Modeling Secure Connectivity of Self-Organized Wireless Ad Hoc Networks

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1 This ull txt ppr ws pr rviw t th irtio o IEEE Commuitios Soity sujt mttr xprts or pulitio i th IEEE INFOCOM 8 prois. Moli Sur Cotivity o Sl-Oriz Wirlss A Ho Ntworks Chi Zh, Y So Yuu F Dprtmt o Eltril Computr Eiri Uivrsity o Flori, Gisvill, FL 36 Emil: {zhhi@, Astrt Wirlss ho tworks (WANETs) or ommuitios ovr shr wirlss hl without y pr-xisti irstrutur. Formi pr-to-pr surity ssoitios i sloriz WANETs is mor hlli th i ovtiol tworks u to th lk o trl uthoritis. I this ppr, w propos ri mol to vlut th rltioship o otivity, mmory siz, ommuitio ovrh surity i ully sl-oriz WANETs. Bs o som rsol ssumptios o o ploymt moility, w show tht wh th vr umr o uthtit ihors o h o is Θ(), with rspt to th twork siz, most o th os surly ot, ormi ot sur ko, i.., th sur twork prolts. This ot sur ko utiliz to rk routi-surity py loop, provi ouh riv sur liks oti isolt os with th sur ko i multi-hop shio, whih ls to th sur otivity o th whol twork. I. INTRODUCTION By iitio, wirlss ho twork (WANET) (moil or sttiory) is roup o wirlss os tht ooprtivly orm twork whih oprts without th support o y pr-stlish or trliz twork mmt irstrutur [] [3]. I th litrtur, thr r two xtrm wys to itrou surity i WANETs: () throuh sil uthority omi, whr rtiits /or kys r issu y sil uthority, typilly, i th systm stup phs [4] [9], or () throuh ull sl-oriztio, whr surity os ot rly o y trust uthority or ix srvr, ot v i th iitiliztio phs [] [3], []. I this ppr, w ollow th so pproh. Our mi motivtio oms rom osrvtios o som WANET srios, whih rquir sloriz twork mmt,.., wh th WANET mrs prtitios i spori wy th twork ot pr-pl; wh h usr hs iiviul itrsts thus is its ow uthority omi whr th ull otrol o th surity sttis o its ow os is sir; wh usrs prr to joi lv th twork t rom without otti y rmot trust uthority; wh th umr o usrs ts lr thus th ky srvrs, i y, will om th ottlk or trl poits o ilur. This work ws support i prt y th U.S. Ntiol Si Foutio ur rt CNS-7744, CNS-7645 CNS All ths situtios hihliht th o surity rhittur with sl-oriztio proprty, i.., th ility o os to stlish surity ssoitios mo thmslvs tr twork ormtio without th i o y orm o o-li or o-li trust thir prty (TTP) []. Althouh ot ll WANETs r rquir to ully sl-oriz, this proprty is osir s o o th il ojtivs i most o th ooi rsrh projts, suh s Trmios, Spott zro-oiurtio tworks [] [3], []. Impromptu, sl-oriz WANETs iormlly visuliz s roup o wirlss ommuitio vis (ll WANET os i this ppr), hl y popl without y prpli, omi tothr to orm twork or ommo purpos (.., mry rspos). Som kyi mtrils or primry surity ssoitios (SAs), whih w will ormlly i ltr, r lry pr-oiur i ommuitio vis, s o th trust rltioships tw th popl ivolv. Th prolm is how to xploit thos primry surity ssoitios to provi sur ommuitio or ritrry o pirs wh. Nihor uthtitio, whih provis hop-y-hop surity, is th irst stp or sur ommuitios i ll kis o tworks. This is spilly ruil or WANETs si vry o to t s th routr to orwr pkts or othrs. I th o ot uthtit its physil ihors, how it trust ll thos physil ihors to hl its pkt orrtly? Oviously, ihori os with primry surity ssoitios uthtit h othr irtly with pr-oiur kyi mtrils. Si th umr th istriutio o primry surity ssoitios r trmi y th m soil twork (.., trust rltios) o usrs, o my ot hv primry surity ssoitios with y o its physil ihors. I this s, Nihor Authtitio Protool (NAP) is rquir to st up riv surity ssoitios with its ihors o th sis with th hlp o lry uthtit ihors. Althouh som xmpls o ully sl-oriz surity rhittur or WANETs hv isuss i th litrtur [] [3], [], my thortil prolms r still rmii:.., wht is th miimum rtio o primry SAs or suri ll th liks? Wht is th ommuitio ovrh or NAPs to provi riv SAs? How o th hrtristis o th trust rph o usrs t th prorm o NAPs th surity o WANETs? Is this ki o sl-oriz /8/$5. 8 IEEE 85

