Fault Toleat Seso Netwoks with Beoulli Nodes Chih-Wei Yi Peg-Ju Wa iag-yag Li Ophi Fiede The Depatmet of Compute Sciece Illiois Istitute of Techology 0 West 3st Steet, Chicago, IL 6066, USA Email: yichihw@iit.edu, wa@cs.iit.edu, li@cs.iit.edu, ophi@cs.iit.edu Abstact Coectivity, powe cosumig ad fault toleace ae thee citical issues i seso etwoks. I this pape, seso etwoks ae modeled by the uit disc gaph, adom poit pocess ad Beoulli odes. A seso etwok is composed of idetical odes adomly located o a uit aea disc with uifom distibutio. Each ode is associated with pobability p (0 < p ) to be active ad with pobability q p to be iactive. is the l +ο tasmissio adius. As goes to ifiite ad let, the the pobability that each ode eighbos at least oe active ode is e (e ο), the Gumbel eteme-value fuctio, fo 8ο R, ad the cadiality of oe coected compoet i the adom gaph G(; ;p) is eithe o tedig to ifiite. Ide Tems seso etwok, ad hoc etwok, geomety adom gaph, adom poit pocess, uit disc gaph, Beoulli ode, coectivity, fault toleace I. INTRODUCTION A seso etwok is composed of may simila seso odes with ited esouces. Two odes have a diect lik if ad oly if both ae withi the othe s tasmissio age. I ode to commuicate with othe odes outside of the tasmissio age, oe ode eeds to ely o its eighbos to elay messages. Thee ae thee citical issues fo the sesos etwok, coectivity, powe cosumig ad fault toleece. If the etwok is ot coected, it is splitted ito seveal disjoit pats ad each pat ca t commuicate with each othes. Oe method is to icease the sigal powe, but that icease powe cosumig. I this pape, the seso etwok is modeled by the uit disc gaph, the Beoulli odes, ad the adom poit pocess o a uit aea disc. The uit disc gaph is a widely used model. Each ode has the same tasmissio age, a disc ceteed at that ode. The Beoulli ode model is fo the fault toleace issue. Each ode is eithe active o iactive with Beoulli model. The pobability that oe ode is active is p ad the pobability that oe ode is iactive is q (p + q ). Thee eists a diect lik betwee two odes if ad oly if both ae active ad thei distace is less tha the tasmissio adius. The adom poit pocess is used to model ode distibutig. D f R jjj < p g is a uit aea disc ceteig at the oigi. Thee ae odes distibuted o D with idepedet ad idetical uifom distibutio. Thei positios ae epeseted by ; ; ;, adom vaiables, ad i 6 j if i 6 j. Coectivity, powe cosumig ad fault toleace ae discussed togethe i this pape. If the tasmissio adius is a fuctio of the umbe of Beoulli odes, what is the pobability of the evet that the adom gaph is coected? Hee the umbe of odes is deoted as ad the tasmissio age is deoted as. If without causig cofusio, we may oly wite fo. We will show that if l +ο, the iduced adom gaph has good coectivity with the pobability e (e ο) fo ay eal umbe ο ad 0 < p as!. Λ(ο) e (e ο) is the Gumbel eteme-value fuctio. Relative poblems ae discussed i diffeet models fo diffeet applicatios. The adom gaph poblems [] coside the popeties of a gaph with odes ad k edges chose adomly fom the complete gaph K. I the geomety adom gaph poblems [][3][4][5][6][7], the positio is oe of the key ifomatios. The locatio of a odes is a adom vaiable ad the eistece of a edge betwee two odes is depedet o the geometic ifomatio. Rectagle (cube) [][3][4][5][6] ad disc (ball) [4][7] ae the two most popula bouded topologies o which the adom vaiable ae defied. If