Analysis of random Boolean networks using the average sensitivity
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Analysis of random Boolean networks using theaverage sensitivity
Steffen Schober ∗ and Martin Bossert Institute of Telecommunications and Applied Information Theory, Ulm UniversityAlbert-Einstein-Allee 43, 89081 Ulm, Germany
November 4, 2018
Abstract
In this work we consider random Boolean networks that provide ageneral model for genetic regulatory networks . We extend the analysis ofJames Lynch who was able to proof Kauffman’s conjecture that in the ordered phase of random networks, the number of ineffective and freezing gates is large, where as in the disordered phase their number is small.Lynch proved the conjecture only for networks with connectivity two andnon-uniform probabilities for the Boolean functions. We show how toapply the proof to networks with arbitrary connectivity K and to randomnetworks with biased Boolean functions. It turns out that in these casesLynch’s parameter λ is equivalent to the expectation of average sensitivity of the Boolean functions used to construct the network. Hence we canapply a known theorem for the expectation of the average sensitivity. Inorder to prove the results for networks with biased functions, we deductthe expectation of the average sensitivity when only functions with specificconnectivity and specific bias are chosen at random. Keywords:
Random Boolean networks, phase transition, average sensitivity
PACS numbers: 02.10.Eb, 05.45.+b, 87.10.+e
In 1969 Stuart Kauffman started to study random Boolean networks as simplemodels of genetic regulatory networks [1]. Random Boolean networks that con-sists of a set of Boolean gates that are capable of storing a single Boolean value.At discrete time steps these gates store a new value according to an initiallychosen random Boolean function, which receives its inputs from random chosengates. We will give a more formal definition later. Kauffman made numerical ∗ Corresponding author. E-Mail: Steff[email protected] RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- studies of random networks, where the functions are chosen from the set of allBoolean functions with K arguments (the so called N K -Networks ). He recog-nised that if K ≤
2, the random networks exhibit a remarkable form of orderedbehaviour: The limit cycles are small, the number of ineffective gates , which aregates that can be perturbed without changing the asymptotic behaviour, andthe number of freezing gates that stop changing their state is large. In contrastif K ≥
3, the networks do not exhibit this kind of ordered behaviour (see [1, 2]).The first analytical proof for this phase transition was given by Derrida andPomeau (see [3]) by studying the evolution of the Hamming distance of randomchosen initial states by means of so called annealed approximation . The firstproof for the number of freezing and ineffective gates was given by James Lynch(see [4], although slightly weaker results appeared earlier [5, 6]). Depending ona parameter λ , that depends on the probabilities of the Boolean functions, heshowed that if λ ≤ λ is less or equal to one. But it turns out that in somecases λ is equal to the expectation of the average sensitivity. Therefore we willfirst study the average sensitivity in Section 3. Afterwards it will be shownin Section 4 how to use the results from the previous section to apply Lynch’sanalysis to classical N K -Networks and biased random Boolean networks . Butfirst we will give some basic definition used throughout the paper in Section 2. In the following F = { , } denotes the Galois field of two elements, whereaddition, denoted by ⊕ , is defined modulo 2. The set of vectors of length K over F will be denoted by F K . If x is a vector from F K , its i th componentwill be denoted by x i . With u ( i ) ∈ F K we will denote the unit vector which hasall components zero except component i which is one. The Hamming weight of x ∈ F K is defined as w H ( x ) = |{ i | x i = 0 , i = 1 , . . . , K }| and the Hamming distance of x , y ∈ F K asd H ( x , y ) = w H ( x ⊕ y ) . A Boolean function is a mapping f : F K → F . A function f may be representedby its truth table t f , that is, a vector in F K , where each component of the truthtable gives the value of f for one of the 2 K possible arguments. To fix an orderon the components of the truth table, suppose that its i th component equalsthe value of the corresponding function, given the binary representation (to K bits) of i as an argument. a definition will be given later RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
In this section we will focus on the average sensitivity . The average sensitivityis a known complexity measure for Boolean functions, see for example [7] . Itwas already used to study Boolean and random Boolean networks for examplein [8, 9]. Definition 1.