2 This ull txt ppr ws pr rviw t th irtio o IEEE Commuitios Soity sujt mttr xprts or pulitio i th IEEE INFOCOM 8 prois. WANETs sll i trms o th rquir mmory siz or kyi mtril i h o or th ommuitio ovrh or th NEP, wh th twork siz oms ritrrily lr? I this ppr, w propos ri mol to vlut th rltios o otivity, mmory siz, ommuitio ovrh surity i ully sl-oriz WANETs. Bs o som rsol ssumptios o o ploymt moility, w show tht wh th vr umr o uthtit ihors o h o is Θ(), with rspt to th twork siz, most o th os surly ot, ormi ot sur ko, i.., th sur twork prolts. This ot sur ko utiliz to rk routi-surity py loop (. Stio II-B), provi ouh riv sur liks oti isolt os with th sur ko i multi-hop shio, whih ls to th sur otivity o whol twork. II. NETWORK MODEL AND PROBLEM FORMULATION A. Ntwork Mol ) Physil Grph G(X, E pl ): Lt X = {X,X,, X } ot th o st o WANET, with twork siz X =. By sliht us o ottio, w lt X i ot th lotio, s wll s th itity, o o. Th os r istriut uiormly t rom o th iv orphil r, whih is, without loss o rlity, uit isk D tr t th orii. Two os X i X j hv physil wirlss lik (X i,x j ) i thir Euli ist is o rtr th r,thommuitio r, X i X j r ll physil ihor with rspt to h othr. Lt E pl ot th physil lik st. Th rph G(X, E lik ) with vrtx st X st E lik is trm rom omtri rph (RGG) [], whih is wily us i th litrtur to mol th physil rph o WANET. A physil pth tw two os,.., X X k, is st o osqutil s i E pl. Two os r si to physilly ot i thr xists physil pth tht strts t o s t th othr. Grph G(X, E pl ) is si to physilly ot i vry pir o os i th rph is physilly ot. I this ppr, w r mily or with vts tht our isi th uit isk D with hih proility (w.h.p.); tht is, with proility ti to o s. For moil WANET, w ssum tht os mov iptly i th uit isk D ori to Browi motio mol (BMM) i []. W ssum tht th iitil positios o os r i.i.. uiormly istriut i th isk D. This implis tht th positios o th os will rmi uiorm t ll tims ur th BMM. Si w r oly itrst i th sttistil proprtis o G(X, E pl ) othr rlt rph mols, s o th BMM, w omit th tim imsio trt th moil WANET s stti i iv tim poit, whih will rtly simpliy our lysis. Not tht th rsults o this ppr lso pply to othr rlt moility mols suh s th rom wlk moility mol [3] th Mrkovi moility mol [4]. This is us th Browi motio mol viw s limiti s o ths othr moility mols []. Commuitio R Fi.. A xmplry WANET with primry SAs. Hr soli lis sh lis rprst physil liks primry SAs, rsptivly. ) Trust Grph G(X, E SA ): Wh w sy two os hv primry surity ssoitio (SA), w m tht two os trust h othr ithr symmtri ky r shr tw thm, or tht th os kow h othrs uthti puli kys. Two os my hv xh thir kys throuh si hl (.., ovr irr hl t th tim o physil outr, or just mully st up thos kys). W urthr ssum tht surity ssoitios r lwys symmtri. I th ii, this ssumptio my sm oliti with th symmtri tur o rtiits o puli kys. Du to th t tht o X i kows th uthti puli ky o o X j os ot ssrily imply tht o X j lso hols th uthti puli ky o o X i, vi vrs, rltios i rtiit rph [], [3], [4] r symmtri. Howvr, irst o ll, i prti th sttistil lysis o th W o Trust mo usrs o Prtty Goo Privy (PGP) [5], th mrkt lr i th worl o sur mil ommuitios, shows tht out /3 o th liks i th lr stroly ot ompot r iirtiol [6]. So, i our shm, w rquir two os to iirtiolly xh thir kyi mtrils i orr to sussully stlish primry SA. Thir, sis th rquirmt o possssi orrt kyi mtril, w lso rquir tht two os trust h othr,.., thy rly o h othr to hl thir pkts, whih is th rso tht this ki o trust must iirtiol. Primry SA m s loil lik oti two os i th WANET. W m two os with primry SA s ris ot th st o primry SAs s E SA. Th rph G(X, E SA ) with vrtx st X st E SA is trm th trust rph, is us to mol th trust rltioships tw usrs i th WANET. Th hrtristis o trust rphs rt i WANETs will p o th xisti soil rltioships tw usrs. I wht ollows, w just itrou o simplii prmtr p to qutiy th xist o SAs i th WANET. W osir homoous trust rph mol, i.., th umr o ris o h o is o th sm orr o. Bsothi.i.. uiorm istriutio o iitil positios o th os th BMM, whvr two os mt s physil ihors, thy will ris (hv SA) with lmost th sm proility, ot s p. 3) Sur Grph G(X, E sl ) or G(X, p ): Oviously, wh o o X i s ris,.., o X j, oms th physil ihor o X i, th os X i X j irtly uthtit h othr. W ll X i th ihori ri o X j, thr xists sur physil lik throuh wirlss om- 86

