2 |=s P {X V ′( n2) 2. (9)P {X V ′1X V ′2= 1} = P {X V ′1= 1|X V ′2= 1}P {X V ′2= 1}= P {X V ′1= 1|X V ′2= 1} E [ X (k)]( n2) (10)to prìblhma upologismoÔ thc diasporˆc tou X (k) anˆgetai ston upologismì thc desmeumènhcpijanìthtac P {X V ′1= 1|X V ′2= 1}, dhlad thc pijanìthtac na eÐnai to V 1 ′ anexˆrthto sÔnolokoruf¸n dedomènou ìti to V 2 ′ eÐnai anexˆrthto sÔnolo koruf¸n, ìpou ta V 1, ′ V 2 ′ eÐnai duo sÔnolak koruf¸n me s koinèc korufèc.Gia na upologÐsoume thn P {X V ′1= 1|X V ′2= 1} ja qrhsimopoi soume mia teqnik sugq¸neushcpoll¸n (apl¸n) koruf¸n se mia uper-koruf . 'Opwc ja doÔme, sthn perÐptwsh pou oi korufècpou sugqwneÔontai apoteloÔn anexˆrthto sÔnolo, h pijanotik sumperiforˆ thc antÐstoiqhc uperkorufc mporeÐ na kajoristeÐ pl rwc.Ac jewr soume loipìn èna sugkekrimèno stoiqeÐo i ∈ M kai èstw v 1 , v 2 duo (uper-)korufècoi opoÐec epilègoun to i anexˆrthta me pijanìthtec p (1)i kai p (2)i antÐstoiqa. 'Estw akìma ìti taS v1 , S v2 eÐnai ta sÔnola twn stoiqeÐwn tou M pou èqoun antistoiqhjeÐ stic v 1 kai v 2 antÐstoiqa.Tìte eÐnai

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 29P {i ∈ S v1 |∄(v 1 , v 2 )} = P {i ∈ S v1 , i /∈ S v2 |∄(v 1 , v 2 )}= P {i ∈ S v 1, i /∈ S v2 , ∄(v 1 , v 2 )}P {∄(v 1 , v 2 )}i p (2)i ) m−1i p (2)i= p(1) i (1 − p (2)i )(1 − p (1)(1 − p (1) ) m= p(1) i (1 − p (2)i )1 − p (1)i p (2)iìpou me (v 1 , v 2 ) sumbolÐzoume mia pleurˆ metaxÔ twn v 1 kai v 2 . Gia thn pr¸th apì tic parapˆnwisìthtec qrhsimopoi same ton orismì tou genikeumènou graf matoc tom c kai sugkekrimèna togegonìc ìti gia na mhn upˆrqei mia pleurˆ (v 1 , v 2 ) ja prèpei ta sÔnola S v1 , S v2 na mhn èqounkanèna koinì stoiqeÐo. Sthn trÐth isìthta qrhsimopoi same thn anexarthsÐa metaxÔ twn epilog¸ndiaforetik¸n koruf¸n kaj¸c kai thn anexarthsÐa metaxÔ epilog¸n diaforetik¸n stoiqeÐwn touM.ParathroÔme ìti h parapˆnw isìthta eÐnai anexˆrthth tou m kai ˆra mporoÔme na sumperˆnoumeta ex c:• Dedomènou ìti den upˆrqei pleurˆ (v 1 , v 2 ), h koruf v 1 sumperifèretai san mia koruf pouepilègei to stoiqeÐo i ∈ M anexˆrthta me pijanìthta p(1) i (1−p (2)i ).1−p (1)i p (2)i• Dedomènou ìti den upˆrqei pleurˆ (v 1 , v 2 ), an jewr soume tic korufèc v 1 , v 2 san mia monadikkoruf , tìte h koruf aut sumperifèretai san mia koruf pou epilègei to stoiqeÐo i ∈ Manexˆrthta me pijanìthta(11)P {i ∈ S v1 ∪ S v2 |∄(v 1 , v 2 )} = P {i ∈ S v1 |∄(v 1 , v 2 )} + P {i ∈ S v2 |∄(v 1 , v 2 )}= p(1) i (1 − p (2)i )1 − p (1)i p (2)i= p(1) i + p (2)i − 2p (1)i p (2)i1 − p (1)i p (2)i+ p(2) i (1 − p (1)i )1 − p (1)i p (2)iìpou i èna sugkekrimèno stoiqeÐo tou M. H pr¸th apì tic parapˆnw exis¸seic prokÔpteiapì thn parat rhsh ìti an den upˆrqei pleurˆ metaxÔ twn v 1 kai v 2 , tìte ta sÔnola S v1 , S v2eÐnai xèna metaxÔ touc kai ˆra kanèna stoiqeÐo tou M de mporeÐ na an kei tautìqrona kaista duo. H deÔterh isìthta prokÔptei lìgw summetrÐac.(12)