the adom vaiable is defied o a bouded aea, the pobability distibutio as poits locatig close to the bouday is diffeet to the pobability distibutio as poits locatig i the iteio aea. This is bouday effects. Thee ae seveal methods to hadlig bouday effects. Heze [] give the asymptotic distibutio of the maimal th-eaest-eighbo o d-dimesio cube ad with poit distibutios. He avoided bouday effects by the delicate defiitio of th-eaest-eighbo. Dette et al. [3][4] discussed the poblem with uifom poit distibutio ad eteded the esult to d-dimesio cube ad ball with diectly hadlig bouday effects. They also show that bouday effects depeds o the space dimesio ad the topology. Peose [5] applied tooidal metics to avoid bouday effects fo the logest edge of the miimal spaig tee poblem. He [6] applied the elatio betwee adom poit pocess ad Poisso poit pocess to solve the pobability of k-coectivity. The esult of cotiuum pecolatio [8][9][0][] is applied to solve the coectivity poblem [5][6][7]. Sectio II is about otatios ad basic ideas. The calculatio of the asymptotic pobability that the adom gaph is -degee is give i sectio III. Sectio IV give the elatio betwee - degee popety ad coectivity. Sectio V is the coclusio ad futue woks. II. PRELIMINARY AND NOTATION The poblem is discussed o R space with L metics i this pape. D () fy j j yj <gis the ope disc with adius ad cete. D D p ((0; 0)) is the uit aea disc with cete at the oigi. f ; ; ; gρdis a istace of adom poits with idetical, idepedet ad uifom
θ π p, ffi si p. D 0 is the disc bouded by the smallest cicle. D is the ig betwee the smallest cicle ad the medium cicle. D is the ig betwee the medium cicle ad the lagest cicles. @D D [ D is the aea outside of the D 0. The aeas of these egios usually is estimated by jd 0 j ( p ) o Fig.. θ π D 0 D D Patitios of the uit aea disc D p δ q j@dj p jd j p ( ffi) p jd j p ffi (as is small eough) The methodology fo the eteme-value poblem [] used by Heze ad Dette i [][3][4] is adapted to fid the asymptotic pobability that G(; ;p) is a -degee gaph. Sice ; ; ; ae adom vaiables with iid, the fo ay f s ; s ; ; sk g f; ; g P( ; ; k ae isolated poits) P( s ; ; sk ae isolated poits) Usig the iclusio-eclusio piciple, we have Fig.. D y How to calculate ffi distibutio o D. Each oe of them epesets the locatio of a seso with same tasmissio adius. The tasmissio adius, deoted by, is a fuctio of. If without aisig ambiguity, it may just be witte as. Each ode is eithe active o iactive with pobability. The pobability that oe ode is active is p ad the pobability that oe ode is iactive is q (p + q ). Thee eists a diect lik betwee two odes if ad oly if both ae active ad thei distace is less tha. G(; ;p) deotes the gaph iduced by a istace of adom poits pocess with tasmissio adius ad Beoulli ode pobability p. The vete set is composed of all the active odes ad the edge set is composed of all the diectly liks. D is divided ito 3 disjoit egios, D 0, D ad D, accodig to the tasmissio adius. See Fig.. Thee ae 3 cicles all with same cete but with diffeet adii p, p ffi ad p. The lagest oe is the bouday of D. The smallest oe is scatched by the cetes of cicles with adius ad taget at the bouday of D. The medium oe is scatched by the cetes of cicles with adius ad whose two itesectio poits with the bouday of D costitutig its diamete. See Fig. fo the calculatio of ffi. o is the cete of D ad p is the cete of