3 This ull txt ppr ws pr rviw t th irtio o IEEE Commuitios Soity sujt mttr xprts or pulitio i th IEEE INFOCOM 8 prois. () () Fi.. Grph mols ostrut rom th oriil wirlss ho tworkifi..() Physil rph; () trust rph; () sur rph; () lol umt sur rph. muitio tw X i X j. W ot th st o sur physil liks s E sl, i th rph with vrtx st X st E sl s th sur rph. Si (X i,x j ) E sl i (X i,x j ) E pl (X i,x j ) E SA, G(X, E sl ) is th oupl rph o G(X, E pl ) G(X, E SA ). Not tht som liks i st E pl \E sl lso sur with th ssist o lol uthtit ihors. W m ll thos liks lolly umt sur liks, ot s E lsl. Th rph G(X, E sl ) with E sl = E sl E lsl is trm lolly umt sur rph. Th rltioship tw G(X, E sl ) G(X, E sl ), th thiqus to lolly umt sur liks will urthr ivstit i Stio III-C. W i th sur otivity i G(X, E sl ) (or G(X, E sl )) i wy similr to physil otivity i G(X, E pl ), i.., thr xists sur pth with liks i E sl (or E sl ) or ritrry o pirs i X X. Hr w iv rliz ormultio to sri th rph mols itrou ov. Giv X s th vrtx st, w ot vry two os X i X j (i j) with proility (X i X j ). Th rsulti rph is ot s G(X,). W trm th ot ompots o G(X,) s lustrs, th umr o os i h lustr s its orr. Th lustr with th mximl orr is ll th it lustr. Th rph G(X,) is ot i thr xists oly o lustr. Thror, G(X, E pl ) lso ot s G(X, ), whr { i x r, (x) = () i x >r. Giv p, G(X, E sl ) lso ot s G(X, p ), whr { p i x r p (x) =, () i x >r. Bs o th proprty o th Poisso pross, th umr o physil ihors th umr o ihori ris or h o i X r rom vrils ollowi Poissoistriutio with xpt vlus o πr p πr, rsptivly (or simpliity, hr w ior th orr t,. th Appix i [7]). Usi th trm o rph thory, ths two vlus r lso ll th vr o r or G(X, ) G(X, p ), rsptivly. B. Nihor Authtitio Pirwis Ky Estlishmt For sl-oriz WANETs, ihor uthtitio osists o thr phss, mly ihori-ri isovry, () () lol sur-lik multi-hop sur-lik umttio. Th ihori-ri isovry phs tks pl wh o isovrs tht w ihori o pprs i its o-hop ihorhoo. Th simplst wy or y two os to isovr i thy r ris is tht h o os its o ID rlvt itiitio iormtio i lr txt ito its o mss (lso ll HELLO mss) whih is prioilly rost to its physil ihors or ihor isovry hl us ooritio i MAC lyr. Altrt mthos xist whih hi itity iormtio rom vrsry thry stlishi privt ihoriri isovry. Wh o o is othr s its ri, thy mutully uthtit th itity o th othr prty s it lims, usi y hll-rspos protools. Wh two ihori os r mutully uthtit, sur lik is stlish, thir surity ssoitio is rliz, orti- vt. Atr th irst phs, h o X i kows ll its physil ihors ihori ris, whih r ot y sts N(X i ) NF(X i ), rsptivly. Th sts uthtit uuthtit ihors o o X i r rprst y NA(X i ) UA(X i ), rsptivly. Atr th irst phs, NA(X i ) = NF(X i ), UA(X i ) = N(X i ) \ NF(X i ) th os with sur liks r mol y th trust rph G(X, E sl ). Du to th limittio o th trust rph, os oly shr primry SAs with sust o th ihori os. Thror, th lol sur-lik umttio phs is to stlish riv SAs with th rmii ihors with th hlp o uthtit ihors. Th thiqus qutittiv lysis o th so phs will til i Stio III-C. Hr, w just mphsiz tht i this phs, h usr oly mks us o its o-hop ihors iormtio, whih ollt i th MAC lyr. O ihor i st UA(X i ) is uthtit th pirwis ky is stlish i th so phs, it is ilu i st NA(X i ) lt rom st UA(X i ), th orrspoi lolly umt sur liks will i th st E lsl. Atr this phs, os with sur liks r mol y th rph G(X, E sl ), whr E sl = E sl E lsl. Not tht th ky ir tw WANETs othr istriut systms is tht, h o i th WANET is rquir to t s routr to orwr pkts or othr os. This uiqu tur itrous th wll-kow routi-surity py loop [8], [9]: yli py riss tw surity srvis routi srvis si multi-hop surity srvis rquir routi lyr surity thmslvs. This loop implis two osqus spilly or o uthtitio. O th o h, th primry SA tw two rmot os ot utiliz i thr is o sur pth tw thm. Ev thouh th pkts -to- rypt or uthtit, thy r t risk to st throuh ls routs, or simply ropp without th sur pth (routi lyr surity). O th othr h, riv SAs tw ihori os ot stlish ovr multipl hops i th routi protool os ot oprt surly. Hr, w rly o th irst two phss to rk this loop, si thy oly th 87

