A Picture for Complex Stochastic Boolean Systems: The Intrinsic Order Graph

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1 A Picture for Complex Stochastic Boolean Systems: The Intrinsic Order Graph Luis González University of Las Palmas de Gran Canaria, Department of Mathematics, Research Institute IUSIANI, 357 Las Palmas de Gran Canaria, Spain Abstract. Complex stochastic Boolean systems, depending on a large number n of statistically independent random Boolean variables, appear in many different scientific, technical or social areas. Each one of the 2 n binary states associated to such systems is denoted by its corresponding binary n-tuple of s and s, u,...,, and it has a certain occurrence probability Pr {u,...,}. The ordering between the 2 n binary n- tuple probabilities, Pr {u,...,}, can be illustrated by a directed graph which scales them by decreasing order, the so-called intrinsic order graph. In this context, this paper provides a simple algorithm for iteratively drawing the intrinsic order graph, for any complex stochastic Boolean system and for any number n of independent random Boolean variables. The presentation is self-contained. Introduction This paper deals with the modelling of complex stochastic Boolean systems, that is, those complex systems which depend on a large number n of random Boolean variables x,...,x n. These systems, with very simple components the Boolean variables x i only take two possible values:, but complex overall behavior, can be found in many different knowledge areas: Wherever a stochastic Boolean phenomenon with n basic variables appears Biology, Biochemistry, Climatology, Computer Science, Networks, Engineering, Economics, Sociology, etc.. According to the usual terminology in Statistics, a stochastic Boolean system can be modeled by the n-dimensional Bernoulli distribution see, e.g., [6]. This distribution consists on n random variables x,...,x n,whichonlytaketwo possible values, or, with probabilities Pr {x i } p i, Pr {x i } p i i n, so that the sample space is the set {, } n of the 2 n binary n-tuples binary strings, u u,...,, of s and s. Throughout this work, we assume that Partially supported by MEC Spain and FEDER. Grant contract: CGL C3-2/CLI. V.N. Alexandrov et al. Eds.: ICCS 26, Part III, LNCS 3993, pp , 26. c Springer-Verlag Berlin Heidelberg 26

2 36 L. González the random variables x i are mutually independent, so that the occurrence probability of each binary n-tuple can be easily computed as the product Pr {u,..., } n Pr {x i u i } i n 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. The following natural question immediately arises in the study of stochastic Boolean systems: How can we order the 2 n binary strings u,..., by decreasing/increasing order of their occurrence probabilities? Obviously, this question has a relevant, theoretical and practical, interest. However, in spite of the simplicity of Equation to compute the probabilities Pr {u,..., }, thisisnot a simple question due to its exponential nature. To avoid this obstacle, in [3, 4] the authors have established a simple positional criterion, the so-called Intrinsic Order Criterion IOC, that aprioriassures us that for certain pairs of binary n-tuples u, v {, } n exactly for those pairs whose bits satisfy IOC, the inequality Pr {u} Pr {v} intrinsically holds, that is, Pr {u} Pr {v} for any values of the parameters p i satisfying certain non restrictive hypothesis. Moreover, in [, 2] the author has roughly constructed the intrinsic order graph for the first values of n, just by direct application of IOC. However, a general method for easily constructing this graph for all n N, has not yet been provided. In this context, the main goal of this paper is to provide a simple algorithm for recursively drawing the intrinsic order graph, for any number n of Boolean variables. This directed graph is a very useful picture for all complex stochastic Boolean systems related to any knowledge area, because it scales the binary n-tuples of s and s, u,..., {, } n, by decreasing order of their occurrence probabilities. In Sect. 2, we describe all previous results required for making this paper self-contained. The rest of the paper is devoted to our new approach. In Sect. 3, we present a simple characterization of the covering relation of the intrinsic order. Finally, in Sect. 4, from this characterization we derive the algorithm for drawing the intrinsic order graph. 2 The Intrinsic Order In [3, 4], we have established the following characterization theorem that allows us to compare two given binary string probabilities, Pr {u} and Pr {v}, without computing them. Theorem The intrinsic order theorem. Let x,...,x n be n independent Bernoulli variables, with parameters p i Pr{x i } i n satisfying: <p p n 2. 2 Then, the probability of the n-tuple u,..., {, } n is intrinsically greater than or equal to the probability of the n-tuple v,..., {, } n that is, for