the disc with adius., y ae the two itesectio poit of these two discs. Sice o oq ad oq?y, we ca get 6 pq 6 oq.so ffi ta.if 0, i.e. P(G(; ;p) is -degee) P(thee eists at least oe isolated poit) + k ( ) k C k P( ; ; k ae isolated poits) ( ) k + Pk k! P( ; ; k ae isolated poits) k Fo ay eal umbe ο, 0 <ο<, thee eists (ο) such that fo all iteges k, the! k P( ; ; k ae isolated poits) e kο! P(G(; (ο);p) is -degee) e (e ο ) The pobability that ; ; k ae isolated poits would be a itegal o D k f( ; ; k ) j ; ; k D g, the k- fold Catesia poduct of the uit aea disk. Depedig o the distace betwee ; ; k, D k will be splitted up. Give D k ad, a equivalet elatio is iduced. i is equivalet to j, if ad oly if, eithe i j o thee eists a itege sequece (i ;i ; ;i m ) such that i ; ;i m k, i i, i m j, ad D ( is ) D ( is+ ) 6 ; fo all s<m. l() (l ();l (); ;l k ()) is a k-tuple. Hee l i () is the umbe of equivalet classes with i elemets. It is obvious that kp S il i () k, 8 D k. Let L k f(l ;l ; ;l k ) j kp il i kg ad D k (L L k )f D k j l() Lg. The D k LL k D k (L) ad D k (L ) D k (L ); if L 6 L. Let
a a t b Fig. 3. Two itesected discs b L 0 to deote (k; 0; ; 0). The D k (L 0 )f D k jj i j j, 8i 6 jg is the space that D ( i ) D ( j );, 8i 6 j. Fo a give L (l ;l ; ;l k ) L k, D k (L) ca be futhe decomposed. We focus o those D k (L) that f g,,f l g fom all l s -elemet equivalet classes, f l+; l+g,, f l+(l )+; l+(l )+g fom all l s -elemet equivalet classes, ad so o. Let ed k (L).deote the space fomed by this kid of. Suppose f () is a fuctio symmetic to all aes, the k! f ()d Q k D k (L) (i!)li l i! f ()d ed k (L) The calculatio of pobability is elated to the measue of discs. A ( D) jd () Dj is the aea of D () withi D, ad A k ( D k )j D ( i ) Dj is the aea of ks D ( i ) withi D. The followig lemma is fo estimatio of those aeas. Lemma : If is small eough, the ) If t j j, thee eists a costat c such that jd ( ) [ D ( )j + ct. ) If D ad t (t [ffi;]) is the distace fom to the bouday of D, thee eists a costat c such that A () + ct. 3) If D 0, D ad t j j, thee eists a costat c such that A (( ; )) A ( )+ct. 4) If @D, j jj j ad t j j, thee eists a costat c such that A (( ; )) A ( )+ ct. Poof: (Pat ) See Fig. 3. Coside f (t) jd ( )D ( )j. Sice f 0 q (t) ja bj ( t ) is deceasig if t [0; ], f (t) is cove. Ad f (0)0ad f (), so let c. This is poved. (Pat ) See Fig. 4. Coside f (t) A (). f 0 (t) ja bj is deceasig if t [ffi;], f (t) is cove if D. Let c. Sice ρ f (ffi) > + ffi f () > + So this is poved. ks Fig. 4. Disc with cete o D (Pat 3) If D 0, it is educed to (Pat ). If @D, coside the wost case, i.e. is o the bouday of D 0 ad is as close to the bouday of D as possible. Let f 3 (t j j) mi j(d ( )D ( )) Dj. If tj j t [0;], f 3 (t) icease ad the decease ad f 3 () p 3 6 c (). Ift [; ], f 3 (t) is a cove fuctio sice f 3 (t) f (t) c (). Hee c () 0. Let p!0 c mi( 3 6 ; ) ". So this is poved. (Pat 4) Fo a give, the miimal value of j(d ( ) [ D ( )) Dj happes as j j j j. Whe is small eough, the aea of (D ( )D ( ))D is just a little moe tha half of D ( )D ( ). Let c ". This is poved. 