4 This ull txt ppr ws pr rviw t th irtio o IEEE Commuitios Soity sujt mttr xprts or pulitio i th IEEE INFOCOM 8 prois. iormtio ollt o th MAC lyr r ipt o oth sur routi othr surity srvis. Upo ompltio o ths two phss, som multi-hop sur pths with hop-y-hop surity will mr, s o whih sur routi implmt. I rl, thr still xists uuthtit physil ihors tr th so phs. I th ollowi multi-hop sur-lik umttio phs, multihop sur pths will us to uthtit stlish pirwis kys with th rmii ihors i st UA(X i ). Th ritrio tht th ihor uthtitio omplts ps o th pplitio rquirmt o prtiulr WANET. For xmpl, i th mi purpos o WANET is to ilitt th ooprtio tw th ihori os/usrs, th mi tri ptt will lol ommuitios whr o s to uthtit s my ihors s possil, or stops util th rtio o uthtit ihors NA(Xi) N(X i) is rtr th systm prmtr, sy. I th mi purpos o WANET is to provi irstrutur to support ommuitios with rmot os, th mi tri pttr will multi-hop ommuitios. I vry sour o is l to i sur pth to th trt stitio, o umt sur liks r rquir. Th uthtitio with th rmii ihors tivt i o-m shio, i.., to uthtit h othr oly wh thy to ommuit. O ihor i st UA(X i ) is uthtit th pirwis ky is stlish, this ihor will ilu i st NA(X i ) lt rom st UA(X i ), th orrspoi lolly umt sur liks will i th st E sl.atth o this phs, os with sur liks r mol y th rph G(X, E sl C. Prolm Formultio ), whr E sl = E sl E sl. Now, w ormlly i th ojtiv o sur WANETs s ollows: Ojtiv: Costruti sur pth tw ritrry pir o os i X X w.h.p. Costrits: (i)physil rph G(X, E pl ) is ot, (ii) trust rph G(X, E SA ) is ot. Mthmtilly, th ojtiv is quivlt to th situtio tht th lolly umt sur rph G(X, E sl ) is ot w.h.p. Hr, w to xpli ths two ostrits i til. I t, ths two ostrits r th ssry oitios to hiv th ojtiv. First o ll, wh th physil rph is isot, it is impossil to provi multi-hop ommuitios or isot os with tritiol routi shms (rtiv or protiv) si or WANETs. Althouh th moility o os utiliz, som routi shms si or ly tolrt tworks (DTNs) us to provi ommuitios tw som isot os, o protools urt th livry o pkts tw thm so r v i th w.h.p. ss. Howvr, it is itrsti to xmi i wht r this ostrit rlx y opti th DTN routi protools, whih will our utur work. So, isot trust rph implis tht usrs ivi ito svrl isolt trust ( x ) p r or G(, r ) p sqush (/p) r x Fi. 3. Squishi Squshi. Th utio is squish sqush to iv th utio p sqush. roups, thr is o trust rltioship tw thos roups. I this situtio, th ojtiv o ostruti sur pth tw two os tht lo to irt trust roups is milss. Thror, wh w mtio th ojtiv o provii sur pth tw ritrry pir o os, w impliitly ilu ths two ostrits. Prvious rsrh o Erös-Réyi rom rph [], [] rom omtri rph [], [7] show tht th ssry oitios or th otivity o th trust rph physil rph r ), rsptivly. p =Ω ( lo ) r =Ω ( lo Ur this ojtiv, w list th prolms tht to rss s ollows: () Wht r th rquirmts o th physil rph G(X, E pl ) trust rph G(X, E SA ) so tht it is possil or G(X, E sl ) to ot? Or mor prisly, wht shoul r p? () Wht is th ommuitio ovrh iurr u to th ihor uthtitio? (3) Wht is th t o th sur lik istriutio o th prorm o sur routi? III. PROPERTIES OF SECURE GRAPHS A. Bkrou o Cotiuum Proltio I this stio, w mk us o th rsults rom otiuum proltio, whih ws irst itrou y Gilrt [] ltr lyz y Mstr Roy [3], Pros [], [4]. I otiuum proltio thory, os r ssum to istriut s homoous Poisso pross P λ o R with sity λ. Howvr, our oriil twork mol ori ix umr o os X romly uiormly istriut i th uit r is D. I most o th proos i this ppr, it is usul to irst osir, ist o X, oupl Poisso pross P λ with sity λ los to, th u th rsults o X rom th rsults o P λ, usi th so-ll Poissoiztio thiqu [, Chptr.7]. W irst i th twork mol s o o istriutio P λ.lt{x,x,x 3, } th st o os ori to P λ i R, so tht th xpt umr o os i y rio is qul to th r o th rio multipli y λ. LtG(P λ,) ot th ollowi rom omtri rph: w ot y two poits X i X j (i j) ip λ with proility (X i X j ), ipt o y othr pirs o os. W trm th ot ompots o G(P λ,) s lustrs, th umr o os i h lustr s its orr. W irst show th xist o otrivil ritil sity t whih proltio ours (tht is, iiit-orr lustr orms) i G(P λ,). Lmm : [Proltio Proprtis o G(P λ,)] Cosir rph G(P λ,) or som iv msurl utio : 88