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 30Ed¸ prèpei na parathr soume ìti an prospajoÔsame na upologÐsoume thn P {i ∈ S v1 |∃(v 1 , v 2 )},dhlad thn pijanìthta èna sugkekrimèno stoiqeÐo i na an kei sto S v1 dedomènou ìti upˆrqei akm(v 1 , v 2 ), tìte ja paÐrname mia èkfrash pou ja exart¸tan apì to mègejoc tou M kai de jamporoÔsame na bgˆloume sumperˆsmata antÐstoiqa me ta parapˆnw (parˆ mìno proseggistikˆ).Ac prospaj soume t¸ra na en¸soume merikèc korufèc tou grˆfou G(n, m, ⃗p), oi opoÐec denèqoun metaxÔ touc akmèc, se mia uper-koruf . Eidikìtera, ac sumbolÐsoume me w j mia uper-korufpou perièqei j aplèc korufèc pou apoteloÔn anexˆrthto sÔnolo, dhlad den èqoun korufèc metaxÔtouc. EÐnai profanèc ìti lìgw summetrÐac h seirˆ sugq¸neushc twn j koruf¸n den ephrreˆzeithn pijanotik sumperiforˆ thc w j . Akìma, an w j1 , w j2 eÐnai uper-korufèc pou antistoiqoÔn seduo anexˆrthta sÔnola koruf¸n pou eÐnai xèna metaxÔ touc, lème ìti upˆrqei akm (w j1 , w j2 ) ankai mìnon eˆn upˆrqei akm metaxÔ miac apl c koruf c thc w j1 kai miac apl c koruf c thc w j2 .Sunep¸c, to gegonìc {∄(w j1 , w j2 )} eÐnai isodÔnamo me to {oi korufèc thc w j1 mazÐ me autèc thcw j2 apoteloÔn anexˆrthto sÔnolo}.Parathr¸ntac ìti mia uper-koruf w 1 eÐnai aplˆ mia koruf kai qrhsimopoi¸ntac thn exÐswsh(12) mporoÔme na deÐxoume me epagwg sto mègejoc j thc uperkoruf c w j ìtiP {i ∈ S wj } =(13)1 + (j − 1)p iìpou i eÐnai èna sugkekrimèno stoiqeÐo tou M kai S wj eÐnai h ènwsh ìlwn twn uposunìlwn tou Mpou antistoiqoÔn stic korufèc pou an koun sth w j , dhladS wj= ⋃v∈w jS v .ParathroÔme ìti apo ton orismì thc w j kai ton orismì tou genikeumènou graf matoc tom c, tasÔnola S v sthn parapˆnw ènwsh eÐnai xèna metaxÔ touc.Lambˆnontac upìyhn ta parapˆnw, èstw V 1 ′ èna opoiod pote sÔnolo k (apl¸n) koruf¸n kaièstw V 2 ′ èna anexˆrthto sÔnolo k koruf¸n pou èqei s koinèc korufèc me to V 1. ′ AfoÔ den upˆrqounakmèc metaxÔ twn koruf¸n tou V 2, ′ mporoÔme na jewr soume tic k − s korufèc tou V 2 ′ pouden an koun sto V 1 ′ kai tic s korufèc pou an koun kai sto V 1 ′ kai sto V 2 ′ san duo xeqwristècuperkorufèc w k−s kai w s antÐstoiqa, oi opoÐec mˆlista den èqoun pleurˆ metaxÔ touc.'Ara, apì tic exis¸seic (11), (12) kai (13), h w s sumperifèretai sa mia monadik koruf ,thn opoÐa sumbolÐzoume me w s, ′ h opoÐa epilègei kˆje stoiqeÐo tou M anexˆrthta me pijanìth-, i = 1, . . . , m} antÐstoiqa, ìpouta {p (w′ s)ip (w′ s)i= p(ws) i (1 − p (w k−s)i )1 − p (ws)i p (w k−s)i=jp i()sp i1+(s−1)p i1 − (k−s)p i1+(k−s−1)p i1 − sp i1+(s−1)p i(k−s)p i=1+(k−s−1)p isp i1 + (k − 1)p i. (14)'Estw t¸ra V ′′ èna sÔnolo pou apoteleÐtai apì k − s aplèc korufèc kai mia koruf w s.′Profan¸c an i eÐnai èna opoid pote stoiqeÐo tou M, tìte kˆje mia apì tic k − s aplèc korufècepilègoun to i anexˆrthta me pijanìthta p i , en¸ h w s ′ epilègei to i anexˆrthta me pijanìthta p (w′ s)i .

