The Dynamics of Conjunctive and Disjunctive Boolean Networks

THE D YNAMICS OF CONJUNCTIVE AND DISJUNCTIVE BOOLEAN NETWORKS ABDUL SALAM JARRAH, REINHARD LAUBENBACHER, ALAN VELIZ-CUBA Abstract. The relationship betw een the prop erties of a dynamical system and the structure of i ts defining equations has long b een studied in many con texts. Here we study this problem f or the class of conjunc tive (resp. dis- junctiv e) Boolean net works, t hat is, Boolean net w orks in which all Bo olean functions are constructed wi th the AND (resp. OR) op erator only . The main results of t his pap er describe net work dynamics i n terms of the structure of the net work dependency graph (topology). F or a given such net work, all p os- sible li mit cycle l engths are computed and l o wer and upper b ounds for the n umber of cycles of eac h length are given. In particular, the exact num ber of fixed p oi n ts i s obtained. The b ounds are in terms of structural features of the dependency graph and its partially ordered set of strongly connected com- ponents. F or net wo rks with strongly connected dependency graph, the exa ct cycle structure is computed. 1. Introduction The understanding of the relationship b etw een structural features of dyna mical systems and the resulting dy namics is an imp or ta n t pro blem that has b een studied extensively in the dynamical systems literature. F or example, work b y Golubitsky and Stew art [1] ab out coupled cell dynamical sys tems given by coupled systems of ODEs attempts to obtain informatio n ab out s ystem dyna mics fro m the top olo gy of the gr aph of co uplings. Alber t and Othemer [2] us ed a Bo olean netw ork mo del to show that the expre s sion pa tterns of the segment p olar it y in Dr osophila are deter- mined by the top ology of its gene re gulatory netw o rk. Thomas et al [3] conjectured that nega tive feedbac k lo ops ar e necessar y for p er io dic dyna mics whereas p ositive feedback lo o ps ar e nece s sary for multistationarity . These conjectures have b een the sub ject o f many published articles [4; 5 ; 6 ]. In [7] we demo ns trated that netw o rks with a large n umber of indep endent negative feedback lo o ps tend to hav e few limit cycles and these cycles a re usually long. In this pap er we study the effect of the netw ork top ology on the dynamics of a family of Boolea n netw o rks. Bo olea n netw orks in general, and cellular automata in particular, ha ve long been used to model and simulate a wide r ange of phenomena, from logic circuits in engineering a nd g e ne regulato ry netw orks in molecula r biology [2; 8; 9; 10; 11; 12] to p opulatio n dy namics and the s pread o f epidemics [13; 14]. Esp ecially for la rge netw orks, e.g., man y a g ent-based simulations, it b ecomes in- feasible to sim ulate the system extensively in or der to obtain information a bo ut its dynamic proper ties, even if s uch s im ulation is po ssible. In suc h instances it Date : Nov em b er 15, 2018. Key wor ds and phr ases. conjunctiv e, disj unctiv e, Bo olean netw or k, phase space, l imit cycle, fixed p oints. This work was supp orted partially by NSF Grant DMS- 0511441 . 1 2 JARRAH, LAUBENBA CHER bec omes impo rtant fro m a practical point of view to b e able to der ive informatio n ab out netw ork dynamics from structural information. But the proble m is o f interest in its own r ight, in particular in the mor e genera l co ntext of time-discr e te dynami- cal systems ov er general finite fields. These have been s tudied extensively , see, for example, [15; 16; 17; 18; 19]. They ha ve a wide r ange of a pplications in eng ineering [12; 20; 21; 22; 23; 24; 25] a nd re c en tly in computationa l biology [26; 27]. Let F 2 := { 0 , 1 } be the Galois field with tw o element s. W e view a Bo o lean net work on n v aria bles as a dynamical system f = ( f 1 , . . . , f n ) : F n 2 − → F n 2 . Here, each of the co ordina te functions f i : F n 2 − → F 2 is a Bo o lean function which can b e written uniquely as a po lynomial where the exp onent o f each v ariable in ea ch term is at most one [28]. In particular AN D ( a, b ) = ab and O R ( a, b ) = a + b + ab . W e use p olyno mial forms of Bo olea n fun ctions throughout this pap er. Two dir ected gra phs are usually assigned to each suc h system: The dep endency gr aph whic h enco des the sta tic relationships among the no des of the netw ork and the phase sp ac e which descr ib e the dyna mic b ehavior of the net work. In this pap er we focus on the q uestion of der iving information ab out the phase space of a Bo olea n net work from the structure of its dep endency gra ph. Next we define these gra phs. The dep endency gr aph D ( f ) o f f ha s n vertices corre s po nding to the Bo olean v ar iables x 1 , . . . , x n of f . There is a directed edge i → j if x i app ears in the function f j . That is, D ( f ) enco des the v ar iable dependencie s in f . It is similar to the coupling gra ph of Stew art and Golubitsky . Example 1.1 . The dep endency graph of f = ( x 2 x 3 , x 1 , x 2 , x 3 x 4 , x 1 x 6 , x 3 x 4 x 5 ) : F 6 2 − → F 6 2 is the directed gr aph in Figure 1. 2 3 4 5 6 1 Figure 1. The dep e ndenc y graph of f from Example 1.1. The dynamics of f is e nc o de d b y its phase sp ac e , denoted by S ( f ). It is the directed gra ph with v ertex set F n 2 and a dir ected edge fro m u to v if f ( u ) = v . F or each u ∈ F n 2 , the s equence { u , f ( u ) , f 2 ( u ) , . . . } is called the orbit of u . If u = f t ( u ) and t is the smallest s uch num b er, the sequence { u , f ( u ) , f 2 ( u ) , . . . , f t − 1 ( u ) } is called a limit cycle of length t denoted by C t , a nd u is called a p erio dic p oint of p erio d t . The p oint u is called a fi xe d p oint if f ( u ) = u . If every limit cycle is of length 1, the system f is called a fixe d-p oint sy stem. Since F n 2 is finite, ev ery orbit m ust include a limit cyc le . W e deno te the c y cle structure of f in the form of the generating function (1.1) C ( f ) = ∞ X i =1 C ( f ) i C i , THE DYNAMICS OF CONJUNCTIVE AND DISJUNCTIVE BOOLEAN NETWORKS 3 where C ( f ) i denotes the num b er of cycles C i of length i in the pha se spa c e o f f . Since the phase space of f is finite, C ( f ) i = 0 for almost all i . The height of f , denoted by h( f ), is the le ast positive in teger s such that f s ( u ) is a per io dic p oint for all u ∈ F n 2 . A c omp onent of the phase space S ( f ) c o nsists of a limit cycle and a ll or bits of f that con tain it. Hence , the pha se spa ce is a disjoint union o f c ompo nent s. W e define the p erio d o f f , deno ted by p( f ), to b e the lea st common mult iple of the leng ths of all limit cycles in the phase space of f . Example 1.2 . Let f : F 3 2 − → F 3 2 be given by f ( x 1 , x 2 , x 3 ) = ( x 2 x 3 , x 1 + x 3 , x 1 x 2 ). The phase space of f has t w o comp onents, co nt aining one fixed p oint and one limit cycle of length t wo, that is C ( f ) = C 1 + C 2 , see Figure 2. It is cle a r from the phase space that h( f ) = p( f ) = 2. The phase s pace in Figur e 2 was gener ated using DVD [29]. 