5 This ull txt ppr ws pr rviw t th irtio o IEEE Commuitios Soity sujt mttr xprts or pulitio i th IEEE INFOCOM 8 prois. R [, ], stisyi: (x) =( x ),x R, (3) < (x)x <, (4) R w hv th ollowi rsults: () Thr xist ritil vlu (otiuum proltio thrshol) λ p suh tht <λ p () =i{λ : iiit-orr lustr w.h.p.} <. Wh λ > λ p w sy tht th G(P λ,) prolts [4, Thorm ]. () I (x) (y) whvr x y, org(p λ,), thr is t most o iiit-orr lustr w.h.p. [3, Thorm 6.3, pp.7]. (3) Giv msurl utio <p<, i th sqush utio p sqush w hv o s sqush p (x) =p ( px), λ p () λ p ( sqush p ), (5) s Fi. 3 or xmpl [5, Thorm.]. (4) Wh (x) = (x) s i i (), w hv spil RGG ot y G(P λ, ). Th xt vlu λ p or this s is ot kow. Simultio stuis iit tht λ p 4.5 πr [, pp.89] whil riorous ous.87 πr λ p.593 πr r iv i Mstr Roy [3, Chptr 3.9]. A rt rsult o Blistr t l. [6] shows tht with 99.99% oi, th ritil vlu λ p lis tw 4.58 πr 4.55 πr, i.., 4.58 πr λ p (6) Gupt Kumr [7] Pros [] show tht, G(P λ, ) is quivlt to G(X, ) wh λ =. Thror, i this ppr w ssum πrλ p =4.5 or G(P λ, ) or G(X, ). W pl o X t th orii. Th th rsulti Poisso poit pross P λ X is oitio to hv poit t, i th ss o Plm msurs X isssumto ritrry poit o th Poisso pross [4]. Lt C() th lustr t th orii, th st o os hvi pth to X i G(P λ X,). Ltp k (λ) ot th proility tht C() hs k poits. Th proltio proility, i.., p (λ), isth proility tht lis i iiit lustr wh λ. W hv th ollowi rsult: Lmm : [Isolt Nos i G(P λ X,)] Assum msurl utio stisyi qutio (3) iqulity (4). I lso loss zro (sstilly, is symmtri, hs ou support, is ou wy rom zro i som op ihorhoo o th sur, s [4]; ot tht ll th utios osir i this ppr loss zro), th lim λ k= p k(λ) =, (7) p (λ) whih implis tht or lr λ, th orii lis i ithr iiit-orr lustr or orr-o lustr (i.., it is isolt) w.h.p. [4, Thorm 3]. B. Thortil Rsults o th Sur Grph Bs o Lmm Lmm (), th ollowi lmm sily riv. Lmm 3: [Isoltio Cotivity o G(P λ X,)] Assum msurl utio stisyi qutio (3) iqulity (4), lso loss zro, (x) (y) whvr x y. Asλ, th proility tht th rph G(P λ X,) is ot is symptotilly qul to th proility tht th rph G(P λ X,) hs o isolt os, i.., lim Pr[G(P λ X,) is ot] λ = lim Pr[G(P λ X,) hs o isolt os]. λ Now w ivstit th otivity proprty o th sur rph G(X, p ). Thorm : [Cotivity o G(X, p )] Assum tht p is y ostt i [, ]. Ltp πr = lo()+(). Th, th rph G(X, p ) is ot w.h.p. i () is isot w.h.p. i (). Proo: This rsult prov rom Lmm 3 usi Poissoiztio thiqu. Th tils o th proo hs omitt u to sp ostrits. Not tht i Thorm, i p is th proility tht two ihori os stlish th SA, th k = p = p πr ots th vr umr o uthtit ihors. Thorm shows tht or G(X, p ), thr xists phs trsitio phomo, i.., thr is ritil thrshol k = lo() or k, orrspoi to miimum umr o uthtit ihors or iiviul os, ov whih sirl lol proprty (.., otivity) xists with hih proility. Wh k is low th thrshol k, th sir lol proprty xists with low proility. This phs trsitio is typilly s to om shrpr s th umr o os i th twork irss. Th ollowi thorm will rvl othr phs trsitio phomo or G(X, p ). Thorm : [Proltio Proprtis o G(X, p )] Di th ritil vlu (proltio thrshol) o th vr o r or G(X, p ) s k p = i{k : iiit-orr lustr w.h.p. i G(X, p ) s }. () <k p 4.5. () Wh k = p πr >k p, w sy tht G(X, p ) prolts, thr xists oly o iiit-orr lustr w.h.p. (3) Wh th rph G(X, p ) prolts, ll o th iit-orr lustrs r o-orr lustrs w.h.p. Proo: Th xist o o-trivil ritil vlu o th vr o r or G(P λ, p ) oms rom Lmm (). W i G(P λ, ) s th rph with vrtx st P λ, (x) = (x) { i x r i x >r, whr r = p r. Oviously, p is th sqush utio o. Aori to Lmm (3) (4), w hv <λ p ( p ) λ p ( )= 4.5 πr. (8) 89

6 This ull txt ppr ws pr rviw t th irtio o IEEE Commuitios Soity sujt mttr xprts or pulitio i th IEEE INFOCOM 8 prois. k p = simult vlus lult vlus p Fi. 4. Proltio thrshols o irt p vlus. Clult vlus p +. ollow th qutio k p =4.5 i (). +. Th vr o r k p or G(P λ, p ) is iv y k p = p λ p ( p ) πr λ p ( ) πr. (9) From (8) (9), w oti <k p 4.5. Ltλ =, w t th sm rsult or G(X, p ). Th so prt o th thorm ollows rom Lmm (). For p, si p (x) p (y) whvr x y, thr xists oly o iiit-orr lustr w.h.p. or G(P λ, p ). Usi Poissoiztio thiqu, w prov tht it lso hols or G(X, p ). Th thir prt o th thorm ollows orm Lmm, usi Poissoiztio thiqu. Lt S th it lustr s siz rtio, i.., th umr o os i th it lustr ivi y. ForErös-Réyi rom rphs [], [], it is wll kow tht S is utio o th vr o r k, is th o-zro solutio to th ollowi qutio S = k S wh. () For RGGs, simultio rsults show tht, S k rltioship lso ollows th sm orm ut pprs to shit lo th vr o r xis (. Fi. 7). Si th proo prour o Thorm iits tht th proility o isolt os