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 31Qrhsimopoi¸ntac t¸ra èna parìmoio trìpo apìdeixhc me autìn tou Jewr matoc 7, mporoÔme nadiapist¸soume ìtiP {X V ′1= 1|X V ′2= 1} = P {to V ′′ eÐnai anexˆrthto sÔnolo koruf¸n}=m∏i=1((1 − p i ) k−s + (k − s)p i (1 − p i ) k−s−1 (1 − p (w′ s)i )'Ara, efarmìzontac to parapˆnw apotèlesma stic (9) kai (10), paÐrnoumeV ar ( X (k)) =s=1) defk∑( )( ) ( n 2k − sγ(k, s) E [ X (k)] [ ) − E2 X (k)]))2k − s s2( nk( nk= γ(k, s).ìpou h E [ X (k)] dÐnetai apì to Je¸rhma 7.□Qrhsimopoi¸ntac th mèjodo thc deÔterhc rop c (blèpe [2, 17]) kai ta apotelèsmata aut cthc enìthtac mporeÐ kaneÐc na breÐ kat¸flia gia thn Ôparxh (me pijanìthta pou teinei sto 1)anexˆrthtwn sunìlwn koruf¸n megèjouc k se genikeumèna tuqaÐa graf mata tom c etiket¸n.4.2 Apodotik eÔresh megˆlwn anexˆrthtwn sunìlwn koruf¸nSthn enìthta aut parousiˆzoume merikoÔc algorÐjmouc pou proteÐname sthn ergasÐa [14] gia thneÔresh anexˆrthtwn sunìlwn koruf¸n. H anˆlush twn algorÐjmwn gÐnetai me bˆsh to montèloG n,m,p . Sugkekrimèna, parousiˆzoume treÐc algorÐjmouc. O pr¸toc algìrijmoc eÐnai o klassikìcˆplhstoc algìrijmoc eÔreshc anexˆrthtwn sunìlwn koruf¸n, o deÔteroc apoteleÐ ousiastikˆ miaparallag tou pr¸tou kai o trÐtoc eÐnai ènac algìrijmoc pou qrhsimopoieÐ tic etikètec tou tuqaÐougraf matoc tom c etiket¸n.4.2.1 Klassikìc ˆplhstoc algìrijmocO ˆplhstoc algìrijmoc pou faÐnetai parakˆtw xekinˆei me èna (arqikˆ) ˆdeio anexˆrthto sÔnolokoruf¸n V ′ . Epiplèon, diathreÐ èna sÔnolo U to opoÐo perièqei tic korufèc pou mporoÔn naeÐnai upoy fia mèlh tou V ′ . Se kˆje epanˆlhyh (while-loop) o algìrijmoc dialègei mia korufelˆqistou bajmoÔ apì to epagìmeno grˆfhma tou G apì to U 6 kai thn prosjètei sto V ′ . Sthsunèqeia sb nei apì to U thn koruf aut kai ìlec tic geitonikèc thc. Me autì ton trìpoexasfalÐzetai ìti den upˆrqei kamÐa akm metaxÔ twn koruf¸n tou V ′ kai ˆra to teleutaÐo eÐnaièna anexˆrthto sÔnolo koruf¸n.6 To epagìmeno grˆfhma tou G apì to U eÐnai èna grˆfhma me sÔnolo koruf¸n U kai sÔnolo akm¸n ekeÐnec ticakmèc tou G twn opoÐwn kai ta duo ˆkra an koun sto U.

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 32Algìrijmoc I:EÐsodoc: 'Ena stigmiìtupo G(V, e(G)) tou G n,m,p .'Exodoc: 'Ena anexˆrthto sÔnolo koruf¸n V ′ tou G.1. jèse V ′ := ∅?2. jèse U := V ?3. while U ≠ ∅ do4. begin5. èstw x := koruf elˆqistou bajmoÔ sto epagìmeno grˆfhma tou G apì to U?6. jèse V ′ := V ′ ∪ {x}?7. diègraye thn x kai ìlec tic geitonikèc thc korufèc apì to U?8. end9. output V ′ ?Gia thn anˆlush tou AlgorÐjmou I qrhsimopoioÔme èna apotèlesma pou mporeÐ na breÐ kaneÐc sto[3]. Ac sumbolÐsoume loipìn me r to mègejoc tou anexˆrthtou sunìlou koruf¸n pou kataskeuˆzeio algìrijmoc (to opoÐo parathroÔme ìti eÐnai Ðso me ton arijmì twn while-loops tou algorÐjmou)kai èstw δ = |e(G)| , ìpou n = |V |. H posìthta δ onomˆzetai puknìthta (density) tou G. TìtenisqÔei to parakˆtwL mma 1 To mègejoc tou anexˆrthtou sunìlou koruf¸n pou kataskeuˆzei o Algìrijmoc I ikanopoieÐth sqèshr(2δ + 1) ≥ n (15)opoiod pote ki an eÐnai to stigmiìtupo eisìdou.Apìdeixh. 'Estw x i h koruf pou epilègei o algìrijmoc sthn epanˆlhyh i kai èstw d i o bajmìcthc sto epagìmeno grˆfhma tou G apì to U. AfoÔ o algìrijmoc diagrˆfei apì to U th x i mazÐme ìlec tic geitonikèc thc korufèc kai afoÔ h x i èqei to mikrìtero bajmì apì ìlec tic geitonikècthc sto epagìmeno grˆfhma, tìte o algìrijmoc diagrˆfei toulˆqiston d i(d i +1)pleurèc tou G sthn2i-st epanˆlhyh.Prosjètontac gia ìlec tic sunolikˆ r epanal yeic paÐrnoume