4 The followig iequality [][] is useful to estimate the uppe boud ad lowe boud. e mz ( z) m (e mz ) < ( z) m e mz () It woks fo all itege m ad eal umbe 0 <z<. I a -poit adom poit pocess, we use the followig otatios fo coveiece. N i is the umbe of i s eighbos. k f ; ; k do t have active eighbosg. Y k f ; ; k fom a equivalet classg. ij f i has j eighbos ad all ae iactiveg. Y k f ; ; k fom a k-elemet coected compoetg. E k f D k j ; ; k fom a equivalet classg: E k f D k j ; ; k fom a k- elemet coected compoetg. I those otatios, is omitted. III. ASYMPTOTIC PROBABILITY OF -DEGREE I this sectio, the pobability that the adom gaph is - degee is give. With the Beoulli ode model, each ode is active o iactive. The active odes should dese eough such that G(; ;p) is -degee ad each iactive ode also has at least oe active eighbo. So G(; ;p) ca always keep - degee eve whe a iactive ode tus to active. The majo goal hee is to figue out the pobability P(each ode has at least oe active eighbo)
The detail calculatio, followig the outlie i Sectio II, is give. The poof is divided ito thee steps, P( does t have active eighbos), P( ; do t have active eighbos), ad k P( ; ; k do t have active eighbos). Lemma : ; ; ; ae adom poits with uifom distibutio o the uit aea disc D. is the tasmissio adius. Each odes idepedetly associates with success pobability p. Hee p, ο ae fied eal umbes ad 0 <p. Let l +ο, the! P( does t have active eighbos) e ο Poof: might have eighbos but all of them must be iactive. The pobability ca be calculated by the umbe of s eighbos. i0 P(all eighbos ae iactive j N i)p(n i) R P(N i) is equal to D C i ( A ()) i A () i d ad P(all eighbos ae iactivej N i) is equal to q i.so P( does t have active eighbos) q i i0 D i0 D ( A ()) i A () i d ( A ()) i (A ()q) i d ( A () +qa ()) d D ( pa ()) d D Depedig o the locatio of, the pobability is calculated o D 0, D ad D.OD 0, usig iequality ad e p d e (l +ο) (( p ) ) D 0 The we have e ο (( p ) )! ( pa ()) d D 0! e p d D 0 e ο O D, apply Lemma ad Eq. D ( pa ()) d p e p( +ct) p dt tffi p p e ( ( e c )) c ( e c p ) p ( p )c pe ο (l + ο) O( pl )! 0 as! O D, A () ( c ( )) ad c ( )0.!0 ( pa ()) d D p e ( c ( ))p p ( ffi ) p e ( c ( )) ( p )e ( c())ο p! 0 as! Combiig these esults, Lemma is poved. l + ο ( c()) The et step is to coside the case i which both ; do t have active eighbos.. Lemma 3: ; ; ; ae adom poits with uifom distibutio o the uit aea disc D. is the tasmissio adius. Each odes idepedetly associates with success pobability p. Hee p, ο ae fied eal umbes ad 0 <p. Let l +ο, the! P( ; do t have active eighbos) e ο Poof: The pobability is calculated i two cases depedig o the distace betwee ad. P( ) P( f g)+ P( f < g) Fist, we show the secod tem teds to 0 as!. Let A () j(d ( )D ( )) Dj, A () j(d ( )D ( )) Dj ad A () j(d ( ) D ( )) Dj. It is obvious that A () A ()+A ()+ A (). Let i,j ad k deote the umbe of odes eceptig ; locatig at A (), A () ad A (). If <, the pobability P( f < g) is equal to D d j j< i+j+k0 C i Cj i C i j k ( A ()) i j A () i A () j A () k q i+j+k
If <, both ; must be iactive ad the pobability P( f < g) is equal to D d j j< i+j+k0 C i Cj i C i j k q ( A ()) i j A () i A () j A () k q i+j+k Combie these two esult ad afte staightfowad calculatio, we have P( f < g) D ( pa ()) d j j< Coside the bouday effect, this itegal ca be evaluated i two cases depedig o D 0 o @D Hee C P( f < g) D ( pa ()) d j j< C(F +F ) ( p ad ) F j j< e pa() d D 0; D F j j< e pa() d @D;j jj j Fo F, apply A () + c j j (Lemma ) j j< e pa() d D0 ; D (( p ) )(e p O(l ) t0 e cpt tdt) Fo F, apply A () A ( )+c j j (Lemma ) j j< e pa() d @D;j jj j e p(a()+cj j) d d ( j j< @D;j jj j e pa() d )( e pct tdt) @D t0 O(l 3 ) We still eed