or G(X, p ) is uppr lowr ou y th proilitis o isolt os or Erös-Réyi rom rph mol G(X, ) RGG mol G(X, ), rsptivly, it is rsol to ssum tht S k rltioship or G(X, p ) rs th sm orm with som shit lo th vr o r xis. Thror, w hv th ollowi ojtur: Cojtur : [Siz o th Lrst Clustr o G(X, p )] Th siz o th lrst lustr o G(X, p ),sys, isth o-zro solutio to th ollowi qutio w.h.p.: S = (k kp+.45) S p +., whr k p = () Not tht () is mrly shit vrsio o (). W to stimt this shit. Oviously it ps o k p. Howvr, Thorm () oly ivs th r o k p.givp, th xt vlu o k p is still op prolm. Simult vlus o k p or irt p vlus r iv i Fi. 4. W hv tri svrl utio orms to stimt th p k p rltioship. A oo p Fi.4 pproximtio is ou to k p = 4.5 illustrts th lult k p rsulti rom this pproximtio, whih shows oo mth tw th simult lult vlus o k p. Two importt prtil osqus () Fi. 5. Lol trust rphs or () o () o i Fi.. Not tht i th lol trust rph, two os r ot i thy r physil ihors thy hv primry SA. ollow rom Cojtur : (i) Giv p, w lult th ritil thrshol o k or proltio; (ii) Giv k, w lult th siz o th it lustr i G(X, p ).W urthr support Cojtur with simultios prorm usi irt vlus o p (s Stio III-D). C. Proprtis o th Grph G(X, E sl ) I this sustio, w wt to stuy th ollowi prolm: iv p, wht is th proility tht two ihori os stlish riv SA tr th lol sur-lik umttio phs? Rll tht tr th ihori-ri isovry phs (. Stio II-B), vry o X i kows its physil ihors N(X i ) ihori ris NF(X i ). I th lol sur-lik umttio phs, vry o X i irst lolly rosts its sts N(X i ) NF(X i ).Usith sts riv rom ihors, o uil lol trust rph s o th ri rltios mo ihors. Th lol trust rph miti y o X i is i s G i (V i,e i ), whr th vrtx st V i = {X j X j N(X i ) j = i} th st E i = {(X j,x k ) X j,x k N(X i ) X k N(X j ) X j N(X k ) (X k,x j ) E SA }. Fi. 5 illustrts lol trust rphs miti y iiviul os. Thr r two possil ss to lolly umt sur liks. I th irst s, two os o ot shr primry SA (ot irtly ot i th trust rph), ut hv ommo physil ihor tht shrs primry SA with h o thm. I Fi. 5 (), or ist, os o ot hv primry SA, ut hv ommo physil ihor o tht hs primry SA with h o thm. Sur ommuitios tw os ow hiv vi th hlp o o. W ot th proility tht this vt ours s p. Wh th symmtri ky s ryptorphy is us, th pirwis ky rt y o is st to o throuh th sur pth i th lol trust rph, sy sur liks (, ) (, ) i Fi. 5 (). Th pirwis ky is rypt/rypt i h hop till it rhs th stitio. Wh th symmtri ky s ryptorph is us, o oti o s vli puli ky rom o, rts th pirwis ky whih is st to o irtly, rypt y o s puli ky. I rl, i o is sur pth i its lol trust rph rom itsl to o o its uuthtit ihors, this o opt th mtho sri ov to uthtit h othr stlish irt sur lik tw thm. I th so s, two os o ot shr primry SA (ot irtly ot i th trust rph), ot i ihor stisyi th irst s. Howvr, thr xists o tht shrs primry SA with h o th two os, is physil ihor o oly o o thm. I Fi. 5 (), os () 83

7 This ull txt ppr ws pr rviw t th irtio o IEEE Commuitios Soity sujt mttr xprts or pulitio i th IEEE INFOCOM 8 prois. o ot hv primry SA, ut o hs primry SA with h o thm, o is th physil ihor o oly o. Sur ommuitios tw os hiv vi th hlp o o. W ot th proility tht this vt ours s p. I this situtio, o will irst i th possiility to umt sur lik (,), tk th rsposiility to iitit this prour. Wh th symmtri ky s ryptorphy is us, th pirwis ky rt y o will st to o, th this ky will st k to whih is rypt y os s pirwis ky i thir primry SA. No th rlys th rypt ky to o. Wh th symmtri ky s ryptorph is us, o oti o s vli puli ky rom o, th pirwis ky rt y o is st to o irtly, whih is rypt y o s puli ky. I rl, i o X i is sur pth rom itsl to o o its uthtit ihors, sy X j, o X j os ot hv primry SA with o o o X i s -hop wy ihors, sy X k, whih is o o o X i s ris, th o X i us this sur pth to hlp ths two os uthtit h othr stlish irt sur lik. Proilitis p p lult s th ollowi: (.5865 (.5865 ) p =( p ) (p ) k k k= ( ( p ).5865 k ( p ) k)). () (.87 (.87 ) p =( p ) ( p ) (p ) k k k= ( ( p ).87 k ( p ) k)). (3) Not tht p p r utios o p, = πr is trmi y r. I rl, r p r lso iv s utio o. Thror, p p r utios o oly. Atr th lol sur-lik umttio phs, th proility tht thr xists sur lik tw two ihori os, ot y p, is iv y, p () =p ()+p ()+p (). (4) Th th rph