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 33r∑i=1d i (d i + 1)2≤ |e(G)| = δn.Epiplèon, afoÔ o Algìrijmoc I termatÐzei ìtan ìlec oi korufèc èqoun diagrafeÐ apì to Uèqoume'Ara2r∑i=1d i (d i + 1)2+r∑(d i + 1) = n.i=1r∑(d i + 1) =i=1r∑(d i + 1) 2 ≤ n(2δ + 1).'Omwc, apì thn anisìthta Cauchy-Schwartz xèroume ìti h posìthta ∑ ri=1 (d i + 1) 2 elaqistopoieÐtaiìtan d i + 1 = n gia ìla ta i = 1, . . . , r. 'Ara eÐnairr∑i=1i=1(d i + 1) 2 ≥ n2rkai sunduˆzontac aut th sqèsh me thn prohgoÔmenhisodÔnaman 2r≤ n(2δ + 1)n ≤ r(2δ + 1).□Prèpei ed¸ na parathr soume ìti epeid o Algìrijmoc I paÐrnei san eÐsodo èna tuqaÐo grˆfhma,to r eÐnai mia tuqaÐa metablht . Epiplèon, tuqaÐa metablht eÐnai kai to pl joc twn akm¸n tou G.'Ara, mporoÔme na pˆroume th mèsh tim kai twn duo mèlwn thc sqèshc (15) kai na prokÔyeiE [r(2δ + 1)] = 2E [rδ] + E [r] ≥ n.Shmei¸noume t¸ra ìti h idiìthta {∃ anexˆrthto sÔnolo koruf¸n megèjouc r} eÐnai fjÐnousaston arijmì twn pleur¸n (dhlad prosjètontac pleurèc sto G to mègejoc twn anexˆrthtwnsunìlwn koruf¸n de mporeÐ na auxhjeÐ), en¸ h idiìthta {h puknìthta tou G eÐnai δ} eÐnai profan¸caÔxousa. Allˆ ìpwc èqoume peÐ se prohgoÔmenh enìthta, mia eidik perÐptwsh thc FKG anisìthtac(blèpe [2]) mac lèei ìti an A eÐnai mia aÔxousa idiìthta kai B eÐnai mia fjÐnousa idiìthta, tìteP (A ∩ B) ≤ P (A)P (B). Sunep¸c, qrhsimopoi¸ntac ton orismì thc mèshc tim c, sthn perÐptwsmac paÐrnoume

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 34E [rδ] ≤ E [r] E [δ] .Qrhsimopoi¸ntac tic duo teleutaÐec anisìthtec paÐrnoumekai ˆran ≤ 2E [rδ] + E [r] ≤ 2E [r] E [δ] + E [r] = E [r] (2E [δ] + 1)n2E [δ] + 1 =E [r] ≥2 E(|e(G)|) + 1 . (16)nGia na upologÐsoume thn posìthta E(|e(G)|) orÐzoume tic deÐktriec tuqaÐec metablhtèc{ 1 an upˆrqei h pleurˆ (u, v) sto GX u,v =0 alli¸c.Profan¸c eÐnai|e(G)| =kai apì th grammikìthta thc mèshc tim c èqoumeAllˆ eÐnaiE(|e(G)|) =∑u,v∈V,u≠v( n2)E [X u,v ] =nX u,v( n2)P {X u,v = 1}.P {X u,v = 1} = P {∃i ∈ M : i ∈ S u ∩ S v } = 1 − (1 − p 2 ) mìpou S u , S v eÐnai ta uposÔnola tou M = {1, 2, . . . , m} pou antistoiqoÔn stic korufèc u, v antÐstoiqa.'Ara telikˆ( ) n (1E(|e(G)|) = − (1 − p 2 ) m) .2Efarmìzontac to parapˆnw sth (16) katal goume sto parakˆtwL mma 2 H mèsh tim tou megèjouc tou anexˆrthtou sunìlou koruf¸n pou kataskeuˆzei o AlgìrijmocI eÐnai toulˆqistonn 22 ( )n2 (1 − (1 − p2 ) m ) + n = n 22E(|E|) + n .To epìmeno apotèlesma eÐnai ˆmesh sunèpeia tou l mmatoc 2.

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 35Pìrisma 2 (Je¸rhma arai¸n G n,m,p ) 'Otan to p eÐnai tètoio ¸ste E(|e(G)|) = Θ(n), tìte hmèsh tim tou megèjouc tou anexˆrthtou sunìlou koruf¸n pou kataskeuˆzei o Algìrijmoc I eÐnaiepÐshc Θ(n).Gia parˆdeigma, an p = √ αnm, gia 0 < α < 1, tìte me bˆsh to Pìrisma 2, o Algìrijmoc I ja kataskeuˆseisÐgoura èna anexˆrthto sÔnolo koruf¸n me toulˆqiston Θ(n) korufèc. Sugkekrimènaja eÐnai E [r] ≥ α.nAxÐzei na shmei¸soume ed¸ ìti an kai h parapˆnw anˆlush ègine me bˆsh to omoiìmorfo montèlo,mporeÐ eÔkola na metaferjeÐ qwrÐc megˆlec allagec sto genikeumèno montèlo tuqaÐwn grafhmˆtwntom c.4.2.2 'Ena apotèlesma gia araiˆ graf mata'Opwc èqoume peÐ se prohgoÔmenh enìthta, endiaferìmaste gia graf mata G n,m,p ìpou h pijanìthtaepilog c etiket¸n p ikanopoieÐ th sqèsh√g(n) 2 log n − g(n)n √ m ≤ p ≤ mgia kˆpoia sunˆrthsh g(n) h opoÐa teÐnei sto ˆpeiro me osod pote argì trìpo ìtan n → ∞ (giaparˆdeigma g(n) = o(log log log n)). To pedÐo autì tim¸n to p apoteleÐ to pedÐo tim¸n gia qr simagraf mata (blèpe enìthta 1.2.3).g(n)1Ac upojèsoume ìti8nm. H perioq aut tim¸n tou p den eÐnai ken afoÔ eÔkola√n √ ≤ p < m√n √ < 1mg(n)epalhjeÔetai ìti8nm, gia n → ∞. Sth sunèqeia upojètoume ìti p = p(n) = c(n) giamkˆpoia sunˆrthsh c(n) → ∞. Gia parˆdeima, afoÔ c(n) = mp, an pˆroume to p sthn perioq tim¸ngia qr sima graf mata, tìte√ mn g(n) ≤ c(n) ≤ √ 2m log n − g(n)m. (17)Mia sunˆrthsh pou ikanopoieÐ th sqèsh 17 eÐnai kai h c(n) = α log n, gia kˆpoia stajerˆ α > 1(upenjumÐzoume ìti to m eÐnai mia dÔnamh tou n√kai ìti g(n) = o(log log log n)).1Shmei¸noume ìti lìgw thc upìjeshc p 8α 2 n log 2 n.'Estw t¸ra v mia koruf kai èstw S v to sÔnolo twn etiket¸n pou thc antistoiqeÐ. Qrhsimopoi¸ntacfrˆgmata Chernoff (blèpe gia parˆdeigma [5]) kai thn anisìthta Boole, gia mp = α log nkai ɛ ∈ (0, 1), paÐrnoumeP {∃v : ∣ ∣ |Sv | − mp ∣ ∣ ≥ ɛmp} ≤∑v∈VP { ∣ ∣ |Sv | − mp ∣ ∣ ≥ ɛmp} ≤ n− αɛ22 +1 .