3 A Picture for Complex Stochastic Boolean Systems 37 all set of parameters {p i } n i such that 2 if, and only if, the matrix Mv u u...u : n v... either has no columns, or for each column there exists at least one corresponding preceding column IOC. Remark. In the following, we assume that the parameters p i always satisfy condition 2. Note that this hypothesis is not restrictive for practical applications because, if for some i : p i > 2, then we only need to consider the variable x i x i, instead of x i. Next, we order the n Bernoulli variables by increasing order of their probabilities. Remark 2. The column preceding to each column is not required to be necessarily placed at the immediately previous position, but just at previous position. Remark 3. The term corresponding, used in Theorem, has the following meaning: For each two columns in matrix M u v, there must exist at least two different columns preceding to each other. In other words, IOC requires that for each column C in matrix M u v,ifwedenotebyn Cn C, resp. the number of columns columns, resp. preceding C, thenn C >n C. Theorem naturally leads to the following partial order relation on the set {, } n. The so-called intrinsic order because it only depends of the relative positions of s and s in the binary n-tuples u, v, but not on the basic probabilities p i, will be denoted by, and we shall denote by P n the partially ordered set poset, for short{, } n,. See [5] for more details about posets. Definition. For all u, v {, } n u v iff Pr {u} Pr {v} for all set {p i } n i s.t. 2 iff M u v satisfies IOC. Example.,,,,,, because matrix satisfies IOC Remark 2. Example 2.,,,,,, and,,,,,, because neither, nor satisfies IOC Remark 3. From now on, we shall indistinctly denote any n-tuple u {, } n by its binary representation u,..., or by its decimal representation u. Also, we shall denote by lex the usual lexicographic order truth-table order between the binary n-tuples, with the convention that <, i.e., n u,..., u : 2 n i u i, u lex v u v. i

4 38 L. González The next corollary see [2] for the proof states the relation between the partial intrinsic order and the total lexicographic order lex. Corollary. For all u, v {, } n u v u lex v, i.e., u v u v. 3 Covering Relation in P n As is well-known [5], every finite poset is completely determined by its cover relations. Specifically, with respect to our poset P n,wesaythatv is covered by u or u covers v, denoted by v u or u v ifv u and there is no w {, } n such that v w u. For instance, in P 2 {, } 2,,wehave,,,, because,,,, Definition, Theorem with no other elements between them. However,,, but,,, because, is between them. In this section, we provide a simple matrix characterization of the covering relation associated to the intrinsic order. This condition for, required for drawing the intrinsic order graph in the next section, will be obviously a particular case of more restrictive than the IOC condition for. First, we need the following lemma. Lemma. Let u, v {, } n and i n. i If v u then v,...,v i u,...,u i. ii If v u and u... u i v... v i has the same number of columns as columns then v i+,..., u i+,...,. iii If v u and the columns u i v i and ui+ v i+ are both different from,then the matrix obtained by permuting these two columns of Mv u,keepingitsother columns, satisfies IOC. iv If v u, u i v i, uj v j for some j>i,andthereisnok such that i<k<jand u k v k, then the matrix obtained by replacing the i-th and j-th columns of Mv u by or, keeping its other columns, satisfies IOC. Proof. Using the matrix description IOC of the intrinsic order Theorem, the proof is straightforward. Theorem 2 Covering relation in P n. Let n and u, v {, } n.then v u if and only if either Mv u u...u n, or 3 u... Mv u u...u i u i+2... i n, 4 u...u i u i+2... where we assume that in case 3 the n first columns of matrix Mv u are deleted if n ;while in case 4 the i first the n i last, resp. columns of matrix Mv u are deleted if i if i n, resp..

5 A Picture for Complex Stochastic Boolean Systems 39 Proof. Sufficient condition. First, note that, for both cases 3 and 4, v u since the two corresponding matrices Mv u obviously satisfy IOC. So, for both cases, we only need to prove that if v w u then w u or w v. If 3 holds then using Corollary, we have v w u u lex w lex v and then w u or w v. If 4 holds then using Corollary, we have v w u u lex w lex v and then two subcases are possible: { u,...,u w,...,w i+ i,, 4-a, u,...,u i,, 4-b. If i n, then 4-a is equivalent to say w u while 4-b is equivalent to say w v. Then, w u or w v. Otherwise, if i<n, then using Lemma -ii, we have: On one hand, since v w and matrix w... w i w i w i+ u... u i has the same number of columns as columns namely, if 4-a holds and if 4-b holds then we get u i+2,..., w i+2,...,w n. On the other hand, since w u and matrix u... u i w... w i w i w i+ has the same number of columns as columns namely, if 4-a holds and if 4-b holds then we get w i+2,...,w n u i+2,...,. So, due to the antisymmetry of the intrinsic order we obtain w i+2,...,w n u i+2,...,, and then w u if 4-a holds or w v if 4-b holds. Necessary condition. We provide a constructive proof by finding w {, } n yielding v w u, for all v u such that the pattern of matrix Mv u is different from 3 and 4. First, note that since v u we assure Corollary that u< lex v, i.e., the first column of Mv u different from and is.let ui v i be this column. Then Mv u u...u i u i+ u i+2..., 5 u...u i v i+ v i+2... where we can assure that i<n, because if i n then Mv u in 5 would be like 3, and we assume that the i first the n i last, resp. columns of matrix 5 are deleted if i ifi n, resp.. The same assumption is established for the i first the n i last, resp. components of the chosen vectors w. We distinguish the following four cases. u i+ v i+.inthiscasewechoose w u,...,u i,,,v i+2,...,, so that v w u and Mv w u... u i v i+2... and Mw u u... u i v i+2... v u... u i u i+2... n u... u i v i+2... satisfy IOC: the first one obviously; the second one due to Lemma -iii. 2 u i+ v i+.inthiscasewechoose w u,...,u i,,,u i+2,...,, so that v w u and