G(X, E sl ) is quivlt to G(X, p ), whr { p p (x) = i x r i x >r. A simpl xmitio o qutios () (3) shows tht p () p () Θ(p ()), whih implis p () Θ(p ()) rom qutio (4). As rsult, w hv th ollowi thorm: Thorm 3: [Et o Lol Sur-Lik Aumttio] Th lol sur-lik umttio shms will ot ltr th orr o th proility or two ihori os to stlish sur lik, i.. p () Θ(p ()). Si w r oly itrst i th symptoti proprtis o th WANET with ti to iiity, Thorm 3 iits tht G(X, E sl ) (i.., G(X, p )) is quivlt to G(X, E sl ) (i.., G(X, p )) si p () Θ(p ()). D. Summry o th Proprtis o Sur Grphs Rsults i Stio III-B show tht th sur rph G(X, ) uros two phs trsitios with th vryi o k, th vr o r o G(X, ). Th two ritil vlus o k, orrspoi to two phs trsitios, sprt th vlu o k ito thr itrvls: () k k p 4.5: sur rph G(X, ) is i th suritil phs, iiti tht G(X, ) os ot prolt. Th tir twork osists o O() smll lustrs, th umr o os i y lustr is ot rtr th O(lo()). () k p <k k = Θ(lo()): sur rph G(X, ) is i th suprritil phs, iiti tht G(X, ) prolts is ot ot w.h.p. I this phs, G(X, ) osists o o iiit-orr lustr som isolt os. Eh o lis i ithr th iiit-orr lustr or o-orr lustr (i.., it is isolt) w.h.p. (3) k>k : th sur rph G(X, ) is ot w.h.p., i.., thr xists oly o lustr. Not tht k = p = p πr, whih iits tht k is trmi y two prmtrs: p rom th trust rph r rom th physil rph. Now, w pro to ivstit th tr-o tw mmory siz, ommuitio ovrh sur otivity i G(X, ). Cot Phs Prvious works [7] [9] sust tht i orr to hiv th ojtiv, t lst G(X, E sl ) shoul ot. Thror, tr th ihori-ri isovry phs, th trust rph G(X, p ) is rt ll th os r ot with sur liks w.h.p. A sur pth tw ritrry o pir stlish with y routi shms oprt o thos sur liks. Nxt, w osir th ommuitio ovrh i this phs. Th sur rph G(X, p ) must hv lowr vr o r ompr to th orrspoi physil rph G(X, ), i tht lowr o r will irs th pth lth whih will t th ommuitio ovrh. Oviously, this ovrh hrtriz y th vr pth lth. Prvious works [7] show tht, i orr to kp th physil rph ot, th vr o r or G(X, E pl ), i.., = πr, shoul o th orr o lo() miimlly, th vr pth lth is O( ). Thollowi thorm iits tht, wh G(X, p ) is ot, th vr pth lth is o th sm orr, whih implis tht th ommuitio ovrh itrou y th sur oprtios is symptotilly liil. Thorm 4: [Avr Pth lth o G(X, p )] Wh G(X, p ) is ot, th vr pth lth ovr ll o pirs (or hop-outs) is O( ). Proo: Cosir th rph G(X, ) with { (x) i x pr = i x > pr. G(X, ) hs th sm vr o r s G(X, p ). Si p is th sqush utio o, th vr pth lth 83

8 This ull txt ppr ws pr rviw t th irtio o IEEE Commuitios Soity sujt mttr xprts or pulitio i th IEEE INFOCOM 8 prois. Fi = p = E[h]=6. p =.9 E[h]=.8 p =.8 E[h]=6.47 p =.7 E[h]= umr o hops, h Et o p o th hop-out istriutio i th sur rph =5 Pro o otivity (lult) 5 5 vr o r, k =5 Pro o otivity (lult) 5 5 vr o r, k = Pro o otivity (lult) 5 5 vr o r, k Fi. 7. Simult lult vlus or th it lustr siz proility o otivity i RGGs (p =) or irt vlus o. =,p =. =,p =.4 =,p =.8 or G(X, p ) is smllr th tht or G(X, ). G(X, ) is lso RGG, hs th sm orr o vr pth lth s G(X, ). Thror, th vr pth lth or G(X, p ) is o th sm orr s tht or G(X, ), i.., is O( ). Fi. 6 isplys th simult vlus o hop-out istriutio or irt p s iv tht th vr o rs rmi th sm. It osrv tht, wh p oms smllr, th vr pth lth rus siiitly. Th rso is tht, to kp th vr o r o th orr o lo(), r shoul orrspoily irs. Thror, th proility o hvi irt lik tw two os t lor ist irss s wll, whih will tivly rs th vr pth lth. Thorm 4 iits tht wh th twork is oprt i th situtio tht G(X, p ) is ot, th sur otivity ojtiv hiv with liil ommuitio ovrh. Th prolm, howvr, is tht it is ot sll, whih is mostrt y th ollowi k-oth-vlop lultio. I orr to kp th physil rph ot, th vr o r or G(X, E pl ), i.., = πr is t lst o th orr o lo() [7]. Thror, i w wt sur rph G(X, E sl ) to ot, p is t lst o th orr o O(), whih ms th vr umr o ris or vry o, i.., p, is i th orr o O(). Oviously, it is urlisti or th trust rph s wll s th rquir mmory siz i h o or primry SAs. Lst, w osir th t o lol sur-lik umttio phs. Thorm 3 shows tht, tr this phs, th proility tht two ihori os shr SA (primry or riv) is p (), whih is o th sm orr o p (). Thror, iv p (), whih ot hiv th otivity o sur rph G(X, p ), lthouh lol sur-lik