0 0 0 0 0 1 0 1 0 0 1 1 1 1 0 1 0 0 1 0 1 1 1 1 Figure 2. The phase space S ( f ) of the system f from Example 1.2. Without exhaus tive iteration and just by a nalyzing D ( f ) wha t can we say a b out S ( f )? Namely , wha t is the per io d of f , the heigh t of f , or the generating function C ( f )? This question is NP-har d in genera l, so it is imp orta n t to limit the class of Bo olean net works considered. Next w e list some of the main known results. • When all co ordinate Bo o le an functions are the XOR function (that is, the functions are linear p olynomials ), th e a bove questions hav e been a nswered completely . In fact the q uestions hav e b een answered for linear sy stems ov er any Galo is field [1 2; 20; 30]. W e ha ve developed and implemen ted algorithms that answer the questions ab ov e, see [31]. • F or Bo olea n netw orks where all co ordina te functions are symmetr ic thres h- old functions , it has b een shown that a ll cycles in the pha se space are either fixed p oints or of length t wo [32], • F or Bo olea n cellular a utomata with the ma jor ity r ule, the n umber of fixed po in ts was determined in [33], and • In [34], the authors studied AND-OR netw orks (Boo lean netw ork where each lo cal function is e ither an OR or AND function) with directed de- pendenc y gra phs. F ormulae for the maximum num ber of fix ed p oints ar e obtained, and • The main result in [35] is an upp er b ound for the num ber of fixed p oints in Bo olean reg ulatory net works. In this pap er we foc us on the class of c o njunctiv e and disjunctive Bo ole an net- works, that is, Bo olean netw orks where all of their co ordinate functions ar e either 4 JARRAH, LAUBENBA CHER the AND function o r the OR function. The following represent main previo us attempts to mathematically analy ze this class of Bo olean netw orks. • A family of this cla ss of net works hav e been analyze d in [36]. The au- thors studied conjunctive Bo olea n net works (whic h they called OR-nets) and disjunctiv e Bo o lean netw orks (AND-nets) on undirected dep endency graph. That is, on gr aphs w he r e each edge is bidirectional a nd hence the depe ndenc y g raph consists o f cycles of length t wo. In particular, their result that OR- nets have only fixed p oints and p oss ibly a limit cyc le of length tw o [36, Lemma 1 ] follows directly from our results as we explain in Remar k 2.8. • In [37], the author s study a smaller family wher e each edge is undirected and each no de in the netw ork has a self lo op. In par ticular, they show ed that disjunctive Bo olea n netw o rks (which they called OR-P DS) have only fixed p oints as limit cycles [37, Theorem 3 .3]; this follows from our results as we s how in Remark 2.8. • Another family consists of the conjunctive Bo olea n cellula r a utomata and was analyzed in [16]. In particula r , upper b ounds for the num b er of cycles are g iven. Here we study the who le clas s of conjunctive Boo lean netw orks and provide low er and upp er b ounds for the num b er of their limit cyc le s. In particula r, we present formulas for the exact num b er of fix ed p oints of any conjunctiv e or disjunctiv e b o olea n net work, s ee Equatio n (7.4). In this pap er we fo cus only o n conjunctive Bo olean netw orks, since for a n y disjunctiv e Bo ole a n netw o rk ther e exists a conjunctive B o olean ne tw ork that has exactly the same dynamics after relab eling the 0 and 1 and hence we ha ve the following theor em. Theorem 1.3. L et f = ( f 1 , . . . , f n ) : F n 2 − → F n 2 wher e f i = x i 1 ∨ · · · ∨ x i j i b e any disjunctive Bo ole an network. Consider the c onjunctive Bo ole an n etwork g = ( g 1 , . . . , g n ) : F n 2 − → F n 2 , wher e g i = x i 1 ∧ · · · ∧ x i j i . Then the t wo pha se sp ac es S ( f ) and S ( g ) ar e isomorphic as dir e cte d gr aphs. Pr o of. Let ¬ : F n 2 − → F n 2 be defined b y ¬ ( x 1 , . . . , x n ) = (1 + x 1 , . . . , 1 + x n ). Then it is easy to see that f ( x 1 , . . . , x n ) = ( ¬ ◦ g ◦ ¬ )( x 1 , . . . , x n ). Th us S ( f ) a nd S ( g ) are isomorphic directed gra phs.  R emark 1.4 . Let G b e a graph on n v ertices such that the in- degree for each vertex is no n-zero ( G has no sour c e vertex ). Then there is one and only one conjunctive net work f on n no des such that D ( f ) = G . T hus there is a one-to- one cor resp on- dence b etw een the set of a ll conjunctive Boo lean net works on n no des where no ne of the loca l functions is co nstant and the set of directed graphs on n vertices whe r e none of the vertices is a source. This cor resp ondence was us e d in [18] to find the p erio d o f a given Bo olean monomial system (co njunctiv e Bo olea n netw o rk) and to decide when that system is a fixed p oint system as we will re c all in the next section. In this pap er we pres e nt upper a nd low er b ounds on the num ber o f cycles of a n y leng th in the phase spa ce of any conjunctive Bo o lean netw ork. F urthermor e , we give upp er bo unds for the height. In the next section, we r ecall some results from gra ph theory as well as r esults ab out p ow ers of p ositive matrices that we will use to obtain upp er bo unds for the lengths of transients. THE DYNAMICS OF CONJUNCTIVE AND DISJUNCTIVE BOOLEAN NETWORKS 5 2. The Rela tionship Between Dependency Graph and Dynamics Let f : F n 2 − → F n 2 be a conjunctiv e B o o lean netw ork, G = D ( f ) and A the adjacency matrix of G . W e will assume here and in the r emainder of the pap er that none of the Bo olean co ordinate functions of f a re constant, that is, a ll vertices of G hav e pos itiv e in-degre e . 