IN many different scientific, technical or social areas, one

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1 Proceedigs of the World Cogress o Egieerig Vol II WCE July 6-8 Lodo U.K. Complex Stochastic Boolea Systems: New Properties of the Itrisic Order Graph Luis Gozález Abstract A complex stochastic Boolea system CSBS is a system depedig o a arbitrary umber of stochastic Boolea variables. The aalysis of CSBSs is maily based o the itrisic order: a partial order relatio defied o the set } of biary -tuples. The usual graphical represetatio for a CSBS is the itrisic order graph: the Hasse diagram of the itrisic order. I this paper some ew properties of the itrisic order graph are studied. Particularly the set the umber of its edges the degree eighbors of each vertex as well as typical properties such as the symmetry fractal structure of this graph are aalyzed. Idex Terms complex stochastic Boolea system Hasse diagram itrisic order itrisic order graph poset. I. INTRODUCTION IN may differet scietific techical or social areas oe ca fid pheomea depedig o a arbitrarily large umber of rom Boolea variables. I other words the basic variables of the system are assumed to be stochastic they oly take two possible values: either or. We call such a system: a complex stochastic Boolea system CSBS. Each oe of the possible elemetary states associated to a CSBS is give by a biary -tuple u u... u } of s s it has its ow occurrece probability Pr u... u }. Usig the statistical termiology a CSBS o variables x... x ca be modeled by the -dimesioal Beroulli distributio with parameters p... p defied by Pr x i } p i Pr x i } p i Throughout this paper we assume that the Beroulli variables x i are mutually statistically idepedet so that the occurrece probability of a give biary strig of legth u u... u } ca be easily computed as Pr u} i p ui i p i ui that is Pr u} is the product of factors p i if u i p i if u i. Example.: Let 4 u } 4. Let p. p. p 3.3 p 4.4. The usig we have Pr } p p p 3 p Mauscript received March 3 ; revised March 4. This work was supported i part by the Spaish Govermet Miisterio de Ciecia e Iovació Secretaría de Estado de Uiversidades e Ivestigació FEDER through Grat cotracts: CGL8-63-C3-/CLI UNLP8-3E-. L. Gozález is with the Research Istitute SIANI & Departmet of Mathematics Uiversity of Las Palmas de Gra Caaria 357 Las Palmas de Gra Caaria Spai luisglez@dma.ulpgc.es. The behavior of a CSBS is determied by the orderig betwee the curret values of the associated biary - tuple probabilities Pr u}. Computig all these probabilities by usig orderig them i decreasig or icreasig order of their values is oly possible i practice for small values of the umber of basic variables. However for large values of to overcome the expoetial ature of this problem we eed alterative procedures for comparig the biary strig probabilities. For this purpose i [] we have defied a partial order relatio o the set } of all the biary -tuples the so-called itrisic order betwee biary -tuples. The itrisic order relatio is characterized by a simple positioal criterio the so-called itrisic order criterio IOC. IOC eables oe to compare to order two give biary -tuple probabilities Pr u} Pr v} without computig them simply lookig at the positios of the s s i the biary -tuples u v. The most useful graphical represetatio of a CSBS is the itrisic order graph. This is a symmetric self-dual diagram o odes deoted by I that displays all the biary -tuples from top to bottom i decreasig order of their occurrece probabilities. Formally I is the Hasse diagram of the itrisic partial order relatio o }. I this cotext the mai goal of this paper is to preset some ew properties of the itrisic order graph. I particular we give the set the umber of edges of I the set the umber of elemets which are eighbors adjacet i the graph to a fixed biary -tuple u } aalyze the properties of symmetry fractal