to show that P( f g) teds to e ο as!. Let i,j deote the umbe of odes located i A ( ) ad A ( ). P( f g) d D (L 0) i+j0 Cj j ( A ()) i j (qa ( )) i (q A ( )) j ( p(a ( )+A ( ))) d D (L 0) Apply Eq. ad A ( )+A ( ) A (), the P( f g) ο D e (A( )+A ( )) d ( Lemma is poved. e A() d) D ο ( P( does t have active eighbos)) e ο as! The last pat is to pove the geeal case k 3. The agumet hee also woks fo the case k. Lemma 4: ; ; ; ae adom poits with uifom distibutio o the uit aea disc D. is the tasmissio adius. Each odes idepedetly associates with success pobability p. Hee p, ο ae fied eal umbes ad 0 <p. Let l +ο, the 8k! k P( ; ; k do t have active eighbos) e kο Poof: The pobability is discussed i two tems. If goes to ifiite, the fist tem teds to e kο ad the secod tem teds to 0. k P( k ) k P( k f D k (L 0 )g) + k P( k f D k D k (L 0 )g) I Lemma 5, we will show that fo ay k! k P( k Y k )0 Fo ay fi k ad L L k fl 0 g, apply this esult, the k P( k f D k (L)g) kq kq k! (i!) li l i! k! (i!) li l i! k P( k f e Dk (L)g) ky ( i P( i Y i )) li 0 as!ad L 6 L 0
The et is to fid out! k P( k f D k (L 0 )g). k P( k f D k (L 0 )g) k k P( k ini f D k (L 0 )g) N +N + N k 0 kp k k k N i ( A k ()) D k (L 0) N +N + N k 0 Sice get ky k k P (C P i k N j N i j D k (L 0) (A ( i )q) Ni )d ( p k A ( i )) k d A ( i ) A k () ad usig pevious esult, we ca! k D k D k (L 0) Ad usig Eq., the ο ο ( ο ( p k k P( k f D k (L 0 )g) k D k (L 0) Lemma is poved. e pa() d) k D e pp k Ai() d A ( i )) k d 0 ( P( does t have active eighbos)) k e kο as! The last lemma is to give the pobability ude the coditio that ; ; ; k fom a equivalet class. Lemma 5: ; ; ; ae adom poits with uifom distibutio o the uit aea disc D. is the tasmissio adius. Each odes idepedetly associates with success pobability p. Hee p, ο ae fied eal umbes ad 0 <p. Let l +ο. The fo ay fied itege k! k P( k Y k )0 Poof: Usig simila agumet i Lemma 3, we ca get k P( k Y k ) k E k ( pa k ()) k d E k is goig to be splitted up. Let M f0; ; g, m M k, ad E k (m) f E k j i D mi g, the E k E k (m). mm k S Fo a give m, let m ma ad m mi deote ma m i ad ik mi m i. We ca get je k (m)j O( m ma ( ) k ) ad ik A k () mmi. Afte staight fowad calculatio, k ( pa k ()) k d E k (m) O( ( m mi + mma ) l k + mma ) If m ma > m mi o (m ma ;m mi ) (; ), the pobability teds to 0 as!. Fo the othe two coditios, i.e. (m ma ;m mi ) is equal to (0; 0) o (; ), the pobability ca be estimated moe tightly. (m ma ;m mi )(0; 0) meas m k l l (0; ; 0). Let k ( l ), S f E k ((0; ; 0)) j 9i 6 j s.t. j i j j k g ad S DS.IfS,we k l l have A k () ( + l ), e pak() O( l k ) ad js j O(( l )k ). The k S ( pa k ()) k d O(l ) 0 as! If S, it meas j i j j < k fo 8i 6 j. k S ( pa k ()) k d O((l l ) k ) O((l l ) k l ) 0 as! D0 ( pa ()) k d j j< k Simila agumet ca apply to the case (m ma ;m mi )(; ). So this is poved. Theoem 6: ; ; ; ae adom poits with uifom distibutio o the uit aea disc D. is the tasmissio adius. Each odes idepedetly associates with success pobability p. Hee p, ο ae fied eal umbes ad 0 < p. Let l +ο. The P(each ode has at least oe active! eighbo) ) e (e ο Poof: Theoem follows Lemma, 3, 4. IV. CONNECTIVITY IN G(; ;p) I this sectio, the coectivity of G(; ;p) is the majo coce. We show that the pobability that thee eists k- elemet coected compoet (k ) ig(; ;p) teds to 0 as!. So oe ode is eithe isolated o belogs to a coected compoet with cadiality teds to ifiite as!.
