Network reliability importance measures : combinatorics and Monte Carlo based computations

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1 7th WSEAS Itratoal Cofrc o APPLIED COMPUTER SCIENCE, Vc, Italy, Novmr 2-2, Ntwork rlalty mportac maur : comatorc ad Mot Carlo ad computato ILYA GERTSAKH Dpartmt of Mathmatc Guro Uvrty PO 65 r Shva 8405 ISRAEL YOSEPH SHPUNGIN Dpartmt of Softwar Egrg Ngv Acadmc Collg of Egrg ad NMCRC PO 950 r Shva 8400 ISRAEL Atract: - I th papr w focu o computatoal apct of twork rlalty mportac maur valuato It a wll kow fact that mot twork rlalty prolm ar NP-hard ad thrfor thr a gfcat gap tw thortcal aaly ad th alty to comput dffrt rlalty paramtr for larg or v modrat twork I th papr w prt two vry ffct comatoral Mot Carlo modl for valuatg twork rlalty mportac maur Ky-Word: - Ntwork, Rlalty, Importac Maur, Mot Carlo, Comatoral Approach Itroducto Computg twork rlalty vry mportat ut t ot th oly prolm rlalty aaly O of th purpo of twork rlalty aaly to dtfy th wak a ytm ad to quatfy th mpact of compot falur o th twork falur Th o calld "rlalty mportac maur" ar ud for th purpo Th mportac maur provd umrcal dcator to dtrm whch compot ar mor mportat for twork rlalty mprovmt or mor crtcal for ytm falur May dffrt mportac maur wr propod ltratur ut thr a gfcat gap tw thortcal aaly ad th alty to comput th maur for modrat or larg twork Thrfor ug Mot Carlo (MC mthod olvg uch prolm vry popular Th c of mot MC applcato th o calld Crud Mot Carlo (CMC Th ma drawack of CMC that t vry ffct two xtrm ca: hghly rlal ad hghly urlal twork (th o calld rar vt phomo Our purpo th papr to dcr how two vry ffct MC modl ca ud for valuatg twork rlalty mportac maur Th commo fatur for th two modl that th approprat mulato chm ar homogou Lt u xpla th lattr oto pla word Codr a ur U wth a larg umr of all t Suppo that ach all markd wth om valu ( ad w wat to calculat th um of ( ovr U: Z = ( ( U Th compltly match th computato of twork rlalty I th ca, th all ar th tat, ad ( ar dfd a 0 for ay ad tat ad qual th proalty of th tat f t Good Thrfor, Z com th rlalty of th twork Sc th umr of all U vry larg, th whol um caot computd prcly, ad w ar forcd to tmat Z y om MC chm W ay that MC chm homogou, f th all ar draw from th ur wth proalty whch do ot dpd o th proalt of th tat (mor o homogou chm [] O mportat fatur of homogou chm that th rlatv rror oudd (A ac xampl of a ohomogou chm th o calld Crud Mot Carlo ad t varato A t wa mtod aov, th ma purpo of th papr to prt ffct Mot Carlo mthod applcal to larg twork Th papr orgad a follow I Scto 2, w gv om ac oto ad dfto I Scto, w prt a computatoally ffct modl [,2] for valuatg rlalty gradt vctor, whch ca ud for