2.1. The Adjacency Matrix. Define the following relation on the vertices of G : a ∼ b if and only if there is a directed pa th from a to b a nd a directed path fro m b to a . It is easy to chec k that ∼ is an eq uiv alence relation. Suppo se there are t equiv ale nc e cla sses V 1 , . . . , V t . F or each equiv alence class V i , the subgraph G i with the vertex set V i is calle d a str ongly c onne cte d c omp onent of G . The gr aph G is called str ongly c onne cte d if G has a unique stro ngly connected comp onent. There exists a p ermutation matrix P that p ermutes the rows and columns of A such that (2.1) P AP − 1 =      A 1 A 12 · · · A 1 t 0 A 2 · · · A 2 t . . . . . . . . . . . . 0 0 · · · A t      where A i is the adjacency matrix of the component G i , a nd A ij represents the edges from the co mpo nent G i to G j , see [38, Theo rem 3.2.4 ]. The for m in (2 .1) is called the F r ob enius Normal F orm of A . R emark 2 .1 . The effect of the matrix P o n the dep endency gr a ph is the relab eling of the v ertices of G s uc h that the diagonal blo cks co rresp ond to strongly connected comp onents of G . In Example 1.1 ab ov e, the adjacency ma trix of the dep endency graph is in the normal for m. Example 1.1 (Cont.). The dep endency g raph G in Figure 1 ha s 3 s tr ongly con- nected comp onents and their vertex sets a re: V 1 = { 1 , 2 , 3 } , V 2 = { 4 } , and V 3 = { 5 , 6 } , s ee Figure 3 (left). F or an y non-empt y strongly co nnec ted comp onent G i , let h i be the conjunctive Bo olean netw ork with dep endency g raph D ( h i ) = G i . Let h : F n 2 − → F n 2 be the conjunctive Bo olean netw ork defined b y h = ( h 1 , . . . , h t ). That is, the dependency graph of h is the disjoint union of the strongly connec ted graphs G 1 , . . . , G t . Example 1 .1 (Con t.). The conjunctive Bo olean net w ork h cor resp onding to the dis- joint union is h : F 6 2 − → F 6 2 and given by h ( x 1 , . . . , x 6 ) = ( h 1 ( x 1 , x 2 , x 3 ) , h 2 ( x 4 ) , h 3 ( x 5 , x 6 )) where h 1 ( x 1 , x 2 , x 3 ) = ( x 2 x 3 , x 1 , x 2 ), h 2 ( x 4 ) = x 4 , and h 3 ( x 5 , x 6 ) = ( x 6 , x 5 ). Now define the following relation among the strongly connected components G 1 , . . . , G t of the dep e ndenc y graph D ( f ) of the netw ork f . (2.2) G i  G j if there is at le a st one edge from a vertex in G i to a vertex in G j . Since G 1 , . . . , G t are the strongly connected comp onents of D ( f ), the set of strongly comp onents with the relation  is a partially ordered set P . In this pap er, we rela te the dynamics of f to the dy namics of its stro ngly connected components a nd their po set P . Example 1.1 (Cont.). The p oset of the stro ngly connected comp onents of f is in Figure 3(right). 6 JARRAH, LAUBENBA CHER G G G 1 2 3 2 3 4 5 1 6 Figure 3. The strongly c onnected comp onents o f f (left) and their p oset (rig ht ). F or a n y no n-negative matr ix A , the sequence { A, A 2 , . . . } has b een studied exten- sively , see, for example, [38; 39]. Next w e use the Bo o lean o per ators AND a nd OR to ex amine the sequence of powers of A and infer results about the co njunctiv e Bo olean net work that corresp onds to an adjacency matrix A . 2.2. P ow ers of B o olean Matrices. Le t A, B b e n × n Bo olea n matrices (all ent ries are either 0 or 1 ). Define A ⊗ B such that ( A ⊗ B ) ij = W n k =1 ( A ik ∧ B kj ), where ∨ (resp. ∧ ) is the Bo olean OR (res p. AND) o per ator. Prop ositio n 2 . 2. L et A, B b e as ab ove and let f , g : F n 2 − → F n 2 b e the t wo c on- junctive Bo ole an n etworks that c orr esp ond to the adjac ency m at ric es A and B , r esp e ctively. That is, f i = x a i 1 1 x a i 2 2 · · · x a in n and g i = x b i 1 1 x b i 2 2 · · · x b in n for al l i . Then the adjac ency matrix c orr esp onding to f ◦ g is A ⊗ B . Pr o of. It is easy to see that, for all i , f i ( g 1 , . . . , g n ) = g a i 1 1 · · · g a in n = ( x b 11 1 x b 12 2 · · · x b 1 n n ) a i 1 · · · ( x b n 1 1 x b n 2 2 · · · x b nn n ) a in = x a i 1 b 11 + ··· + a in b n 1 1 · · · x a i 1 b 1 n + ··· + a in b nn n = x P n j =1 a ij b j 1 1 · · · x P n j =1 a ij b jn n Since we are working over F 2 , for all 1 ≤ k ≤ n , we have x q k = x k for all p ositive int eger s q . Thus x k divides f i ( g 1 , . . . , g n ) ⇐ ⇒ n X j =1 a ij b j k ≥ 1 ⇐ ⇒ a ij 0 b j 0 k = 1 for some j 0 ⇐ ⇒ a ij 0 ∧ b j 0 k = 1 for some j 0 ⇐ ⇒ n _ j =1 a ij ∧ b j k = 1 ⇐ ⇒ ( A ⊗ B ) ik = 1 . Thu s the matrix ( A ⊗ B ) is the a djacency matrix of f ◦ g .  Throughout this pap er, we use A s to denote A ⊗ · · · ⊗ A | {z } s time s . THE DYNAMICS OF CONJUNCTIVE AND DISJUNCTIVE BOOLEAN NETWORKS 7 Corollary 2 .3. L et f b e a c onjunctive Bo ole an network and let A b e the adjac en cy matrix of its dep endency gr aph D ( f ) . Then A s is the adjac ency matrix of f s . 2.3. The Lo op Number. An inv ariant of a strongly connected g r aph, called the lo op n umber, was defined in [18]. W e genera lize this de finitio n to any dire cted graph. Definition 2.4. The lo op numb er of a strongly connected g raph is the g reatest common diviso r o f the lengths of its simple (no rep eated vertices) directed cycles. W e define the loop num b er of a trivial s tr ongly connected g r aph to b e 0. The loop nu mber of an y directed g raph G is the least common m ultiple of the lo op n umbers of its non-trivia l strongly co nnected compo nent s. R emark 2.5 . Let G b e a dir e cted graph and A its adjacency matr ix. (1) The lo op num be r of G is the same a s the index of cyclicity of G a s in [40] and the index of imprimitivity of A as in [38; 41]. (2) The lo op num b er can b e computed in p olynomia l time, see [18] for an algorithm. (3) If the lo op num ber of G is 1, the adjacency ma trix A of G is called primitive , see [38]. Example 1.1 (Cont .). The lo o p n umbers o f V 1 , V 2 , V 3 are 1,1,2, r esp ectively . See Figure 3(left). In particular, the lo op num b er of G is 2. Definition 2 .6. The exp onent of an irreducible matrix A with lo op num ber c is the least p ositive in teger k such tha t A k + c = A k . The following lemma follo ws from Pro po sition 2.2 and the definition ab ov e. Lemma 2.7. L et f b e a c onjunctive Bo ole an network and let A b e t he adjac ency matrix of its dep endency gr aph D ( f ) . Supp ose the lo op numb er of A is c and its exp onent is k . Then h( f ) = k and p( f ) divi des c . In p articular, C ( f ) i = 0 for every i ∤ c and henc e Equation (1.1 ) b e c omes (2.3) C ( f ) = X i | c C ( f ) i C i , Example 1.1 (Cont.). The phase spa ce S ( f ) ha s 2 6 vertices, its cycle structure is C ( f ) = 4 C 1 + 1 C 2 . In particular, the p erio d of f is 2 which is the same as the lo op nu mber of its dep endency g raph G . R emark 2.8 . It is clea r that if x i app ears in f i for all i , then the lo o p num ber of D ( f ) is one a nd hence S ( f ) has o nly fixed points as limit cy cles, this was shown in [37, Theor em 3.3]. Also if each edge in the dep e ndenc y gr aph is undirected, then the depe ndenc y graph is made up o f simple cycles of length 2 a nd hence the lo op nu mber of D ( f ) is either tw o or o ne. Thus S ( f ) has only fixed points a nd possibly cycles of length tw o, which was shown in [36, Lemma 1]. F or the case when D ( f ) is strong ly c o nnected p( f ) = c a s was s hown in [1 8, Theorem 4.13]; in par ticular, S ( f ) has a s imple cy cle of le ng th l if and only if l divides c . In general, how ever, this is not the case. Example 2 .9 . Consider the Bo olea n net work f = ( x 2 , x 1 , x 2 x 5 , x 3 , x 4 ) : F 5 2 − → F 5 2 . The graph D ( f ) has tw o stro ngly connected c o mpo nent s with lo op nu mbers 2 and 8 JARRAH, LAUBENBA CHER 3 resp ectively , and hence the lo op n umber of D ( f ) is 6. Ho wev er, it is easy to see, using DVD [29], that C ( f ) = 3 C 1 + 1 C 2 + 2 C 3 , and hence f has no cycle of length 6. The exp onent has b een studied extensively and upper b ounds a re known, [40, Theorem 3.11] presents an upp er b ound for the exp onent of any ir reducible matrix. Using the lemma ab ov e, we rewrite this upper bound for the heigh t of an y Bo o lean monomial system with a stro ngly connected dep e ndenc y graph. Theorem 2.10. L et f b e a c onjun ctive Bo ole an network with str ongly c onne cte d dep endency gr aph D ( f ) , and su pp ose the lo op numb er of D ( f ) is c . Then h( f ) ≤    ( n − 1 ) 2 + 1 , if c = 1; max { n − 1 , n 2 − 1 2 + n 2 c − 3 n + 2 c } , if c > 1 . Pr o of. The pro of follows from Propo sition 2.2 above and [40, Theorem 3.11].  F urther mo re, [40, Theo rem 3.20 ] als o implies an upp er b ound for the height of any conjunctive Bo olean net work. Theorem 2.11. L et f b e any c onjunctive Bo ole an network on n n o des. Then h( f ) ≤ 2 n 2 − 3 n + 2 . Pr o of. The pro of follows from Propo sition 2.2 above and [40, Theorem 3.20].  W e clo se this section with a classical theore m about p ositive p ow ers of Bo olean matrices. This theo rem has the remark able coro llary that almost a ll c o njunctiv e Bo olean netw orks have a strong ly co nnec ted dependency gr aph a nd, furthermore, hav e only fixed points as limit cycles. Theorem 2.12. [38, Th e or em 3.5.11] L et N ( n ) b e t he numb er of c onjunctive Bo ole an networks on n no des with str ongly c onne cte d dep endency gr aph and lo op numb er 1. Then lim n →∞ N ( n ) 2 n 2 = 1 . In p articular, sinc e ther e ar e 2 n 2 differ ent c onjunctive Bo ole an networks on n no des, as n → ∞ , almost al l c onjunctive Bo ole an networks on n no des have a str ongly c onne cte d dep endency gr aph and have only fixe d p oints as limit cycles. Although ther e is a wealth of informa tio n ab out p ow ers o f non-negative ma trices such as the transient leng th or possible cycle length, very little seems to be known ab out the num b er of cycles of a given length in the phas e spa ce and that is the main goal o f this pap er. In the next section we give a complete a nswer to this problem for conjunctive Bo olea n net works with s trongly c o nnected dep endency gra ph. F o r this class of systems, we find the n umber of cy c le s of any p ossible length in the phase space. 3. Networks with S tr ongl y Connected Dependency Graph In this sec tion we give an exact formula fo r the cycle struc tur e of conjunctive Bo olean netw orks with strong ly connected dep endency graphs. Since ”almost a ll” conjunctive B o olean netw o rks have this pr op erty by Theo rem 2 .12, one may co n- sider this result as giv ing a complete answer for c onjunctiv e Bo ole a n netw o rks in the limit. Howev er, in the next section w e will also co nsider net works with genera l depe ndency graphs and give upper a nd low er b ounds for the cycle structure. THE DYNAMICS OF CONJUNCTIVE AND DISJUNCTIVE BOOLEAN NETWORKS 9 Before deriving the desired state space r esults we prov e so me needed facts ab out general finite dynamical systems. Lemma 3.1. L et f : X − → X b e a finite dynamic al system, with X a finite set, and let u ∈ X b e a p erio dic p oint of p erio d t . (1) If f s ( u ) = u , then t divides s . (2) The p erio d of f j ( u ) is t , for any j ≥ 1 . (3) If f s ( u ) = f j ( u ) , then t divide s s − j . Pr o of. Since t is the per io d o f u and f s ( u ) = u , by definition s ≥ t . Th us s = q t + r , where 0 ≤ r < t . Now u = f s ( u ) = f r ( f qt ( u )) = f r ( u ). Since r < t and t is the per io d of u , r = 0 and henc e t divides s . That prov es (1). The pro of of (2) is straightforward, since f t ( f j ( u )) = f j ( f t ( u )) = f j ( u ). No w we prov e (3). Supp ose s ≥ j . Since f s ( u ) = f j ( u ), we hav e f s − j ( f j ( u )) = f s ( u ) = f j ( u ). Thus, by (1), the p erio d of f j ( u ) divides s − j . But the pe r io d of f j ( u ) is t , by (2).  Lemma 3. 2. L et f : X − → X b e a finite dynamic al system. Then, p( f ) = c and h( f ) = d if and only if c and d ar e the le ast p ositive inte gers such that f m + c ( u ) = f m ( u ) for al l m ≥ d and u ∈ X . Pr o of. Suppo s e that p( f ) = c and h( f ) = d . Then for all u ∈ X and m ≥ d , f m ( u ) is a per io dic p oint and hence its p erio d divides c . Th us f m + c ( u ) = f c ( f m ( u )) = f m ( u ). Now supp ose c and d are the lea st positive integers such that f m + c ( u ) = f m ( u ) for all m ≥ d and u ∈ X . W e wan t to sho w that p( f ) = c and h( f ) = d . It is clear that f d ( u ) is p er io dic for all u ∈ X and d is the lea st such positive n umber. Thus the height of f is d . Also, the perio d f is c , since c is the least p ositive integer s uch that f c ( u ) = u for a n y p erio dic p oint u ∈ X .  