umttio thiqus hiv p rtr th p,th rsulti sur rph G(X, p ) is still isot. Put it i othr wy, si p () Θ(p ()), it is impossil to rly o lol sur-lik umttio thiqus to ru th rquirmt o th orr o p () or th sur otivity. Suprritil Phs Thorm shows tht i orr to kp G(X, p ) ot, th vr o r must row pproximtly lik lo(), wh th umr o os i th twork irss, whih is xpsiv rsults i poor slility. W show, howvr, tht it is o lor th s i w oly slihtly loos th otivity rquirmt, y just imposi tht G(X, p ) is i th suprritil phs. Mor Pro o otivity (lult) 5 5 vr o r, k Pro o otivity (lult) 5 5 vr o r, k Pro o otivity (lult) 5 5 vr o r, k Fi. 8. Simult lult vlus or th it lustr siz proility o otivity i trust rphs or irt vlus o p. prisly, w wt it lustr to ppr i G(X, p ), tht otis vst mjority o th os, ut w lv smll umr o os out o it (i othr wors, w wt th surly ot os to prolt). This pprtly i h ivs muh mor optimisti prsptiv o th slility o our shm. Fi. 7 8 show th lult simult rsults out th siz o th it lustr th proility o otivity or irt p vlus. Th irst olusio w rw tr lyzi simultio t is tht Cojtur ivs quit urt stimtio o th it lustr siz wh is lr ouh. Soly, i ll simult ss w s tht th it lustr siz is rowi stply towrs or thos vlus o th m r tht th proility o -otivity is vry low. For rltivly lr sp o th m r vlus th it lustr is lry ovri most o th twork ut -otivity is ot hiv yt. This is u to oly w isolt os or smll o lustrs outsi th it lustr. Not tht th it lustr o G(X, p ) orms ot sur ko, i.., ll os i th it lustr r ot with sur liks or pths. It is sy to show tht th vr sur pth lth o o pirs i th it lustr is lso o th orr o O( ), whih ms th ommuitio ovrh (or routi strth) is symptotilly liil. Thror, w ru th mmory siz or h th slility o th twork y tor o lo() with th tr-o o isoltio o ritrry smll rtio o os. Hr, isoltio oly ms tht thr is o sur liks oti isolt os with th sur ko. W still urt th physil otivity y stti tht = πr is t lst o th orr o lo() [7]. Hli Isolt Nos Althouh smll rtio o isolt os is rsol tr-o, i rti viromts w miht to ot ths isolt os to th twork. To ot isolt os to th twork, th isolt os to tt it is isolt. Existi twork prtitio ttio lorithms my us or this purpos. But ori to Thorm, wh th rph ts ito th 83

9 This ull txt ppr ws pr rviw t th irtio o IEEE Commuitios Soity sujt mttr xprts or pulitio i th IEEE INFOCOM 8 prois. suprritil phs, xpt or th it lustr, th siz o th rmii lustrs is vry smll, usully i our simultio rsults. Thus th ost or th prtitio ttio lorithm is ot xpsiv, or wh th lustr siz is, it roiz itsl s isolt without y prtitio ttio lorithm. Th isolt os surly ot to th it lustr (sur ko) y ii o o th os i th it lustr whih shrs primry SA with it. To o this it shoul ithr irs trsmissio r or mov rou or rost th mss to i ri with or mor hops. Th routi-surity py loop voi si tht h isolt o ommuit with th sur ko (th it lustr i G(X, p )) with oly o-hop w.h.p. th ommuitio ovrh ior si th umr o isolt os m ritrrily smll. Suritil Phs It is itrsti to sk whthr w urthr ru th rquir umr o uthtit ihors, i.., k, suh tht k<4.5 th sur rph G(X, p ) is i th suritil phs. Th prolm is tht i this phs, th tir twork osists o O() smll isolt trust roups (lustrs), th umr o os i y trust roup is ot rtr th O(lo()). Th, wh w to stlish pth tw two os, this pth will o throuh O( ) isolt trust roups w.h.p., whih is lmost o th sm orr o th pth lth. Thror, wh G(X, p ) is i th suritil phs, w will mt routi-surity py loop i, lol sur-lik umttio is ot ouh to rk this loop. Our olusio is tht, th sur rph G(X, p ) is t lst i th suprritil phs, k ot urthr ru. IV. CONCLUSION I this ppr, w propos ri mol to vlut th rltioship o otivity, mmory siz, ommuitio ovrh surity i ully sl-oriz WANETs. Bs o som rsol ssumptios o o ploymt moility, w show tht w hiv sur otivity wh th vr umr o uthtit ihors o h o is t lst Ω(). W utiliz otiuum proltio thory to ostrut th sur ko, ot th isolt os to th sur ko with multi-hop sur lik umttio shm. Th ommuitio ovrh iurr u to th ihor uthtitio th routi strth r symptotilly liil. REFERENCES [] J. Huux, L. Butty, S. Cpku, Th qust or surity i moil ho tworks, i MoiHo, Lo Bh, CA, Ot.. [] L. Fy, B. Ahlr, A. 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