2 7th WSEAS Itratoal Cofrc o APPLIED COMPUTER SCIENCE, Vc, Italy, Novmr 2-2, computg raum Importac Maur gral ca of twork wth o-dtcal lmt For twork wth dtcal lmt w propo Scto 4 a hghly ffct pctrum approach [-5] It' worth otg that th approach provd aly mplmtd computato ad allow otag wth mmal dffcult dffrt topologcal fatur of th twork Scto 5 prt a umrcal xampl 2 ac oto ad dfto All twork hav vrtc (od ad dg Thr ar may typ of twork varyg thr prformac dfto ad thrfor wth dffrt cocpt of thr rlalty Lt K-twork a udrctd graph N = ( V, E, K wth a od-t V, a dg-t E ad a t K V of pcal od calld trmal Alo lt V = m ad E = I our modl, od ca vr fal, whl dg ca If a dg fal, w ay that t dow; othrw w ay t up y tat of a twork w call a ary vctor ( x,, x, whr ach compot x = f up ad x = 0 othrw A tat of th twork dfd a g Good f ay two trmal ar coctd y a path cotg of dg th up tat Othrw t ad Th trmal coctvty crtro ha th proprty of g mooto: ach ut of a ad tat a ad tat ad ach uprt of a Good tat a Good tat Thr ar two twork rlalty modl: tatc ad dyamc I th papr w rtrct our attto to tatc twork Each dg aocatd wth proalty p of g up ad a proalty q = p of g dow W ay that dg ar dtcal f thy all hav th am proalty of g up, that for ach w hav p = p = p W df th twork rlalty R= R( p,, p a th proalty that th twork a Good tat Th followg ar th ma compot rlalty mportac maur propod ltratur: raum Importac Maur (IM [6], Full- Vly Importac Maur (FVIM [7], Crtcalty Importac [8], Rlalty Achvmt Worth [9], Rlalty Rducto Worth [9] I our papr w focu o th two frt of thm Th IM of lmt dfd a ( p,, p I = (2 It xpr th rat of cra of th twork rlalty wth rpct to th lmt' rlalty cra Rmark For qual p p, frt th drvatv ar computd ad oly aftrward all qual p p t to Th FVIM of lmt dfd y FV R( p,, p,0, p+,, p I = ( R( p,, p It quatf th dcrmt ytm rlalty caud y a partcular compot falur Ug Rlalty Gradt for IM valuato A t wa mtod Itroducto, th rlalty gradt vctor ca ud for IM valuato Th purpo of th cto to dcr a pcal form of th gradt vctor whch allow ug a hghly ffct Graph Evoluto Modl [2] for t computato Smlar form of th gradt wa outld [] W wll gt hr a pcal form of gradt for mor gral ca of mooto ytm Lt u codr a mooto ytm of lmt Suppo that ach lmt may two tat : up wth proalty p ad dow wth proalty q Th tat of a ytm dfd a a ary vctor ( x,, x, whr ach compot x = f up ad x = 0 othrw All 2 ary tat ar dvdd to two cla: Good ad ad Dfto Rlalty gradt vctor R dfd a R= (,,, compot of p p th rlalty gradt vctor IM of lmt Dfto 2 Sytm tat w= ( w,, w ad calld drct ghor or mply ghor of tat v= ( v,, v UP f w dffr from v xactly o poto Th t of all ghor tat of DOWN calld ordr t ad dotd a DN Ovouly, DN DOWN It tur out that th rlalty gradt vctor tmatly rlatd to ordr tat To rval th cocto, w troduc a artfcal voluto proc o ytm lmt At t= 0 all lmt ar dow Elmt "or" aftr radom tm τ ~ xp( λ, whr λ cho o that th followg qualty tak plac: p = P( τ = λ Aftr th "rth", lmt rma up forvr Codr two ytm tat v= ( v, v,, v,0, v, v ad 2 +

3 7th WSEAS Itratoal Cofrc o APPLIED COMPUTER SCIENCE, Vc, Italy, Novmr 2-2, w= ( v, v2,, v,, v+, v Suppo that at tm t th ytm tat v W look for th proalty that durg a mall tm trval t th ytm mov from v to w Ovouly, t wll happ ff th lmt or durg th trval, ad all othr compot whch ar tat 0 wll ot com alv durg th am trval Th frt vt ha proalty λ t+ o( t, ad th cod vt ha proalty o( t Th th proalty that durg [ t, t+ t] thr wll th trato v w qual λ t+ o( t Lt v a ordr tat of ytm, v DN Dot y Γ( v th um of λ ovr th t of all dc uch that v+ (0,,,0 Good Call Γ( v th flow from v to Good Formally, Γ ( v = λ W d { v DN, v+ (0,,0,,0,,0 UP } two othr otato Lt R( p ( t,, p ( t th proalty that th ytm Good tat at th tat t Lt P( v; t th proalty that th ytm tat v at tm t Now lt u codr th vt "th ytm Good tat at tm t+ t " Th vt tak plac f at tm t th ytm wa alrady th Good t or at tm t t wa o of t ordr tat ad wt durg th trval from a ordr tat to Good All othr polt whch volv mor tha o trato durg [ t, t+ t] hav proalty o( t Formally, R( p ( t+ t,, p ( t+ t = R( p ( t,, p( t + P( v; t Γ( v t+ o( t v DN Trafr R( p ( t,, p ( t to th lft-had d, dvd oth d y t ad t t 0 W arrv at th followg rlatohp: dr( p ( t,, p ( t = P( v; t Γ( v (4 dt v DN Now, rprt th lft-had d of (4 a altratv form: dr( p ( ( t,, p ( dp t dr t = = dt = dp dt dr ( p ( t =, q ( t = = q λ = λ t t λ = dp R { qλ,, q λ } (5 Comparg (4 ad (5 w arrv at th drd rlatohp tw th gradt vctor ad th ordr tat proalt: R { qλ,, q λ } = P( v; t Γ( v (6 v DN From th lattr formula w ca gt th xpro for IM of ytm lmt th followg mar It follow from th aov proof that f tad of gral vctor { qλ,, q λ } w tak pcfc vctor {0,, qλ,,0} th w gt th followg formula: R q λ = R {0,, q λ,,0} = P( v; λ (7 { v DN, v+ (0,,,,0 Good } Exampl Lt u tak th twork gv Fg ad comput th IM for dg Th approprat ordr tat v uch that v+ (,0,0,0 Good ar: S = (0,, 0, 0, S 2 = (0,,,0, S = (0,, 0, Th w gt y (7: q λ = λ ( P( S + P( S 2 + P( S = λ ( p2 q q q4+ p2 p q q4+ p2 p4 q q Dvdg th oth d of th lattr xpro y qλ, w arrv to th IM of S Fg 4 2 Th aov xampl dmotrat computato va formula (6 It ovou that th ma tchcal dffculty l dtfyg th ordr tat ad fdg thr proalt Computato mlar to how th aov xampl ar dffcult to carry out for larg or modrat twork Thr howvr a powrful computatoal Mot Carlo tchqu ad o troducg a pcal chm calld Evoluto ad Mrgg proc, whch allow ffct tmato of xpro of typ (6 It wa frt uggtd th prcpal papr [2] W wll lav th rlvat dtal outd th papr ad prt thm ad th corrpodg umrcal rult our forthcomg papr 4 Spctral approach to computg twork rlalty mportac maur I th cto w wll drv th IM ad th FVIM for twork wth dtcal lmt y ma of o calld twork comatoral pctrum Th oto T