character of I. For this purpose this paper has bee orgaized as follows. I Sectio II we preset some prelimiaries about the itrisic order the itrisic order graph to make this paper self-cotaied. Sectio III is devoted to preset the ew properties of the itrisic order graph. Fially i Sectio IV we preset our coclusios. II. BACKGROUND IN INTRINSIC ORDER Throughout this paper we idistictly deote the -tuple u } by its biary represetatio u... u or by its decimal represetatio we use the symbol to idicate the coversio betwee these two umberig systems. The decimal umberig the Hammig weight i.e. the umber of -bits of u will be respectively deoted by u u i u i w H u u i. i i Give two biary -tuples u v } the orderig betwee their occurrece probabilities Pr u Pr v obviously depeds o the Beroulli parameters p i as the followig simple example shows. ISBN: ISSN: Prit; ISSN: Olie WCE

2 Proceedigs of the World Cogress o Egieerig Vol II WCE July 6-8 Lodo U.K. Example.: Let 3 u v. For p. p. p 3.3 usig we have: Pr }.54 < Pr }.56 for p. p.3 p 3.4 usig we have: Pr }.96 > Pr }.84. However as metioed i Sectio I i [] we have established a itrisic positioal criterio to compare the occurrece probabilities of two give biary -tuples without computig them. This criterio is preseted i detail i Sectio II-A while its graphical represetatio is show i Sectio II-B. A. The Itrisic Order Criterio Theorem. The itrisic order theorem: Let. Let x... x be mutually idepedet Beroulli variables whose parameters p i Pr x i } satisfy < p p p.5. The the occurrece probability of the biary -tuple v i.e. v v... v } is itrisically less tha or equal to the occurrece probability of the biary -tuple u i.e. u u... u } that is for all set p i } i satisfyig if oly if the matrix Mv u u... u : v... v either has o colums or for each colum i Mv u there exists at least oe correspodig precedig colum IOC. Remark.: I the followig we assume that the parameters p i always satisfy coditio. Fortuately this hypothesis is ot restrictive for practical applicatios. Remark.: The colum precedig each colum is ot required to be ecessarily placed at the immediately previous positio but just at previous positio. Remark.3: The term correspodig used i Theorem. has the followig meaig: For each two colums i matrix Mv u there must exist at least two differet colums precedig each other. I other words for each colum i matrix Mv u the umber of precedig colums must be strictly greater tha the umber of precedig colums. Claim.: IOC ca be equivaletly reformulated i the followig way ivolvig oly the -bits of u v with o eed to use their -bits. Matrix Mv u satisfies IOC if oly if either u has o -bits i.e. u is the zero -tuple or for each -bit i u there exists at least oe correspodig -bit i v placed at the same or at a previous positio. I other words either u has o -bits or for each -bit i u say u i the umber of -bits i v... v i must be greater tha or equal to the umber of -bits i u... u i. The matrix coditio IOC stated by Theorem. or by Claim. is called the itrisic order criterio because it is idepedet of the basic probabilities p i it oly depeds o the relative positios of the s s i the biary strigs u v. Theorem. aturally leads to the followig partial order relatio o the set } [] [3]. The so-called itrisic order will be deoted by whe v u we say that v is itrisically less tha or equal to u or u is itrisically greater tha or equal to v. Defiitio.: For all u v } v u iff Pr v} Pr u} for all set p i } i iff matrix M u v satisfies IOC. s.t. I the followig the partially ordered set poset for short for variables } will be deoted by I ; see [] for more details about posets. Example.: For 3: 3 4 & sice do ot satisfy IOC Remark.3. Therefore are icomparable by itrisic order i.e. the orderig betwee Pr } Pr } depeds o the basic probabilities p i as Example. has show. Example.3: For 4: 3 sice satisfies IOC Remark.. For all < p p 4 Pr } Pr }. B. The Itrisic Order Graph I this subsectio the graphical represetatio of the poset I } is preseted. The usual represetatio of a poset is its Hasse diagram see [] for more details about these diagrams. Specifically for our poset I its Hasse diagram is a directed graph digraph for short whose vertices are the biary -tuples of s s whose edges go upward from v to u wheever u covers v deoted by u v. This meas that u is itrisically greater tha v with o other elemets betwee them i.e. u v u v w } s.t. u w v. A simple matrix characterizatio of the coverig relatio for the itrisic order is give i the ext theorem; see [4] for the proof. Theorem. Coverig relatio i I : Let u v }. The u v if oly if the oly colums of matrix Mv u differet from are either its last colum or just two colums amely oe colum immediately preceded by oe colum i.e. either Mv u u... u or 3 u... u M u v u... u i u i+... u. 4 u... u i u i+... u i Example.4: For 4 we have 6 7 sice M7 6 has the patter 3 ISBN: ISSN: Prit; ISSN: Olie WCE

3 Proceedigs of the World Cogress o Egieerig Vol II WCE July 6-8 Lodo U.K. sice M has the patter 4. The Hasse diagram of the poset I will be also called the itrisic order graph for variables deoted as well by I. For small values of the itrisic order graph I ca be directly costructed by usig either Theorem. or Theorem.. For istace for : I } its Hasse diagram is show i Fig.. Fig.. The itrisic order graph for. Ideed I cotais a dowward edge from to because see Theorem. sice matrix has o colums! Alteratively usig Theorem. we have that sice matrix has the patter 3! Moreover this is i accordace with the obvious fact that Pr } p p Pr } sice p / due to! However for large values of a more efficiet method is eeded. For this purpose i [4] the followig algorithm for iteratively buildig up I for all from I depicted i Fig. has bee developed. Theorem.3 Buildig up I from I : Let. The the graph of the poset I... } o odes ca be draw simply by addig to the graph of the poset I... } o odes its isomorphic copy + I... } o odes. This additio must be performed placig the powers of at cosecutive levels of the Hasse diagram of I. Fially the edges coectig oe vertex u of I with the other vertex v of + I are give by the set of vertex pairs u v u + u u }. Fig. illustrates the above iterative process for the first few values of deotig all the biary -tuples by their decimal equivalets. Basically after addig to I its isomorphic copy + I we coect oe-to-oe the odes of the secod half of the first half to the odes of the first half of the secod half : A ice fractal property of I! Fig.. The itrisic order graphs for 3 4. Each pair u v of vertices coected i I either by oe edge or by a loger descedig path from u to v meas that u is itrisically greater tha v i.e. u v. For istace lookig at the Hasse diagram of I 4 the right-most oe i Fig. we observe that 3 i accordace with Example.3. O the cotrary each pair u v of o-coected vertices i I either by oe edge or by a loger descedig path meas that u v are icomparable by itrisic order i.e. u v v u. For istace lookig at the Hasse diagram of I 3 the third oe from left to right i Fig. we observe that 3 4 are icomparable by itrisic order i accordace with Example.. The edgeless graph for a give graph is obtaied by removig all its edges keepig its odes at the same positios. I Fig. 3 the edgeless itrisic order graph of I 5 is depicted Fig. 3. The edgeless itrisic order graph for 5. For further theoretical properties practical applicatios of the itrisic order the itrisic order graph we refer the reader to [5] [6] [7] [8] [9]. III. NEW PROPERTIES OF THE INTRINSIC ORDER GRAPH Whe viewed as a udirected graph the Hasse diagram is called the cover graph of the poset. We refer the reader to [] for stard otatio termiology cocerig graphs. Usig Theorems...3 we ca derive may differet properties of the cover graph of I. Here we have selected oly a few of them. A. Edges Let V E be the sets of vertices edges respectively of I. As usual A deotes the cardiality of the set A. As metioed the umber of odes of I is obviously V }. Our first property gives the umber of edges of I. Propositio 3.: For all the umber of edges i the itrisic order graph I is E +. 