Theorem 3.3. L et f : X − → X and g : Y − → Y b e t wo finite dynamic al systems. Define the system h : X × Y − → X × Y by h ( u , u ′ ) = ( f ( u ) , g ( u ′ )) . Then S ( h ) = S ( f ) × S ( g ) . Pr o of. This follows from the fact that h ( u , u ′ ) = ( v , v ′ ) if and only if f ( u ) = v and g ( u ′ ) = v ′ for all u ∈ X a nd u ′ ∈ Y .  Corollary 3.4. L et f , g and h b e as in The or em 3.3. Then t he p erio d of h is p( h ) = lcm { p( f ) , p( g ) } and its height is h( h ) = max { h( f ) , h( g ) } . Pr o of. It is easy to s ee that a set A ⊂ X × Y is a cycle in S ( h ) if and o nly if A X (resp. A Y ) is a cycle in S ( f ) (resp. S ( g )), wher e A X := { u ∈ X : ( u , v ) ∈ A for some v ∈ Y } . F urthermor e, it is clea r that the leng th of the c ycle A is the least common mult iple of the lengths of A X and A Y . Thus p( h ) = lcm { p( f ) , p( g ) } . The pro of of h( h ) = ma x { h( f ) , h( g ) } follows from the definition of height.  Recall Equation (1.1), in particular , that C ( f ) m is the num b er o f cycles of length m in the phase spa c e o f f , the following co rollary follows directly from Theor em 3.3. 10 JARRAH, LAUBENBA CHER Corollary 3.5. L et f , g and h b e as ab ove. Then, t he cycle structu r e of h is C ( h ) = C ( f ) C ( g ) := P m | p( h ) C ( h ) m C m , wher e (3.1) C ( h ) m = X s | p ( f ) t | p ( g ) lcm { s,t } = m gcd { s, t } C ( f ) s C ( g ) t . That is, t he gener ating fu n ction for the cycle structur e of h is the pr o duct of the gener ating fun ct ions of f and g , wher e C s · C t = gcd { s , t }C lcm { s,t } . Let f : F n 2 − → F n 2 be a conjunctive Bo olean netw ork. Assume that the dep endency graph D ( f ) o f f is strongly connected with lo o p num ber c . F or any divisor k o f c , it is well-known that the set of vertices of D ( f ) can b e partitione d int o c non-empty sets W 1 , . . . , W k such that each edge of D ( f ) is an edg e from a vertex in W i to a vertex in W i +1 for some i with 1 ≤ i ≤ k a nd W k +1 = W 1 . F or a pro of of this fact see [38, Lemma 3.4.1(iii)] or [1 8, Lemma 4.7]. By definition the length of any cyc le in the phase s pa ce S ( f ) of f divides c, the p erio d of f . F or any positive integers p, k that div ide c , let A ( p ) b e the set of per io dic po in ts of p erio d p a nd let D ( k ) := [ p | k A ( p ). Lemma 3. 6. The c ar dinality of the set D ( k ) is | D ( k ) | = 2 k . Pr o of. Let Φ : F k 2 − → D ( k ) b e defined by Φ( x 1 , . . . , x k ) = ( x 1 , . . . , x 1 | {z } s 1 times , x 2 , . . . , x 2 | {z } s 2 times , . . . , x k , . . . , x k | {z } s k times ) , where, without lo ss of genera lit y , W i = { v i, 1 , . . . , v i,s i } for all 1 ≤ i ≤ k . W e will sho w that Φ is a bijection. First w e show tha t Φ is w ell-defined. Let z = Φ( x 1 , . . . , x k ), by [18, Theorem 4.10], there exists a p ositive in teger m such that f mk + j ( z ) = ( x j +1 , . . . , x j +1 | {z } s 1 times , x j +2 , . . . , x j +2 | {z } s 2 times , . . . , x j , . . . , x j | {z } s k times ) . In particular, f k ( z ) = f mk + k ( z ) = z . Thus z = Φ( x 1 , . . . , x k ) ∈ D ( k ). Since it is clear that Φ is one-to one , it is left to show that Φ is ont o. Let z ∈ D ( k ) which means that f k ( z ) = z . It is enough to show that z i,h = z i,g for all 1 ≤ h, g ≤ s i k and 1 ≤ i ≤ k . Supp ose no t. With out loss of g enerality , let z i,h = 1 and z i,g = 0. Since k divides c , f c ( z ) = z . But ther e is a path from v i,h and v i,g of length divides c , contradiction. Hence Φ is a bijection a nd | D ( k ) | = 2 k .  Corollary 3. 7. If p is a prime numb er and p k divides c for some k ≥ 1 , then | A ( p k ) | = 2 p k − 2 p k − 1 . Pr o of. It is clea r that | D (1) | = 2. Namely , (0 , . . . , 0) a nd (1 , . . . , 1) a re the only t wo fix e d p oints. Now if p is prime and k ≥ 1, then the pro of follows fr om the fa c t that D ( p k ) = D ( p k − 1 ) U A ( p k ), where U is the disjoint union.  The follo wing theorem gives the exact num b er of perio dic points of any p ossible length. THE DYNAMICS OF CONJUNCTIVE AND DISJUNCTIVE BOOLEAN NETWORKS 11 Theorem 3.8 . L et f b e a c onjun ct ive Bo ole an network whose dep endency gr aph is str ongly c onne cte d and has lo op numb er c . If c = 1 , then f has the two fixe d p oints (0 , 0 , . . . , 0 ) and (1 , 1 , . . . , 1) and no other limit cycles of any length. If c > 1 and m is a divisor of c , then (3.2) | A ( m ) | = 1 X i 1 =0 · · · 1 X i r =0 ( − 1) i 1 + i 2 + ··· + i r 2 p k 1 − i 1 1 p k 2 − i 2 2 ...p k r − i r r , wher e m = Q r i =1 p k i i is the prime factorization of m , t hat is p 1 , . . . , p r ar e distinct primes and k i ≥ 1 for al l i . Pr o of. The sta tement f or c = 1 is part of the previous cor ollary . Now supp ose that c > 1. F o r 1 ≤ j ≤ r , let m j = p k j − 1 j Q r i =1 ,i 6 = j p k i i . T he n D ( m ) = A ( m ) U ( S r j =1 D ( m j )), where U is a disjoint union, in particular, A ( m ) = D ( m ) \ r [ j =1 D ( m j ) . The formula 3.2 fo llows from the inclusion-ex clusion principle and the disjoint union ab ov e.  Corollary 3. 9. If m divides c , then the n umb er of cycles of lengt h m in t he phase sp ac e of f is C ( f ) m = | A ( m ) | m . Henc e the cycle st ructur e of f is C ( f ) = X m | c | A ( m ) | m C m . R emark 3.10 . Notice that the cy cle str ucture of f depe nds o n its lo op num b er only . In particular , a co njunctive Bo olean netw o rk with lo op num b er 1 on a strongly co n- nected dep endency graph only ha s as limit cycles the tw o fixed po ints (0 , 0 , . . . , 0) and (1 , 1 , . . . , 1), regardless of how man y vertices its dependency graph has. 4. Networks with general dependency graph Let f : F n 2 − → F n 2 be a co njunctiv e Bo olean netw ork with dep endency gra ph D ( f ). Without loss of gener ality , the adjacency matrix o f D ( f ) is in the form (2.1). Let G 1 , . . . , G t be the stro ngly c onnected compo ne nts of D ( f ) corre spo nding to the matrices A 1 , . . . , A t , resp ectively . F urthermor e, suppose that none of the G i is trivial (i.e., A i is the zero matrix). F or 1 ≤ i ≤ t , let h i be the co njunctiv e Boolea n net work that has G i as its dep endency graph and suppose that the lo op n um b er of h i is l i . In particular, the lo op num b er o f f is l := lcm { l 1 , . . . , l t } . In the remainder o f the pap er, we present lo wer and upp er b ounds for the num ber of cycles of a g iven length. W e use the strongly connected comp onents of the depe ndency graph and their p oset to infer the cycle structure of f . Let G 1 and G 2 be tw o strongly connected comp onents in D ( f ) and supp ose G 1  G 2 . F urthermore, assume the vertex set of G 1 (resp. G 2 ) is { x i 1 , . . . , x i s } (resp. { x j 1 , . . . , x j t } ). Without loss o f generality , let x i 1 − → x j 1 be a dire c ted edge in D ( f ) and let D ′ be the gr aph D ( f ) after deleting the edge ( x i 1 , x j 1 ). Let g be the conjunctive Boolea n netw ork such that D ( g ) = D ′ . Theorem 4. 