4 7th WSEAS Itratoal Cofrc o APPLIED COMPUTER SCIENCE, Vc, Italy, Novmr 2-2, wa troducd [] ad [4,5] to tmat twork lftm dtruto ad /or t tatc rlalty For radr covc w rmd hortly th prcpal da of twork comatoral charactrtc calld pctrum For mplcty w dmotrat th mthod for th ca of rlal od ad urlal dg It wa how [4] that th approach applcal alo to th ca of rlal dg ad urlal od, or whch mor complcatd to th ca of oth urlal od ad dg Lt Π E th t of all dg prmutato E Lt π a partcular prmutato y u-prmutato π ( of π w dot a quc cotructd of th frt dg π For ach u-prmutato π ( w df a twork tat S( π (, whr all th dg π ( ar up ad all othr dg π ar dow For ach dg ad prmutato π dot y π ( th dx of th dg π Exampl 2 Lt u tak th twork Fg ad lt our prmutato π = (,, 2, 4 Th, for xampl, π ( = (,, 2 ad S( π ( a tat whch dg,,2 ar up, ad dg 4 dow W hav alo: π ( =, π ( 2 =, π ( = 2, π ( 4 = 4 Nxt w df a achor Th oto play a ctral rol our raog Dfto Lt r= r( π th frt dx prmutato π o that N( π ( r Good W ay that r( π th achor of th prmutato π Dfto 4 Dot y x th umr of all prmutatoπ uch that th achor ofπ W ay that th t SP={{ x}, } (8 th comatoral pctrum of th twork Exampl W dmotrat th dfto o a twork gv Fg Th total umr of prmutato of 4 dg th twork 24 Lt π = (,,2,4 W that th frt dx uch that th twork tat com Good Thrfor r( π = r(,,2,4 = th achor of th prmutato Aftr gog ovr all prmutato w arrv at th followg comatoral pctrum of th gv twork: 2 4 x It wa how [4] that gv a twork pctrum SP= {{ x}, }, th twork rlalty may xprd th followg form: R= xr (9 r= = r! (! I our xampl, th twork rlalty : R= xr = p + p q+ p q r= = r! (4! Rmark Somtm t mor covt to u th cumulatv form of th pctrum: SP = { y : y = x, } (0 k= Th valu y xpr th umr of prmutato π uch that r( π, or, othr word, that N( π ( Good Exampl 4 For th twork o Fg, w hav from th aov xampl that thr ar 4 prmutato wth achor r= ad 4 prmutato wth achor r= 2 So, w gt y = 8 It ay to chck from (9 that th ca of th cumulatv pctrum th twork rlalty gv y R= y ( =! (! Clarly, th ca of larg or modrat twork w ca ot gt th xact valu of th pctrum W ca howvr try to tmat thm y a Mot Carlo mulato [,4] It worth to mto th ma advatag of th comatoral approach: (a lmatg th rar vt phomo Th fact rult oudg th rlatv rror, o th mthod pcally ffct for hghly rlal twork ( oc computd, th comatoral pctrum rv for a may valu of od or dg falur proalt a dd (c polty to u for olvg dffrt rlalty prolm dyamc twork Dfto 5 Dot y, th umr of all prmutato π uch that S( π ( Good ad π ( W call th t {,,, } - th IM pctrum Dfto 6 Dot y v, th umr of all prmutatoπ uch that S( π ( Good ad π ( > W call th t { v,,, } - th FVIM pctrum W from th dfto that, + v, = y Tal y,,2,,