Theoem 7: If U is coected compoet i G(; ;p) with l +ο, Cad(U ) is eithe o as!. Poof: Suppose Cad(U ) 6.Ifk is a fied itege, P(f9U s.t. Cad(U )kg) Ck P(W k) k k E k i0 k E k ( pa k ()) k d k (A k ()q) i ( A k ()) k i d Use the esult i Lemma 5 ad E k ρ E k, the 8k Lemma is poved. P(f9U s.t. Cad(U )kg) 0! As teds to ad G(; ;p) is -degee, the thee ae o isolated poits i G(; ;p) ad the cadiality of coected compoets teds to ifiite. V. CONCLUSION The fault toleace of seso etwoks is ivestigate by the pobability of ode failue. Let l +ο. The as!, the pobability of each odes has at least oe active ode is e (e ο) ad the cadiality of a coected compoet is eithe o tedig to. We believe that G(; ;p) is almost sue coected if G(; ;p) is -degee. But this still eeds to be poved. Beside the ode failue model, othe failue models ae also iteestig. I the lik failue model, we also have simila esult. REFERENCES [] B. Bollobas, Radom Gaphs. Academic Pess, 985. [] N. Heze, The it distibutio fo maima of weighted th-eaesteighbou distaces, Joual of Applied Pobability, vol. 9, pp. 344 354, 98. [3] H. Dette ad N. Heze, The it distibutio of the lagest eaesteighbou lik i the uit d-cube, Joual of Applied Pobability, vol. 6, pp. 67 80, 989. [4] H. Dette ad N. Heze, Some peculia bouday pheomea fo etemes of th eaest eighbo liks, Statistics & Pobability Lettes, vol. 0, pp. 38 390, 990. [5] M. D. Peose, The logest edge of the adom miimal spaig tee, The aals of applied pobability, vol. 7, pp. 340 36, 997. [6] M. D. Peose, O k-coectivity fo a geometic adom gaph, Radom Stuctues ad Algoithms, vol. 5, pp. 45 64, Septembe 999. [7] P. Gupta ad P. R. Kuma, tical powe fo asymptotic coectivity i wieless etwoks, i Stochastic Aalysis, Cotol, Optimizatio ad Applicatios: A Volume i Hoo of W.H. Flemig (W. McEeaey, G. Yi, ad Q. hag, eds.), 998. [8] H. Keste, Pecolatio Theoy fo Mathematicias. Bikhause, 98. [9] R. Meeste ad R. Roy, Cotiuum Pecolatio. Cambidge Uivesity Pess, 996. [0] K. S. Aleade, Pecolatio ad miimal spaig foests i ifiite gaphs, Aal of Pobability, vol. 3, pp. 87 04, Ja. 995. [] M. D. Peose, O a cotiuum pecolatio model, Adv. i Appl. Pobab., vol. 3, pp. 536 556, 99. [] J. Galambos, The Asymptotic Theoy of Eteme Ode Statistics. Wiley, 978.