1. Any cycle in the phase s p ac e of f is a cycle in t he phase sp ac e of g . In p articular C ( f ) ≤ C ( g ) c omp onentwise. 12 JARRAH, LAUBENBA CHER Pr o of. Let C := { u , f ( u ) , . . . , f m − 1 ( u ) } be a cycle of length m in S ( f ). T o show that C is a cy c le in S ( g ), it is enough to show that, whenever x i 1 -v alue in u is 0, there exists x j w ∈ G 2 , suc h that there is an edge from x j w to x j 1 and the x j w -v alue in u is 0. Thus, the v alue of x j 1 is determined already by the v alue of x j w and the edge ( x i − 1 , x j 1 ) do es not make a difference here and hence C is a cycle in S ( g ). Suppo se the lo op num b er of G 1 (resp. G 2 ) is a (resp. b ). Now, any path from x i 1 (resp. y j 1 ) to itself is o f leng th pa (resp. q b ) where q , p ≥ T and T is larg e enough, see [18, Coro llary 4.4]. Th us there is a path fr om x i 1 to x j 1 of leng th q a + 1 for any q ≥ T . Also , there is a directed path fr o m x j 1 to x j w of length q b − 1 for any q ≥ T . This implies the existence of an edge from x i 1 to x j w of length q ( a + b ) for all q ≥ T . Now u = f m ( u ) = f mk ( u ), for all k ≥ 1. Choos e q , k large eno ugh such that q ( a + b ) = km . Then the v alue of x j w in f q ( a + b ) ( u ) is equal to zero, since there is a path fr om x i 1 to x j w of length q ( a + b ) and the v alue o f x i 1 is zer o. Therefor e, the v alue o f x j w in u is zero , since u = f mk ( u ) = f q ( a + b ) ( u ).  Corollary 4.2. L et f and D ( f ) b e as ab ove. F or any t wo stro ngly c onne cte d c omp onents in D ( f ) that ar e c onn e cte d, dr op al l but one of t he e dges. L et D ′ b e the new gr aph and let g b e the c onjunctive B o ole an network such that D ( g ) = D ′ . Then any cycle in S ( f ) is a cycle in S ( g ) . In p articular C ( f ) ≤ C ( g ) c omp onentwise. Notice that there ar e many different p ossible g one can get and ea ch one of them has the same poset as f and provides an upper b ound for the cycle structur e o f f . How ev er, we do not hav e a po lynomial for m ula for the cycle structur e of g . When we delete all edges b etw een an y tw o str ongly connected comp onents we get a n easy formula for the c ycle structure for the corres po nding system as w e describ e be low. In Section 7 w e will pr esent an improv ed upper b ound for the cycle structure o f f . Let h : F n 2 − → F n 2 be the conjunctive Bo olea n netw ork with the disjoint union of G 1 , . . . , G t as its dep endency gra ph. That is, h = ( h 1 , . . . , h t ). F or h , there are no e dg es b etw een any tw o strongly connected co mpo nen ts of the dep endency gra ph of the netw ork. Its cycle structure can b e co mpletely determined from the cycles structures of the h i alone. Theorem 4.3. L et C ( h i ) = P j | l i a i,j C j b e the cyc le structur e of h i . Th en the cycle structur e of h is C ( h ) = Q t i =1 C ( h i ) and the numb er of cycles of length m (wher e m | l ) in the phase sp ac e of h is (4.1) C ( h ) m = X j i | l i lcm { j 1 ,...,j t } = m j 1 · · · j t m t Y i =1 a i,j i . Pr o of. This follows directly from Corollary 3.5.  Corollary 4.4. L et f and h b e as ab ove. The numb er of cycles of any length in the phase sp ac e of f is less than or e qual to t he numb er of cycles of that length in the phase sp ac e of h . That is C ( f ) ≤ C ( h ) c omp onentwise. Example 2 .9 (Co n t.). Here h = ( h 1 , h 2 ), wher e h 1 ( x 1 , x 2 ) = ( x 2 , x 1 ) and h 2 ( x 3 , x 4 , x 5 ) = ( x 5 , x 3 , x 4 ). It is e a sy to c heck that C ( h 1 ) = 2 C 1 + 1 C 2 , and C ( h 2 ) = 2 C 1 + 2 C 3 . By Theorem 4.3, C ( h ) = 4 C 1 + 2 C 2 + 4 C 3 + 2 C 6 . In particular, C ( f ) ≤ C ( h ). THE DYNAMICS OF CONJUNCTIVE AND DISJUNCTIVE BOOLEAN NETWORKS 13 5. Bound s on the cycle structure Let f , h , G 1 , . . . , G t be as ab ov e a nd let l i be the lo op n umber of G i . F urther- more, let P be the p oset of the str o ngly connected comp o nent s. Let Ω b e the set of all maximal antic hains in P . F o r J ⊆ [ t ] = { 1 , . . . , t } , let x J := Q j ∈ J x j and let J := [ t ] \ J . Definition 5.1. Le t A be a subset of the set of limit cycles of h and let s i be the nu mber of limit cy cles of length i in A . W e define and denote kAk by kAk := X i s i C i . R emark 5.2 . Using the inclusio n-exclusion principle, if A , A 1 , ..., A s are subs e ts of the set of limit cycles of h and A = S s i A i , then, kAk = X J ⊆ [ s ] ( − 1) | J | +1 k \ j ∈ J A j k . Definition 5. 3. F o r any subset J ⊆ [ t ], let J  := { k : G j  G k for some j ∈ J } , J  := { k : G j  G k for some j ∈ J } , J ≺ := { k : G j ≺ G k for some j ∈ J } , and J ≻ := { k : G j ≻ G k for some j ∈ J } . A limit cycle C in the phase s pace of f is J 0 (resp. J 1 ) if the G j comp onent of C is 0 (resp. 1) for all j ∈ J . Notice that the sets defined ab ove are closely related to upper and low er order ideals in p osets, see [42, p. 100]. Example 1.1 (Cont.). Let J = { 2 } . Then J  = { 2 , 3 } , J  = { 1 , 2 } , J ≺ = { 3 } , and J ≻ = { 1 } . Definition 5.4. A limit cycle C in S ( h ) is J − r e gular if C is b oth ( J ≺ ) 0 and ( J ≻ ) 1 , for some maximal antic hain J in P . A limit cycle is ca lled r e gular if it is J − regular for some maximal a ntic hain J . That is, an element of a J − reg ula r limit cycle has 0 in all en tries corres po nding to strongly connec ted components ly ing ab ov e comp onents in J and 1 in all entries co r resp onding to co mpo nen ts lying b elow comp onents in J . Lemma 5 .5. L et J ⊆ Ω b e a set of maximal antichains. F or J ∈ J , let A J b e the set of al l J − r e gular limit cycles in the phase sp ac e of f . Then, k \ J ∈J A J k = Y j ∈ T J ∈J J C ( h j ) . Pr o of. It is eas y to see that a regula r limit cycle C ∈ T J ∈J A J if a nd o nly if the only non trivial comp onents of C a re in j ∈ T J ∈J J .  