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 36An epilèxoume thn parˆmetro α ètsi ¸ste αɛ22− 1 > 2, tìte ìlec oi korufèc tou G n,m,p jaèqoun ènan arijmì etiket¸n {kontˆ} sto mp me pijanìthta toulˆqiston 1 − 1 n 2 . 'Estw ìti togegonìc autì eÐnai dedomèno gia to G n,m,p .Ac jewr soume t¸ra thn ex c parallag tou AlgorÐjmou I:Algìrijmoc II:EÐsodoc: 'Ena stigmiìtupo G(V, e(G)) tou G n,m,p .'Exodoc: 'Ena anexˆrthto sÔnolo koruf¸n V ′ tou G.1. jèse V ′ := ∅?2. jèse U := V ?3. while U ≠ ∅ do4. begin5. èstw u := mia tuqaÐa koruf tou U?6. U := U − {u}?7. èstw S(V ′ ) := ∪ u∈V ′S u ?8. if (S u ∩ S(V ′ ) ≠ ∅) then apìrriye th u9. else V ′ := V ′ ∪ {u}?10. endH basik diaforˆ metaxÔ twn AlgorÐjmwn I kai II eÐnai ìti ston teleutaÐo h koruf pouepilègetai gia prosj kh sto sÔnolo V ′ den eÐnai aut me to mikrìtero bajmì, allˆ epilègetaituqaÐa apì tic enapomeÐnasec korufèc sto sÔnolo U. Apì ed¸ kai sto ex c ja sumbolÐzoume mer 1 kai r 2 to mègejoc twn anexˆrthtwn sunìlwn koruf¸n pou kataseuˆzoun oi Algìrijmoi I kai IIantÐstoiqa.Ac epikentr¸soume t¸ra to endiafèron mac sthn ektÐmhsh thc tim c tou r 2 (me megˆlh pijanìthta).EÐnai eÔkolo na diapist¸sei kaneÐc ìti metˆ apì i epituqhmènec prosj kec sto sÔnolo V ′(pou telikˆ ja apotelèsei to sÔnolo anexˆrthtwn koruf¸n sthn èxodo tou algorÐjmou) isqÔounta ex c:• |S(V ′ )| ∈ (1 ± ɛ)imp = (1 ± ɛ)ic(n), ìpou S(V ′ ) eÐnai h ènwsh twn sunìlwn etiket¸n pou

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 384.2.3 'Enac diaforetikìc algìrijmocSthn enìthta aut parousiˆzoume ènan algìrijmo kataskeu c anexˆrthtwn sunìlwn koruf¸n oopoÐoc blèpei to grˆfhma tom c apì thn pleurˆ twn etiket¸n tou. O Algìrijmoc III xekinˆei me ènapl rec sÔnolo koruf¸n (dhlad A 0 = V ) kai se kˆje epanˆlhyh diathreÐ èna sÔnolo apì korufècpou eÐnai upoy fiec na apotelèsoun to anexˆrthto sÔnolo koruf¸n sthn èxodo tou algorÐjmou.Se kˆje epanˆlhyh o algìrijmoc epilègei mia (diaforetik kˆje forˆ) etikèta kai diagrˆfei apìto sÔnolo autì ìlec tic korufèc pou èqoun epilèxei th sugkekrimènh etikèta, ektìc apì mia. Oprosektikìc anagn¸sthc ja katalˆbei ìti o trìpoc me ton opoÐo o algìrijmoc kataskeuˆzei toanexˆrthto sÔnolo koruf¸n èqei pollˆ koinˆ qarakthristikˆ me thn teqnik pou qrhsimopoi samesthn apìdeixh tou Jewr matoc 7 thc enìthtac 4.Algìrijmoc III:EÐsodoc: 'Ena tuqaÐo grˆfhma tom c G n,m,p .'Exodoc: 'Ena anexˆrthto sÔnolo koruf¸n A m .1. jèse A 0 := V ? jèse L := M?2. for i = 1 to m do3. begin4. epèlexe mia tuqaÐa etikèta l i ∈ L? jèse L := L − {l i }?5. jèse D i := {v ∈ A i−1 : l i ∈ S v }?6. if (|D i | ≥ 1) then epèlexe mia tuqaÐa koruf u ∈ D i kai jèse D i :=D i − {u}?7. jèse A i := A i−1 − D i ?8. end9. output A m ?To epìmeno Je¸rhma aforˆ thn orjìthta tou algorÐjmou.Je¸rhma 10 (Orjìthta) H èxodoc tou AlgorÐjmou III eÐnai pˆnta èna anexˆrthto sÔnolo koruf¸n.