5 Proof: The edges goig dowward from u to v i a Hasse diagram are exactly the coverig relatios u v. Hece usig Theorem. we obtai E u v V V u v } u v V V M u v has the patter 3} + u v V V Mv u has the patter 4} } u... u + u... u } u... u i u i+... u u... u i u i+... u + + ISBN: ISSN: Prit; ISSN: Olie WCE

4 Proceedigs of the World Cogress o Egieerig Vol II WCE July 6-8 Lodo U.K. as was to be show. Remark 3.: Usig propositio 3. we get for all E E + a recurrece relatio for the umber E of edges of I which could be also obtaied directly from Theorem.. Whe we use the biary represetatio the set E of all the + edges i I is give by Theorem.. The followig propositio gives this set usig the decimal umberig for the pairs of adjacet odes see Fig.. Propositio 3.: For all E u u + } u p p m u u + m u q + m + 4r q m r m. Proof: The edges goig dowward from u to v i a Hasse diagram are exactly the coverig relatios u v. So usig Theorem. we obtai E u v V V u v } u v V V M u v has the patter 3 } u v V V M u v has the patter 4 }. O oe h if Mv u has the patter 3 the we have that v u + u u... u u... u p p. O the other h if Mv u has the patter 4 the makig the chage of variable m i we get v u + i with i i.e. v u + m with m u u... u i u i+... u u... u i u i+... u i+ u... u i + i + u i+... u m+ r + m + q q + m + 4r where q m r m. Example 3.: Let 4. Usig Propositio 3. we get A 4 u u + } u p p 7 } B 4 m u u + m u q + m + 4r q m r m where the three above rows respectively correspod to: m : q r 3 v u + m : q r v u + m : q 3 r v u + Thus E 4 A 4 B 4 cotais all the edges pairs of adjacet odes of the graph I 4 as oe ca cofirm lookig at the right-most diagram i Fig.. Note that usig 5 for 4 we ca also cofirm that the cardiality of E 4 is E B. Shadows Neighbors Degrees The eighbors of a give vertex u i a graph are all those odes adjacet to u i.e. coected by oe edge to u. I particular for the cover graph of a Hasse diagram the eighbors of vertex u either cover u or are covered by u. This aturally leads to the followig defiitio []. Defiitio 3.: Let P be a poset u P. The i The lower shadow of u is the set u v P v is covered by u} v P u v }. ii The upper shadow of u is the set u v P v covers u} v P v u}. Particularly for our poset P I regardig the lower shadow of u } usig Theorem. we have u v } u v } v } M u v has the patter 3} v } M u v has the patter 4} hece the cardiality of the lower shadow of u is exactly u patter 3 plus the umber of pairs of cosecutive bits u i u i i u patter 4. Formally: u u + max u i u i }. 6 i Similarly for the upper shadow of u } usig agai Theorem. we have u v } v u} v } M v u has the patter 3} v } M v u has the patter 4} hece the cardiality of the upper shadow of u is exactly u patter 3 plus the umber of pairs of cosecutive bits u i u i i u patter 4. Formally: u u + max u i u i }. 7 i Next propositio provides the total umber of eighbors of each ode u of the itrisic order graph I the so-called degree of u deoted as usual by δ u. Propositio 3.3: Let u }. The degree δ u of u i.e. the umber of eighbors of u is δ u + u i u i. 8 i ISBN: ISSN: Prit; ISSN: Olie WCE