6. A Lower Bound It is easy to s ee that a regular cycle is actua lly a limit cycle in the phase spa ce of f , a nd hence the num b ers of regular cycle s of different length provide low er bo unds for the num b er o f limit cycles of different lengths in the pha se s pace of f . Next we expre s s these n um b ers in ter ms of the cycle str ucture C ( h i ) of the strongly connected comp onents. 14 JARRAH, LAUBENBA CHER Lemma 6. 1. The cycle stru ct ur e of the r e gular limit cycles in S ( h ) is (6.1) X J ⊆ Ω ( − 1) |J | +1 Y j ∈ T J ∈J J C ( h j ) . Pr o of. By definition, the set of regular limit cycles is S J ∈ Ω A J . No w, using the inclusion-exclusio n principle and Lemma 5.5, it follows that [ J ∈ Ω A J = X J ⊆ Ω ( − 1) |J | +1 k \ J ∈J A J k = X J ⊆ Ω ( − 1) |J | +1 Y j ∈ T J ∈J J C ( h j ) .  Theorem 6. 2. Consider the function L ( x 1 , . . . , x t ) := X J ⊆ Ω ( − 1) |J | +1 Y j ∈ T J ∈J J x i . Then, for any c onjunctive Bo ole an net work f with subnetworks h 1 , . . . , h t and Ω it’s set of maximal antichains in the p oset of f , we have (6.2) L ( C ( h 1 ) , . . . , C ( h t )) ≤ C ( f ) . Her e, the function L is evaluate d using the “mult iplic ation” describ e d in Cor ol lary 3.5. This ine qu ality pr ovides a sharp lower b ound on the numb er of limit cycles of f of a given length. Pr o of. The inequality follows from the previous lemma. Now to show that this low er bo und is sharp, it is enough to prese nt a Bo olean monomial dynamical sy stem f such that C ( f ) = L ( C ( h 1 ) , . . . , C ( h t )). Let f b e such tha t w he never there is a directed edge from G i to G j in the p ose t p , there is a dir ected edge fr o m e very vertex in G i to every vertex in G j in the depe ndency graph of f . It is enough to check that any cycle in the phase space of f is regular. Let C b e a cycle in the phas e space of f . Le t C j be the pr o jection of C on the strong ly connected comp onent G j . It is clear that C j is cy cle in the C ( h j ). Suppo se that C j is non tr ivial. No w if G i  G j , then all entries cor resp onding to G i m ust be one. On the o ther hand, if G i  G j , then all entries corr esp o nding to G i m ust b e zer o. Therefore, C is regular.  Note that the left side of the inequality (6.2) is a p olyno mial function in the v ar iables C ( h i ), with co efficient s that are functions of the cy c le n umbers of v ario us lengths of the h i and the anti-chains of the p oset of strongly connected co mpone nts. That is , the sharp lower bound is a po lynomial function dep ending exc lus ively on measures of the netw ork top ology . 7. An Upper Bound Next we present a p olynomial whose co efficients provide upp er b ounds fo r the nu mber of p ossible limit cy cles in the phase space of f . Similar to the low er bound po lynomial, this upp er b ound p olynomial is in terms o f the stro ngly connected comp onents of f and their p oset. How ever, this upper b ound is not sharp. In fact, we will show that it is not p ossible to give a sharp upp er b ound that is in THE DYNAMICS OF CONJUNCTIVE AND DISJUNCTIVE BOOLEAN NETWORKS 15 a p olynomial for m with co nstant co efficie nts. First we prov e the following set- theoretic equality whic h w e need in this section. Lemma 7 .1. L et { A i : i ∈ ∆ } b e a finite c ol le ction of finite s ets of limit cycles in S ( h ) , A = T i A i , and B i ⊆ A i . Then, for K ⊆ ∆ , k \ k ∈ K ( A k \ B k ) k = X L ⊆ K ( − 1) | L | k A ∩ \ k ∈ L B k k . Pr o of. T o b egin w ith, observe that \ k ∈ K ( A k \ B k ) = \ k ∈ K ( A \ B k ) = A \ [ k ∈ K ( A ∩ B k ) . Then, using the inclusion- exclusion principle we obtain: k \ k ∈ K ( A k \ B k ) k = k A \ [ k ∈ K ( A ∩ B k ) k = X L ⊆ K ( − 1) | L | k \ k ∈ L ( A ∩ B k ) k = X L ⊆ K ( − 1) | L | k A ∩ \ k ∈ L B k k .  Definition 7. 2. A limit c ycle C in S ( h ) is called admissible if, fo r any j ∈ [ t ], (1) If the co o r dinates of s tates in C corres po nding to component G j are 0, then C is ( j  ) 0 ; and (2) If the co o r dinates of s tates in C corres po nding to component G j are 1, then C is ( j  ) 1 . Notice that a ny limit cycle in the phase space of f is admissible. In particula r , we have the following lemma. Lemma 7.3. The nu mb er of admissible limit cycles in S ( h ) is an upp er b ound for the numb er of limit cycles in S ( f ) . F or J ⊆ [ t ], let Z J be the set of all J 0 limit cycles and let O J be the set o f all J 1 limit cycles. Let K , L ⊆ [ t ], then it is eas y to s ee tha t Z K ∩ Z L = Z K ∪ L , O K ∩ O L = O K ∪ L , and the cycle structure o f Z K ∩ O L is k Z K ∩ O L k = h K , L i Y j ∈ K ∪ L C ( h j ) , where h K, L i = ( 0 , if K ∩ L 6 = ∅ , 1 , if K ∩ L = ∅ . Theorem 7.4. The cycle structu r e of the admissible limit cycles in S ( h ) is e qu al to X I ⊆ N ⊆ [ t ] J ⊆ M ⊆ [ t ] ( − 1) | N | + | M | + | I | + | J | h I N , J M i Y k ∈ I N ∪ J M C ( h k ) , wher e I N = I  ∪ N , J M = J  ∪ M . 16 JARRAH, LAUBENBA CHER Pr o of. By definition, it follows that the set of admissible limit cycle s is Adm := { limit cycles in S ( h ) }\ [ i,j ∈ [ t ] ( Z { i } \ Z { i }  ) ∪ ( O { j } \ O { j }  ) . Using the inclusio n- exclusion principle, it follows that (7.1) k Adm k = Y k ∈ [ t ] C ( h k ) + X N ⊆ [ t ] M ⊆ [ t ] N ∪ M 6 = ∅ ( − 1) | N | + | M | k \ i ∈ N ( Z { i } \ Z { i }  ) ∩ \ j ∈ M ( O { j } \ O { j }  ) k . On the other hand, since T i ∈ N Z { i } = Z N , T j ∈ M O { j } = O M , Z i ⊆ Z { i }  and O j ⊆ O { j }  it follows fro m Lemma 7.1 that k \ i ∈ N ( Z { i } \ Z { i }  ) ∩ \ j ∈ M O { j } \ O { j }  ) k = X I ⊆ N J ⊆ M ( − 1) | I | + | J | k ( Z N ∩ O M ) ∩ ( \ i ∈ N Z { i }  ∩ \ j ∈ M O { j }  ) k = X I ⊆ N J ⊆ M ( − 1) | I | + | J | k ( Z N ∩ O M ) ∩ ( Z S i ∈ N { i }  ∩ O S j ∈ M { j }  ) k = X I ⊆ N J ⊆ M ( − 1) | I | + | J | k Z N ∩ O M ∩ Z I  ∩ O J  k = X I ⊆ N J ⊆ M ( − 1) | I | + | J | k Z N ∪ I  ∩ O M ∪ J  k = X I ⊆ N J ⊆ M ( − 1) | I | + | J | k Z I N ∩ O J M k = X I ⊆ N J ⊆ M ( − 1) | I | + | J | h I N , J M i Y k ∈ I N ∪ J M C ( h k ) . Equation (7.1) then b ecomes k Adm k = Y k ∈ [ t ] C ( h k ) + X N ⊆ [ t ] M ⊆ [ t ] N ∪ M 6 = ⊘ ( − 1) | N | + | M | X I ⊆ N J ⊆ M ( − 1) | I | + | J | h I N , J M i Y k ∈ I N ∪ J M C ( h k ) = Y k ∈ [ t ] C ( h k ) + X I ⊆ N ⊆ [ t ] J ⊆ M ⊆ [ t ] N ∪ M 6 = ⊘ ( − 1) | N | + | M | + | I | + | J | h I N , J M i Y k ∈ I N ∪ J M C ( h k ) = X I ⊆ N ⊆ [ t ] J ⊆ M ⊆ [ t ] ( − 1) | N | + | M | + | I | + | J | h I N , J M i Y k ∈ I N ∪ J M C ( h k ) .  