Let f denote a Boolean function F K → F and u ( i ) a unit vector.1. The sensitivity s f ( w ) is defined as: s f ( w ) = (cid:12)(cid:12)(cid:12)n i | f ( w ) = f ( w ⊕ u ( i ) ) , i = 1 , . . . , K o(cid:12)(cid:12)(cid:12) .
2. The average sensitivity s f is defined as the average of s f ( w ) over all w ∈ F K : s f = 2 − K X w ∈ F K s f ( w )Now consider the random variable F K : Ω → F K , where F K denotes the seta all 2 K Boolean function with K arguments. The probability measure is givenby P ( F K = f ) = K . The expected value of the average sensitivity of thisrandom variable is denoted by E F K ( s f ), and is given by E F K ( s f ) = X f P ( F K = f ) s f The expected value was already derived in [10], and is given by:
Theorem 1 (Bernasconi [10]) . Let the random variable F K be defined as above, then E F K ( s f ) = X f P ( F K = f ) s f = K . We will now concentrate on biased Boolean functions. The bias of a Booleanfunction f : F K → F is defined as the number of 1 in the functions truth tabledivided by 2 K . To define the bias of a random Boolean function two definitionsare possible. First we can assumes that the truth tables of the Boolean functionsare produced by independent Bernoulli trials with probability p for a one (Thisshould be called mean bias, used for example in [3, 8] ). Therefore consider therandom variable F K,p . The probability of choosing a function f is given by P ( F K,p = f ) = p w H ( t f ) (1 − p ) K − w H ( t f ) For p = 1 / F K . here it is called critical complexity RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
As a second possibility, we can only choose functions which have bias p whereas to all other functions we assign probability 0 (we will call this fixed bias). Therefore consider the random variables F fixed K,p : Ω → F K . Denote thetruth table of a function f by t f . Further denote the set of all Boolean functions f with K arguments and w H ( t f ) = p K with F K,p . The probability for a certainfunction chosen according F fixed K,p is given by P ( F fixed K,p = f ) = ( |F K,p | if f ∈ F K,p f / ∈ F
K,p
Both definitions ensure that the expectation to get a one is equal to p if theinput of a function is chosen at random (with respect to uniform distribution).But it will turn out that these two different methods of creating biased Booleanfunctions, have a major impact on the average sensitivity.The expectation of the average sensitivity of F K,p was derived in [8]:
Theorem 2 ([8]) . Let the random variable F K,p be defined as above: E F K,p ( s f ) = 2 Kp (1 − p )For the random variable F fixed K,p we will now proof the following theorem:
Theorem 3.
Let the random variable F fixed K,p be defined as above: E F fixed K,p ( s f ) = 2 K +1 Kp (1 − p )(2 K − . Proof.
To find E F fixed K,p ( s f ) we will first consider the random variable F K,t : Ω →F K where t ∈ { , , · · · , K } and the probability of a function is given by P ( F K,t = f ) = ( Kt ) if w H ( t f ) = t . Consider the Boolean functions as functions into R by identifying 0 , ∈ F with 0 , ∈ R . Then we get or the function f : s f = 2 − K X w ∈ F K (cid:12)(cid:12)(cid:12)n i | f ( w ) = f ( u ( i ) ⊕ w ) , i = 1 , . . . , K o(cid:12)(cid:12)(cid:12) = 2 − K X w ∈ F K K X i =1 ( f ( w ) − f ( w ⊕ u ( i ) )) = 2 − K X w ∈ F K K X i =1 ( f ( w ) + f ( w ⊕ u ( i ) ) − f ( w ) f ( w ⊕ u ( i ) )) . RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- where u ( i ) again denotes the unit vector with i th component set to 1. Hence bythe linearity of the expectation E F K,t ( s f ) = 2 − K X w ∈ F K K X i =1 (cid:16) E F K,t ( f ( w )) + E F K,t ( f ( w ⊕ u ( i ) )) − E