5 7th WSEAS Itratoal Cofrc o APPLIED COMPUTER SCIENCE, Vc, Italy, Novmr 2-2, Exampl 5 Lt u tak th dg from th twork Fg ad lt u comput, It ay to that thr ar 6 prmutato π uch that S( π ( Good ad π ( > So w gt y, = 6 From th prvou xampl, y = 8 Hc, w gt that, = 2 Th IM pctrum for th twork gv Tal Clam (a Th IM for dg gv y th followg formula:, ( y, I = (2 =! (! ( Th FVIM for dg gv y th followg formula: v, FV I = ( R( p =! (! Proof (a Rmd that IM for dg qual = R( p,,, p R( p,,0,, p Th valu,, y dfto, th umr of prmutato π uch that S( π ( Good ad th dg up For fxd prmutatoπ th proalty of a approprat tat wth g up, qual p q Tak to accout that a pcfc tat wth dg g up ad - dg g dow w ota! (! tm (from dffrt prmutato Th th ummary proalty of all Good tat wth dg g up ad - dg g dow qual, For th ca of th dg g! (! dow w gt th xpro of th approprat ( y, proalty a, ad (a follow! (! FV ( Ug (a, th dfto of IS ad th aov mtod fact that, + v, = y, w arrv at th drd xpro I ordr to rak th lmt accordg to thr mportac maur thr o d to comput th partal drvatv Th followg mpl clam tak plac Clam 2 Lt {, } ad {, } th IM pctrum lmt for th dg rpctvly Th: (a If for all th qualty th ad hold, Morovr, f for at lat o dx t hold >, th > ( Suppo that th codto of (a do ot tak plac Tha lt th k th maxmal dx uch that Suppo that k > k Th thr xt om valu p 0 uch that for all p p0 th qualty > hold Proof (a From (2 w hav:, ( y, = =! (! ( ( =! (!,,,, (,, ad (a follow =! (! ( From th dfto of k ad th lattr xpro w ota: k (,, = =! (! k (,, k k q k ( ( ad for p, th =! (! p arto follow W u th followg Mot Carlo chm to ota uad tmat for th y,, ad v, Smulato chm Stp Ital all a,, ad c, to 0 Stp2 Smulat th prmutatoπ Π Stp Fd r= r( π - th mmal dx of dg π o that th tat N( π ( r Good Stp4 Lt ar : = ar + Stp5 For all uch that π ( r lt r, : = r, + Stp6 For all uch that π ( > r lt cr, : = cr, + Stp7 Lt r : = r+ If r Go to Stp4 Stp8 Rpat tp 2-7 M tm a,!,!! c Computg yˆ, ˆ ˆ =, =, v, = w M M M ca from (, (2, ( ota th uad FV tmat for R, I ad I accordgly 5 Numrcal Exampl I th cto w prt a xampl, whch xpla how w ca rak dg accordac to thr IM y ug pctrum approach W choo 4 two-trmal hyprcu H4 - a twork wth 2 =6 od ad 2 dg (umrcal rult for largr twork w wll prt our forthcomg papr Th hyprcu how o Fg 2 It trmal ar =

7th WSEAS Itratoal Cofrc o APPLIED COMPUTER SCIENCE, Vc, Italy, Novmr 2-2, 2007 9 od ad 6 Edg wll dotd y (k,, whr k ad ar od umr Lt u ot that hyprcu cofgurato ar wdly ud computr twork [0] Fg 2 ad 6,