THE DYNAMICS OF CONJUNCTIVE AND DISJUNCTIVE BOOLEAN NETWORKS 17 Corollary 7. 5. F or K, L ⊆ [ t ] , let φ ( K,L ) ( x 1 , . . . , x t ) =  0 , if K ∩ L 6 = ∅ , x K ∪ L , if K ∩ L = ∅ . Consider the function (7.2) U ( x 1 , . . . , x t ) = X I ⊆ N , J ⊆ M ( − 1) | N | + | M | + | I | + | J | φ ( I N ,J M ) ( x 1 , . . . , x t ) then (7.3) C ( f ) ≤ U ( C ( h 1 ) , . . . , C ( h t )) , which pr ovides ther efor e an upp er b ou n d for the cycle st r u ctur e of f . This upp e r b ound is clearly not sharp, since it is easy to find admiss ible limit cycles in the phas e space of h that a re not cycles in the phase space of f . The adv antage of this upp e r b ound, how ever, is that it is a p olynomial in terms of the strongly connected comp onents and their po set. F urthermor e, we will show next that this bo und gives the exact n umber of fixed p oints. Theorem 7.6. L et f b e a c onjunctive Bo ole an network. Then any a dmissible fixe d p oint is r e gu lar and henc e the numb er of fixe d p oints C ( f ) 0 in the phase sp ac e of f is (7.4) C ( f ) 0 = X J ⊆ Ω ( − 1) |J | +1 2 | T J ∈J J | . Pr o of. It follows from the definitions that any admissible fixed p oint is regula r, and hence U (2 C 1 , . . . , 2 C 1 ) = L (2 C 1 , . . . , 2 C 1 ). In particular, the num b er of fixed p oints C ( f ) 0 is the co efficient of C 1 in L (2 C 1 , . . . , 2 C 1 ). By Equation (6.1), we get L (2 C 1 , . . . , 2 C 1 ) = X J ⊆ Ω ( − 1) |J | +1 Y j ∈ T J ∈J J 2 C 1 = X J ⊆ Ω ( − 1) |J | +1 2 | T J ∈J J | C 1 .  7.1. An example. E ach system g from Co rollary 4.2 has the s ame p oset as f and the cycle structure o f each g is an uppe r b o und fo r the cycle structure o f f . How ever, using the p oset alone, o r using the cy c le structure o f the s trongly connected comp onents alone, one ca n not ex pect to find a p olyno mial form with constant co efficients that would pro vide a sharp upp er b ound, as we no w show. Let p be the p oset on t wo no des G 1 and G 2 where G 1  G 2 . Let f b e a conjunctive Boo le an net work with a dep endency gr aph that has only tw o strongly connected co mponents which are connected by one edge. Let F b e the set of all such systems. W e will show that there is no p oly no mial W ( x 1 , x 2 ) = P i,j a ij x i 1 x j 2 such that, for all f ∈ F , W ( C ( h 1 ) , C ( h 2 )) = C ( f ), where h 1 and h 2 are the conjunctive Bo olean net works corresp onding to the tw o s trongly connected comp onents of f . Suppo se the lo o p num b er of h 1 is 1 and that of h 2 is q wher e q is prime. Then C ( h 1 ) = 2 C 1 and C ( h 2 ) = 2 C 1 + 2 q − 2 q C q . It is easy to chec k that C ( f ) = 3 C 1 + 2 q − 2 q C q . Since C ( f ) = W ( C ( h 1 ) , C ( h 2 )), we ha ve 18 JARRAH, LAUBENBA CHER 2 q − 2 q C q + 3 C 1 = W (2 C 1 , 2 C 1 + 2 q − 2 q C q ) = X i,j a ij (2 C 1 ) i (2 C 1 + 2 q − 2 q C q ) j = X i,j 2 i a ij (2 C 1 + 2 q − 2 q C q ) j = X i,j 2 i a ij [2 j C 1 + [(2 + 2 q − 2 q ) j − 2 j ] C q ] = X i,j 2 i + j a ij C 1 + X i,j 2 i a ij [(2 + 2 q − 2 q ) j − 2 j ] C q . Equating c o efficient s, for a ny q , we have 2 q − 2 q = P i,j 2 i a ij [(2 + 2 q − 2 q ) j − 2 j ]. Therefore, a ij = 0 for all i, j ≥ 1 and hence W ( x 1 , x 2 ) must b e o f the form W ( x 1 , x 2 ) = a 11 x 1 x 2 + a 10 x 1 + a 01 x 2 − a 00 . Now suppose the lo op num ber of h 1 is p and that of h 2 is q . Then any cycle in the phase space of f is reg ular and hence it is eas y to chec k that C ( f ) = C ( h 1 ) + C ( h 2 ) − C 1 . In particular, a 11 = 0 and henc e W ( x 1 , x 2 ) = a 10 x 1 + a 01 x 2 − a 00 . How ev er, for the case when the lo op num b e r o f h 1 is 4 and that of h 2 is 6, we hav e C ( f ) = 3 C 1 + 3 C 2 + 2 C 3 + 4 C 4 + 11 C 6 + 2 C 12 , C ( h 1 ) = 2 C 1 + C 2 + 3 C 4 and C ( h 2 ) = 2 C 1 + C 2 + 2 C 3 + 9 C 6 . In particular, C ( f ) 6 = W ( C ( h 1 ) , C ( h 2 )). 8. Discussion In this pap er we have fo cused on the clas s of co njunctiv e and disjunctive Bo olean net works and hav e treated the problem of predicting net work dynamics from the net work topo logy . Such net works are entirely determined by the top olog y of their depe ndency graphs , so it should b e p ossible in pr inciple to extract a ll information ab out the dynamics from this top olog y . This pr oblem has be e n the sub ject of ma n y resear ch a r ticles and b een solved for the cla ss of XOR Bo olean netw ork s , that is, net works where all no des use the XOR Bo olea n op er ator. Determining the dynam- ics o f a network from its topolog y ha s practical impo rtance, fo r instance in the study o f dynamical pro cesses on so cial netw o rks, such as the sprea d of an infectious disease. P ublic health interv entions, such a s quara ntine of selected individuals or closure of certain institutions or means of travel, often attempt to alter ne tw ork dynamics by alter ing netw ork top ology . The cur rent study can b e seen as a the- oretical study that b egins to elucidate the role of certain fea tur es of the netw ork top ology in suppor ting cer tain types of dynamics. Another example is discussed in [2], where it is shown that the dynamics of a cer tain gene regula to ry netw ork in the fruit fly Dr osophi la melano gaster is determined b y the top olog y of the net work. W e ha ve shown that if the dependency graph of the netw ork is strongly con- nected, then the key top olog ical deter minant of dynamics is the lo op num b er, o r index of imprimitivity , of the gr aph, a nd one can describ e the cycle structure ex- actly . The approach here is to look at p ow ers of the incidence matrix of the graph in the ma x-plus algebra . In light of Theorem 2.1 2 one could argue that this is the prev a len t cas e for la rge netw orks. What is more, this theorem shows that large THE DYNAMICS OF CONJUNCTIVE AND DISJUNCTIVE BOOLEAN NETWORKS 19 net works tend to have loop num b er one, so that their dyna mics is e x traordinar ily simple, consisting only of tw o fixed p oints. 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(Abdul Salam Jarrah, Reinhard Laub enba cher, Alan V eliz-Cuba) Virginia Bioinforma tics Institute, Virginia Tech, Black sburg, V A 24061-0 477, USA E-mail addr ess : { ajarrah,reinha rd,alanavc } @vbi.vt.edu