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 39Apìdeixh. Gia na apodeÐxoume thn orjìthta tou AlgorÐjmou III, ac jewr soume duo geitonikèckorufèc v 1 kai v 2 . Apì ton orismì tou montèlou tuqaÐwn grafhmˆtwn tom c, autì shmaÐnei ìtiupˆrqei toulˆqiston mia etikèta j ∈ M tètoia ¸ste j ∈ S v1 ∩ S v2 , dhlad kai oi duo korufècèqoun epilèxei th j. EÐnai eÔkolo na diapist¸soume ìti to polÔ mia apì tic korufèc v 1 , v 2 mporeÐna an kei telikˆ sto A m . Prˆgmati, ac upojèsoume ìti o algìrijmoc ftˆnei sto shmeÐo na exetˆseithn etikèta j sthn k-ost epanˆlhyh. Tìte, an kai oi duo korufèc an koun sto A k−1 (dhladkamÐa apì tic duo den èqei aporrifjeÐ lìgw kˆpoiac prohgoÔmenhc etikètac), metˆ thn epanˆlhyhk mporeÐ na upˆrqei sto A k to polÔ mia apì tic duo. Autì apodeiknÔei ìti metaxÔ twn koruf¸ntou sunìlou A m den upˆrqei kamÐa pleurˆ, kai ˆra, to sÔnolo A m apoteleÐ anexˆrthto sÔnolokoruf¸n tou tuqaÐou graf matoc tom c sthn eÐsodo tou Algorijmou III.□To epìmeno Je¸rhma aforˆ to mègejoc tou sunìlou anexˆrthtwn koruf¸n pou kataskeuˆzeio Algìrijmoc III.Je¸rhma 11 (Apodotikìthta [14]) 'Estw èna grˆfhma G n,m,p , me m ≥ n kai mp = α log n,gia kˆpoia stajerˆ α > 1. Tìte, sqedìn bèbaia èqoume ìti, gia mia stajerˆ β > 0,1. an np → ∞, tìte |A m | ≥ (1 − β) nlog n.2. an np → b, ìpou b > 0 eÐnai mia stajerˆ, tìte |A m | ≥ (1 − β)n(1 − e −b ).3. an np → 0, tìte |A m | ≥ (1 − β)n.Apìdeixh. OrÐzoume tic deÐktriec tuqaÐec metablhtèckaiProfan¸c eÐnaiX (i)v ={ 1 ,an h koruf v ∈ A i−1 den perièqei thn etikèta l i0 ,alli¸c|A i | = ∑I Di =v∈A i−1X (i){ 1 ,an |D i | ≥ 10 ,alli¸c.v + I Di ,gia i = 1, 2, . . . , m.ExaitÐac thc anexarthsÐac ìson aforˆ sthn epilog twn stoiqeÐwn tou M, oi metablhtèc X v(i)eÐnai anexˆrthtec apì to sÔnolo A i−1 . 'Omwc, h isìthta tou Wald mac lèei ìti an Y 1 , Y 2 , . . . eÐnaianexˆrthtec kai omokatanemhmènec tuqaÐec metablhtèc me peperasmènh mèsh tim kai an N eÐnaiènac qrìnoc perˆtwshc 9 twn Y 1 , Y 2 , . . ., tètoioc ¸ste E[N] < ∞, tìte9 Lème ìti mia tuqaÐa metablht N pou paÐrnei mh arnhtikèc akèraiec timèc apoteleÐ qrìno perˆtwshc (stoppingtime) miac akoloujÐac tuqaÐwn metablht¸n {Y i } ∞ i=0 , an to gegonìc {N = n} exartˆtai mìno apì tic tuqaÐecmetablhtèc Y 1 , Y 2 , . . . , Y n kai eÐnai anexˆrthto twn Y n+1 , Y n+2 , . . ..