6 7th WSEAS Itratoal Cofrc o APPLIED COMPUTER SCIENCE, Vc, Italy, Novmr 2-2, od ad 6 Edg wll dotd y (k,, whr k ad ar od umr Lt u ot that hyprcu cofgurato ar wdly ud computr twork [0] Fg 2 ad 6, ad th thrd all othr dg Th cocluo may m to tutvly ovou, ut for th am hyprcu wth k> 2 oymmtrcal trmal or for othr, oymmtrcal twork mlar cocluo ar ot o clar Hr th rakg ovou pt of radom fluctuato of valu I mor volvd ca thr wll th,( k, a d to u tattcal tool for ttr dcrmato tw qually mportat dg group Tal 2 prt a fragmt of mulato rult, ad o 0000 rplcato y a w markd th mulatd valu of pctrum ad y,( k, - mulatd valu of IM pctrum IS for dg (k, Rmd that for dg rakg thr o d to comput thr IM' W from Tal 2 that th valu of IS for dg (,9 ad (8,6 ar vry clo to ach othr O th othr d, th valu ar cottly gratr tha th IM pctrum valu for dg (,2 ad th lattr tha tho of (,4 So, w rak th dg y thr mportac th followg ordr (rad tal 2 horotal drcto (,9 = (8,6 > (,2 > (, 4 Tal 2 a,(,9,(8,6,(,2,(, Not that from th whol data fl o ca fr that th gv twork thr ar thr, y thr IM', dffrt group of dg Th frt th par of dg (,9 ad (8,6 Th cod th group cotg of all othr dg cdt to th trmal 4 Cocluo ( To th t of our kowldg, vry fw work wr coductd o computatoal prolm of rlalty mportac valu valuato for larg or modrat twork (2 Th propod mthod for th gral ca of dffrt dg ffct, c t ad o Mot Carlo modl wth wll-talhd ffccy Not that o of th advatag of th mthod that o mulato o w ca valuat mportac maur for dffrt dg ad for dffrt gv dg proalt ( Th pctrum approach hghly ffct ad ha may advatag O of thm that cotructd pctrum do ot dpd o th dg proalt ad rflct th topologcal fatur of a twork (4 I may practcal tuato, th rakg of dg y thr mportac maur may do wthout computg approprat proalt, ut oly ug th IM pctrum (5 Th two propod mthod may aly mplmtd for twork wth rlal dg ad urlal od, ad alo (th tchcally mor dffcult for twork wth oth urlal od ad dg (6 Our mthod ca codrd a th frt tp toward optmal twork dg Rfrc: [] T Elpr, I Grtakh, M Lomooov, A voluto modl for Mot Carlo tmato of qulrum twork rwal paramtr, Proalty th Egrg ad Iformatoal Scc, Vol 6, 992, pp [2] T Elpr, I Grtakh, M Lomooov, Etmato of twork rlalty ug graph voluto modl, IEEE Traacto o Rlalty, Vol 40, No5, 99, pp [] Grtakh ad Y Shpug, Comatoral approach to Mot Carlo tmato of twork

7 7th WSEAS Itratoal Cofrc o APPLIED COMPUTER SCIENCE, Vc, Italy, Novmr 2-2, lftm dtruto, Appl Stochatc Modl u Id, Vol 20, 2004, pp [4] Y Shpug, Comatoral Approach to Rlalty Evaluato of Ntwork wth Urlal Nod ad Urlal Edg, Itratoal Joural of Computr Scc, Vol, No, 2006, pp 77-8 [5] Y Shpug, Ntwork wth urlal od ad dg: Mot Carlo lftm Etmato, Itratoal Joural of Appld Mathmatc ad Computr Scc, Vol 4, No, 2007, pp 68-7 [6] Z W raum, O th mportac of dffrt compot a multcompot ytm, Multvarat Aaly 2, Nw York, Acadmc Pr, 969 [7] J Ful, How to calculat ytm rlalty ad afty charactrtc, IEEE Traacto o Rlalty, Vol 24, No, 975, pp [8] F C Mg, Comparg th mportac of ytm lmt y om tructural charactrtc, IEEE Tracto o Rlalty, Vol 45, No, 996, pp [9] A Gad, Importac ad tvty aaly ag ytm rlalty, IEEE Traacto o Rlalty, Vol 9, No,990, pp 6-70 [0] M Mtmachr, E Upfal, Proalty ad Computg Radomd Algorthm ad Proaltc Aaly, Camrdg Uvrty Pr, 2005

Total Prime Graph. Abstract: We introduce a new type of labeling known as Total Prime Labeling. Graphs which admit a Total Prime labeling are

Total Prime Graph. Abstract: We introduce a new type of labeling known as Total Prime Labeling. Graphs which admit a Total Prime labeling are Itratoal Joural Of Computatoal Egrg Rsarch (crol.com) Vol. Issu. 5 Total Prm Graph M.Rav (a) Ramasubramaa 1, R.Kala 1 Dpt.of Mathmatcs, Sr Shakth Isttut of Egrg & Tchology, Combator 641 06. Dpt. of Mathmatcs,