5 Proceedigs of the World Cogress o Egieerig Vol II WCE July 6-8 Lodo U.K. Proof: Deotig by N u the set of eighbors of a vertex u } i the graph I obviously we have N u u u from 6 7 we immediately obtai δ u N u u + u + u i u i i as was to be show. Next propositio provides us with the set of eighbors of each ode u of the itrisic order graph I usig decimal represetatio. Propositio 3.4: Let let u } with Hammig weight m. Write u as sum of powers of i icreasig order of the expoets i.e. u i u i p + p + + pm 9 i p < p < < p m. i The lower shadow u of u is characterized as follows: i-a If u is eve i.e. if u the u + u i.e. u u +. i-b For ay power p p i 9 s.t. p+ does ot appear i 9 the u + p u i.e. u u + p. ii The upper shadow u of u is characterized as follows: ii-a If u is odd i.e. if u the u u i.e. u u. ii-b For ay power p p i 9 s.t. p does ot appear i 9 the u p u i.e. u p u. Proof: The assertios i-a ii-a immediately follow usig patter 3 i Theorem. for matrices Mv u Mu v respectively. The assertios i-b ii-b immediately follow usig patter 4 i Theorem. for matrices Mv u Mu v respectively. Example 3.: Let 4 u. The u u + 3. Usig Propositio 3.4-i we get ote that u is eve i.e. u 4 + } + } } usig Propositio 3.4-ii we get } 6 9}. Thus see the graph I 4 the right-most oe i Fig. N 6 9 } usig 8 we cofirm that the cardiality of N is 4 δ N + u i u i i + u u + u 3 u + u 4 u C. Complemetarity Symmetry Lookig at ay of the graphs i Figs. & 3 we observe a certai symmetry i these diagrams. Let us formalize this fact. Defiitio 3.: Let u }. i The complemetary -tuple of u is the -tuple obtaied by chagig all its s ito s vice versa i.e. u... u c u... u. ii The complemetary set of a subset S } is the set of complemetary -tuples of all the -tuples of S i.e. S c u c u S }. Remark 3.: Note that for all u... u } u... u + u... u c.... Hece the simplest way to verify that two biary -tuples are complemetary whe we use their decimal represetatios is to check that they sum up to. For istace the biary 3-tuples 5 are complemetary sice Similarly the complemetary of the biary 4-tuple 4 is sice The reaso uderlyig the symmetry of the itrisic order graph is the duality property of the itrisic order stated by the followig propositio. Propositio 3.5: Let u v }. The u v v c u c u v v c u c. Proof: Clearly the colums i matrix Mv u respectively become colums i matrix Mu vc c. Hece usig Theorem. we have that u v iff Mv u has either the patter 3 or the patter 4 iff Mu vc respectively has either the patter 3 or the c patter 4 iff v c u c. Fially the right-h equivalece immediately follows from the left-h oe from the trasitive property of the itrisic order. May ice cosequeces ca be derived from Propositio 3.5. Next corollary states oly a few of them. Corollary 3.: Let. Let u v be ay two biary -tuples placed at symmetric positios with respect to the cetral poit i the graph I. The i u v are complemetary -tuples i.e. v u c u v c. ii The Hammig weights of u v sum up to. iii u c u c c v u c u c c v. iv The sets of eighbors of u v are complemetary. I particular u v have the same degree. v u is eve odd v is odd eve. Proof: i It is a direct cosequece of Propositio 3.5. ii It suffices to use i the obvious fact that w H u + w H u c. iii Usig Defiitio 3. Propositio 3.5 i we get: w u u w w c u c w c u c w c u c w c v thus takig complemetaries we get u c u c c u u c u c u c c v. ISBN: ISSN: Prit; ISSN: Olie WCE

6 Proceedigs of the World Cogress o Egieerig Vol II WCE July 6-8 Lodo U.K. iv Usig iii we get cosequetly N u u u c v c v [ v v] c N c v δ u N u N c v N v δ v. v Usig i we get u is eve odd u v v is odd eve this cocludes the proof. Example 3.3: Let 4. The biary 4-tuples u 4 v are placed at symmetric positios with respect to the cetral poit i the graph I 4 see Fig.. Therefore: i 4 c c ii w H + w H iii 4 5 8} 7 } 5 8} c 7 }. 4 } 3} } c 3}. iv N 4 5 8} N 7 3} 5 8} c 7 3} δ 4 3 δ. v 4 is eve is odd. D. Isomorphic Subgraphs Fractal Structure A bisectio of a graph is a partitio of its vertex set ito two disjoit subsets with half the vertices each []. The most atural way of bisectig the itrisic order graph I is the followig. The first secod half respectively of } will be the subsets of biary -tuples whose first compoet is u u respectively. This procedure ca be reiterated by successively bisectig i the same way each of the so-obtaied subgraphs. Let k let ū... ū k } be k fixed biary digits. From ow o Iū...