6 3 L. González Mv w u... u i u... u i u i+2... v i+2... and Mw u u... u i u i+2... u... u i u i+2... satisfy IOC: the first one due to Lemma -iii; the second one obviously. 3 u i+ v i+. Here we distinguish two subcases: 3. M u v has no columns. In this subcase we choose w u,...,u i,,,v i+2,...,, so that v w u and Mv w u... u i v i+2... and M u u... u i w u... u i v i+2... u... u i u i+2... v i+2... satisfy IOC, since they have no columns. 3.2 Mv u has at least one column. Let uj v j,i+2 j n, bethe left-most column of M u v. In this subcase we choose w u,...,u i,,,v i+2,...,v j,,v j+,...,, so that v w u and with the assumption that the substring v i+2,...,v j ofw is deleted if j i +2.Then Mv w u... u i v i+2... v j v j+..., u... u i v i+2... v j v j+... Mw u u... u i u i+2... u j u j+... u... u i v i+2... v j v j+... satisfy IOC: the first one obviously; the second one due to Lemma -iv. 4 u i+ v i+. Recall that for all possible cases -4 we assured that i<n. Moreover, in this fourth case we can assure that i<n and v i+2,..., u i+2,...,, because if i n orifi<n with v i+2,..., u i+2,..., then, in both cases, Mv u in 5 would has the pattern 4. In this last case we choose w u,...,u i,,,v i+2,...,, so that v w u and Mv w u... u i v i+2... and Mw u u... u i v i+2... u... u i u... u i u i+2... v i+2... satisfy IOC: the first one obviously; the second one due to Lemma -iv. 4 The Intrinsic Order Graph The classical picture of a poset P is its Hasse diagram. This is a directed graph digraph, for short whose vertices are the elements of P,whoseedgesarethe cover relations, and with the usual convention that if v u then u is drawn

7 A Picture for Complex Stochastic Boolean Systems 3 above v [5]. For instance, the digraph of P is,since. The next theorem provides us with a fast algorithm for iteratively building up the Hasse diagram of P n, for all n>, from the Hasse diagram of P. The binary n-tuples are represented by their decimal numbering. Theorem 3 Iterative construction of P n from P. For all n>, the digraph of P n {,...,2 n } can be drawn simply by adding to the digraph of P n {,...,2 n } its isomorphic copy 2 n +P n { 2 n,...,2 n }. This addition must be performed placing the powers of 2 at consecutive levels of the Hasse diagram of P n. Finally, the edges u v connecting one vertex u of P n with one vertex v of 2 n + P n is given by the set of vertex pairs {u, v /u v} { u, 2 n 2 + u / 2 n 2 u 2 n }. Proof. For all n>, consider the following partition of {, } n {, } n {} {, } n {} {, } n, i.e., {,...,2 n } {,...,2 n } { 2 n,...,2 n }. On one hand, note that according to IOC Theorem, the subposets of P n, {} {, } n, and {} {, } n,, are both isomorphic to P n in the following sense. For all u,...,, v,..., {, } n u,..., v,...,,u,...,,v,...,,u,...,,v,...,. 6 On the other hand, note that according to Theorem 2, we have for all n>,...,,,...,,,,...,, i.e., n. 7 Equations 6 and 7 prove that P n can be iteratively built up by adding to the digraph of P n its isomorphic copy 2 n + P n, placing the powers of 2 at consecutive levels of the Hasse diagram of P n. Finally, we prove the last assertion about the edges connecting the two mentioned subposets of P n.let u {} {, } n P n, v {} {, } n 2 n + P n and put u,u,...,, v,v,...,. Using Theorem 2 and taking into account that the pattern 3 is not possible for matrix Mv u because n 2, we have u v Mv u u... has the pattern 4 v... u and, if n>2thenv i u i for all i 2,...,n v { {, if n 2 u,,u 2,..., if n>2,v, if n 2,,u 2,..., if n>2

8 32 L. González and,,, n 2..., 2n 2 u 2 n,,, n 2..., v u 2 n 2 +2 n u +2 n 2 and this concludes the proof. Fig. illustrates the iterative construction of the digraph of P n n, 2, 3, 4 from P, denoting all the binary n-tuples by their decimal numbering Fig.. The intrinsic order graph for n, 2, 3, 4 References. González, L.: A new method for ordering binary states probabilities in Reliability and Risk Analysis. Lect. Notes Comput. Sc González, L.: N-tuples of s and s: Necessary and sufficient conditions for intrinsic order. Lect. Notes Comput. Sc González, L., Galván, B., García, D.: Sobre el análisis computacional de funciones Booleanas estocásticas de muchas variables. In: González, L., Sendra, J.R. eds.: Proc. Primer Encuentro de Álgebra Computacional y Aplicaciones EACA-95. Santander González, L., García, D., Galván, B.J.: An intrinsic order criterion to evaluate large, complex fault trees. IEEE Trans. Reliability Stanley, R.P.: Enumerative Combinatorics. Volume. Cambridge University Press, Cambridge Stuart, A., Ord, J.K.: Kendall s Advanced Theory of Statistics. Volume : Distribution Theory. Oxford University Press, New York 998