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 40[ N]∑E Y i = E[N]E[Y ].j=1Efìson oi tuqaÐec metablhtèc X v(i) kai to sÔnolo A i−1 (akribèstera to |A i−1 |) ikanopoioÔnprofan¸c tic anagkaÐec sunj kec thc isothtac tou Wald (gia th mèsh tim tou ajroÐsmatoc enìctuqaÐou arijmoÔ anexˆrthtwn tuqaÐwn metablht¸n blèpe [18]) kai apì th grammikìthta thc mèshctim c, paÐrnoumeE(|A i |) = E(|A i−1 |)(1 − p) + P {|D i | ≥ 1} ,gia i = 1, 2, . . . , m.Qrhsimopoi¸ntac thn parapˆnw sqèsh mporoÔme na deÐxoume epagwgikˆ ìtiE(|A m |) = n(1 − p) m +m∑i=1(1 − p) m−i P {|D i | ≥ 1}. (18)(ShmeÐwsh: EÐnai eÔkolo na deÐ kaneÐc ìti o ìroc n(1−p) m sth sqèsh (18) eÐnai o mèsoc arijmìctwn koruf¸n pou de dialègoun kamÐa etikèta. Epilègontac mp ≥ αlogn, gia kˆpoia stajerˆ α > 1o arijmìc autìc teÐnei sto 0.)'Estw t¸ra L i = {v ∈ V : l i ∈ S v }, dhlad to L i eÐnai to sÔnolo twn koruf¸n pou èqounepilèxei thn etikèta l i . H diaforˆ tou sunìlou L i apì to D i eÐnai ìti to pr¸to eÐnai anexˆrthtoapì tic epilogèc tou AlgorÐjmou III. EÐnaiP {|D i | ≥ 1} = 1 − P {|D i | = 0} = 1 − (P {v /∈ D i }) n (19)ìpou veÐnai mia sugkekrimènh koruf . H deÔterh isìthta isqÔei lìgw summetrÐac kai lìgw thcanexarthsÐac metaxÔ twn epilog¸n twn diˆforwn koruf¸n. 'Omwc eÐnaiP {v /∈ D i } = P {v /∈ L i ∩ v /∈ D i } + P {v ∈ L i ∩ v /∈ D i }= 1 − p + P {v ∈ L i ∩ {v ∈ L 1 ∪ L 2 ∪ · · · ∪ L i−1 }}.ExaitÐac thc anexarthsÐac metaxÔ twn stoiqeÐwn tou M, ta gegonìta {v ∈ L i } kai {v ∈L 1 ∪ L 2 ∪ · · · ∪ L i−1 } eÐnai epÐshc anexˆrthta. Sunep¸c,P {v /∈ D i } = 1 − p + P {v ∈ L i }P {v ∈ L 1 ∪ L 2 ∪ · · · ∪ L i−1 }= 1 − p + p ( 1 − (1 − p) i−1) = 1 − p(1 − p) i−1 .Apì th sqèsh (19) tìte eÐnai P {|D i | ≥ 1} = 1 − (1 − p(1 − p) i−1 ) n . Antikajist¸ntac sthsqèsh (18) paÐrnoumeE(|A m |) = n(1 − p) m + 1 p (1 − (1 − p)m ) −m∑(1 − p) ( m−i 1 − p(1 − p) i−1) n.i=1

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 41PÐnakac 3: Apìdosh twn AlgorÐjmwn I, II kai III, gia mp = α log n, p → 0np T I T II , T III r I r II r IIInp < 18α log nnp → 0np → bO( n log np) O( n log np) (≥) 8 9 n n4(1 − β)nO( n log np) O( n log np) (1 − β)nO( n log np) O( n log np) (∼)nαb log n(1 − β)n(1 − e −b )np → ∞O( n log np) O( n log np) (∼)1αp log n(1 − β) nlog nnp > 1 pO(n 2 p log n) O( n log np) (∼)1αp log n(1 − β) nlog nSthn endiafèrousa perÐptwsh ìpou m ≥ n kai mp ≥ α log n, gia kˆpoia stajerˆ α > 1 (opìteeÐnai profan¸c p → 0) paÐrnoumeE(|A m |) ∼ n(1 − p) m + 1 p (1 − (1 − p)m ) −∼ 1 p (1 − (1 − p)n ) .m∑(1 − p) m−i (1 − p) ni=1DiakrÐnoume t¸ra treÐc peript¸seic, anˆloga me tic timèc pou paÐrnei h posìthta np:1. An np → ∞, tìte E(|A m |) ∼ p. 1 H megalÔterh tim pou mporeÐ na pˆrei to p ètsi ¸ste np →∞, mp ≥ α log n kai m ≥ n eÐnai p = log n . SumperaÐnoume loipìn ìti E(|Anm |) = Ω( n ). log n2. An np → b, ìpou b > 0 eÐnai mia stajerˆ, tìte E(|A m |) ∼ n(1 − b e−b ) = Θ(n).3. An np → 0, tìte E(|A m |) ∼ 1 (1 − 1 + np) = Θ(n).pTelei¸noume thn apìdeixh me thn parat rhsh ìti afoÔ se ìlec tic peript¸seic èqoume E(|A m |) →∞, tìte mporoÔme na qrhsimopoi soume frˆgmata Chernoff gia na deÐxoume ìti |A m | ≥ (1 −β)E(|A m |), gia kˆje stajerˆ β > 0, me megˆlh pijanìthta.□

4 ANEXŸARTHTA SŸUNOLA KORUFŸWN: NŸEA APOTELŸESMATA 42O pÐnakac 3 sunoyÐzei thn apìdosh twn AlgorÐjmwn I, II kai III, sthn perÐptwsh mp = α log nkai p → 0. Ston pÐnaka, sumbolÐzoume me T I , T II kai T III touc qrìnouc ektèleshc twn AlgorÐjmwnI, II kai III antÐstoiqa. Akìma, sumbolÐzoume me r I , r II kai r III kˆtw frˆgmata gia ta megèjh twnanexˆrthtwn sunìlwn koruf¸n pou kataskeuˆzontai (sqedìn sÐgoura) apì touc AlgorÐjmouc I, IIkai III antÐstoiqa. Oi stajerèc α > 1, β > 0 kai b > 0 eÐnai oi Ðdiec stajerèc pou qrhsimopoi jhkansta Jewr mata 9 kai 11. ParathroÔme epÐshc ìti an kai, ìpwc mporoÔme eÔkola na diapist¸soume,oi qrìnoi ektèleshc twn AlgorÐjmwn I, II kai III sth qeirìterh perÐptwsh eÐnai O(mn + n +e(G)), O(nm) kai O(mn) antÐstoiqa, oi pragmatikoÐ qrìnoi ektèleshc touc mporoÔn na eÐnai arketˆmikrìteroi apì autèc tic timèc, epeid exart¸ntai apì thn puknìthta tou graf matoc. Tèloc, gialìgouc eukolÐac qr shc tou pÐnaka 3, den emfanÐzontai se autìn oi pijanìthtec epituqÐac twnalgorÐjmwn. AxÐzei omwc na parathr soume ìti en¸ oi duo teleutaÐoi algìrijmoi èqoun mia mikrpijanìthta apotuqÐac (h opoÐa teÐnei sto 0 kaj¸c n → ∞), o Algìrijmoc I katafèrnei pˆntote (mepijanìthta akrib¸c 1) na kataskeuˆsei èna sÔnolo anexˆrthtwn koruf¸n tou opoÐou to mègejoceÐnai toulˆqiston ìso h tim pou faÐnetai ston pÐnaka.