ū k deotes the subset of bitstrigs of } whose first or left-most k compoets are fixed amely u ū... u k ū k ; while its last or right-most k compoets u k+... u take all possible values or. More precisely Iū...ū k ū... ū k u k+... u is the set } u k+... u } k its cardiality is Iū...ū k } k k. Let us recall that two graphs G V E G V E are said to be isomorphic if there exists a isomorphism of oe of them to the other i.e. a edge-preservig bijectio []. That is a graph isomorphism is a oe-to-oe mappig betwee the vertex sets Φ : V V which preserves adjacecy i.e. u v are adjacet i G if oly if Φ u Φ v are adjacet i G. The self-similarity property or fractal structure that oe ca observe i Figs. & 3 is a immediate cosequece of the followig propositio. Propositio 3.6: Let k. The k equalsized subgraphs Iū...ū k each with k odes obtaied after k successive bisectios of the itrisic order graph I are pair-wise isomorphic ideed all of them are isomorphic to the itrisic order graph I k. Proof: Cosider the followig oe-to-oe mappig Iū...ū k I k ū... ū k u k+... u u k+... u Usig Theorem. we have ū... ū k u k+... u ū... ū k v k+... v if oly if u k+... u v k+... v so that Φ is a isomorphism of graphs sice it preserves the edges coverig relatios. For istace let 5 k 3. After k 3 successive bisectios of the itrisic order graph I 5 the k 8 subgraphs are the 8 isomorphic colums each cotaiig k 4 odes depicted i Fig. 3. Moreover ay of these colum -subgraphs of I 5 5-tuples is isomorphic to I - tuples the secod graph from the left i Fig.. Φ IV. CONCLUSION The behavior of a CSBS depeds o the curret values of the biary -tuple probabilities o the orderig betwee them. I this sese the itrisic order graph I provides us with a useful represetatio of a CSBS by displayig all the bitstrigs i decreasig order of their occurrece probabilities. I this paper several ew properties of the digraph I have bee stated rigorously proved e.g. umber of edges eighbors degrees of each vertex symmetry fractal structure etc.. Each of these properties has bee illustrated with a simple example with the correspodig graph. Sice may differet techical systems i Reliability Egieerig are ideed CSBSs the our results ca be applied to develop ew or to improve already kow algorithms based o the itrisic order for evaluatig the uavailability system. From a theoretical poit of view this paper suggests the search of ew graph-theoretic ordertheoretic properties of the itrisic order graph I. REFERENCES [] R. Diestel Graph Theory 3rd ed. New York: Spriger 5. [] L. Gozález A New Method for Orderig Biary States Probabilities i Reliability Risk Aalysis Lect Notes Comp Sc vol. 39 o. pp [3] L. Gozález N-tuples of s s: Necessary Sufficiet Coditios for Itrisic Order Lect Notes Comp Sc vol. 667 o. pp [4] L. Gozález A Picture for Complex Stochastic Boolea Systems: The Itrisic Order Graph Lect Notes Comp Sc vol o. 3 pp [5] L. Gozález Algorithm comparig biary strig probabilities i complex stochastic Boolea systems usig itrisic order graph Adv Complex Syst vol. o. Suppl. pp [6] L. Gozález Complex Stochastic Boolea Systems: Geeratig Coutig the Biary -Tuples Itrisically Less or Greater tha u i Lecture Notes i Egieerig Computer Sciece: World Cogress o Egieerig Computer Sciece 9 pp [7] L. Gozález Partitioig the Itrisic Order Graph for Complex Stochastic Boolea Systems i Lecture Notes i Egieerig Computer Sciece: World Cogress o Egieerig pp [8] L. Gozález Rakig Itervals i Complex Stochastic Boolea Systems Usig Itrisic Orderig i Machie Learig Systems Egieerig Lecture Notes i Electrical Egieerig vol. 68 B. B. Rieger M. A. Amouzegar S.-I. Ao Eds. New York: Spriger pp [9] L. Gozález D. García B. Galvá A Itrisic Order Criterio to Evaluate Large Complex Fault Trees IEEE Tras o Reliability vol. 53 o. 3 pp [] R. P. Staley Eumerative Combiatorics vol.. Cambridge MA: Cambridge Uiversity Press 997. ISBN: ISSN: Prit; ISSN: Olie WCE