5 EUQARISTŸIES 435 EuqaristÐecJa jela na euqarist sw to sÔmboulo kajhght mou Lèktora Swt rh Nikoletsèa gia th bo jeiapou mou prìsfere ¸ste na èrjei eic pèrac aut h diplwmatik . Akìma euqarist¸ ton KajhghtPaÔlo Spurˆkh gia tic anektÐmhtec sumboulèc tou. Oi suzht seic pou èginan kai me touc duo tananektÐmhtec kai eÐqan san apotèlesma th dhmosÐeush [14]. Tèloc ja jela na euqarist sw tonM. Karoński kai thn K.B. Singer-Cohen gia th bo jeia pou mac prìsferan sta pr¸ta mac b matasta montèla tuqaÐwn grafhmˆtwn tom c etiket¸n, dÐnontˆc mac prìsbash se sqetikì ulikì.

ANAFORŸES 44Anaforèc[1] D. Aldous and J. Fill, “Reversible Markov Chains and Random Walks on Graphs”, Unpublishedmanuscript. http://stat-www.berkeley.edu/users/aldous/book.html, 1999.[2] N. Alon and J.H. Spencer, “The Probabilistic Method”, Second Edition, John Wiley & Sons,Inc, 2000.[3] G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela and M. Protasi,“Complexity and Approximation”, Springer-Verlag Berlin Heidelberg, 1999.[4] B. Bollobás, “Random Graphs”, Second Edition, Cambridge University Press, 2001.[5] J. Díaz, J.Petit and M. Serna, Chapter titled “A Guide to Concentration Bounds”, in the“Handbook of Randomized Computing - Volumes I & II (Combinatorial Optimization 9)”,pp.457-507, Kluwer Academic Publishers, Volume I, 2001.[6] P. Erdös, A. Goodman and L. Poza, “The Representation of a Graph by Set Intersections”,Canadian Journal of Mathematics, 18:106-112, 1966.[7] P. Erdös and A. Rényi, “On Random Graphs”, Publ. Math. Debrecen, 6:290-297, 1959.[8] P. Erdös and A. Rényi, “On the Evolution of Random Graphs”, MTA Mat. Kut. Int. Kozl.,5:17-61, 1960.[9] J.A. Fill, E.R. Sheinerman and K.B Singer-Cohen, “Random Intersection Graphs when m =ω(n): An Equivalence Theorem Relating the Evolution of the G(n, m, p) and G(n, p) models”,http://citeseer.nj.nec.com/fill98random.html[10] M. Karoński, E.R. Scheinerman and K.B. Singer-Cohen, “On Random Intersection Graphs:The Subgraph Problem”, Combinatorics, Probability and Computing journal (1999) 8, 131-159.[11] T. Luczak, “On the Equivalence of Two Basic Models of Random Graphs”, Random Graphs’87, pages 151-157, 1990.[12] E. Marczewski, “Sur Deux Propriétés des Classes d’ Ensembles”, Fund. Math., 33:303-307,1945.[13] R. Motwani and P. Raghavan, “Randomized Algorithms”, Cambridge University Press, 1995.[14] S. Nikoletseas, C. Raptopoulos and P. Spirakis, “The Existence and Efficient Constructionof Large Independent Sets in General Random Intersection Graphs”, in the Proceedings ofthe 31st International Colloquium on Automata, Languages and Programming (ICALP),Lecture Notes in Computer Science (Springer Verlag), pp.1029-1040, 2004.

ANAFORŸES 45[15] S. Nikoletseas, C. Raptopoulos and P. Spirakis, “Large Independent Sets in General RandomIntersection Graphs”, invited paper in the Theoretical Computer Science (TCS) Journal,Special Issue on Global Computing, under review, to appear in 2005.[16] S. Nikoletseas and P. Spirakis, ”Expander Properties in Random Regular Graphs with EdgeFaults”, in the Proceedings of the 12th Annual Symposium on Theoretical Aspects of ComputerScience (STACS), Lecture Notes in Computer Science Vol. 900 (Springer Verlag), pp.421-432,1995.[17] S. Nikoletsèac kai P. Spurˆkhc, {StoiqeÐa thc Pijanotik c Mejìdou}, Tìmoc I, Gutenberg,1996.[18] S.M. Ross, “Stochastic Processes”, Second Edition, John Wiley & Sons, Inc., 1996.[19] E.R. Scheinerman, “Random Interval Graphs”, Combinatorica 8 (4) 1988, pp. 357-371.[20] E.R. Scheinerman, “An Evolution of Interval Graphs”, Discrete Mathematics 82 (1990), pp.287-302.[21] K.B. Singer-Cohen, “Random Intersection Graphs”, PhD thesis, John Hopkins University,1995.