F K,t ( f ( w ) f ( w ⊕ u ( i ) )) (cid:17) . (1)Now we form a matrix with the truth tables of all functions with Hammingweight t as column vectors: M = (cid:16) c (1) , c (2) , · · · , c ( ( kt ) ) (cid:17) where c ( i ) ∈ F n M has exactly (cid:0) K t (cid:1) columns and 2 K rows. Each entry M i,j in the i th row and j th column equals the value of function f j given the binary representation of i as input.Hence E F K,t ( f ( w )) is determined by the number of 1 in the row associatedwith w divided by the length of the row. Consider an arbitrary row i . This rowhas a one at position j if the corresponding column c ( j ) has a one at position i .But there are (cid:0) K − t − (cid:1) column vectors with a 1 at position i . It follows: ∀ w ∈ F K : E F K,t ( f ( w )) = (cid:0) K − t − (cid:1)(cid:0) K t (cid:1) = t K . (2)As this holds for all w , we have ∀ w , u ( i ) ∈ F K : E F K,t ( f ( w ⊕ u ( i ) )) = t K . (3)To find an expression for E F fixed K,p ( f ( w ) f ( w ⊕ u ( i ) )) we consider two arbitraryrows l, m ( l = m ). Define the following sum: γ l,m = ( Kt ) X i =1 M l,i M m,i . Obviously M l,i M m,i = 1 only if we have a 1 in both rows at position i . Thismeans for the column vectors c ( i ) of M , we have c ( i ) l = c ( i ) m = 1. But there areexactly (cid:0) K − t − (cid:1) such column vectors in M . Therefore we have ∀ l, m, l = m : γ l,m = (cid:18) K − t − (cid:19) . As w = w ⊕ u ( i ) for all w , u ( i ) it follows: E F K,t ( f ( w ) f ( w ⊕ u ( i ) )) = (cid:0) K − t − (cid:1)(cid:0) K t (cid:1) = t ( t − K (2 K − . (4)5 RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
Hence substituting Equations (2), (3) and (4) into Equation (1) leads to E F K,t ( s f ) = K (2 K − t ) t K − (2 K − . Finally the claimed expression for E F fixed K,p ( s f ) can be obtained from the aboveequation by a substitution of t : t → p K .It should be noted, that the Theorems 1 and 2 can be proved using in asimilar way. Also worth noting is the fact, that if the functions are chosenaccording F K , F fixed K,p or F K,p the expectation of the sensitivity of a fixed vector w (namely the expectation of s f ( w )) is independent of w (see Equation (1),(2),(3) and (4)). Hence the following lemma holds Lemma 1. If F = F K , F fixed K,p or F K,p , then ∀ w , v ∈ F K : E F ( s f ( w )) = E F ( s f ( v ))Before proceeding to the next section, it should be noted, that using the samearguments as in the proof of Theorem 3, we can also prove the expectation ofaverage sensitivity of order l , defined as s ( l ) ( f ) = 2 − K X w ∈ F K (cid:12)(cid:12)(cid:8) x ∈ F K | w H ( x ) = l and f ( w ) = f ( w ⊕ x ) (cid:9)(cid:12)(cid:12) . In this case, instead of summing up all unit vectors in Equation (1), we sumup all vectors of Hamming weight l . As the equations (2) and (4) hold for all w ∈ F K we conclude that E ( s ( l ) ( F fixed K,p )) = (cid:18) Kl (cid:19) K +1 p (1 − p )(2 K − E ( s ( l ) ( F K,p )) = (cid:18) Kl (cid:19) p (1 − p )respectively E ( s ( l ) ( F K )) = 12 (cid:18) Kl (cid:19) . As already mentioned James Lynch gave a very general analysis of randomlyconstructed Boolean networks (see [4]). Before stating his results we give aformal definition for Boolean networks A Boolean network B is a 4-tuple6 RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- ( V, E, ˜ F , x ) where V = { , ..., N } is a set of natural numbers, E is a set oflabeled edges on V , ˜ F = { f , ..., f N } is a ordered set of Boolean functions suchthat for each v ∈ V the number of arguments of f v is the in-degree of v in E , these edges are labeled with 1 , ..., in-degree ( v ), and x = (x , . . . x n ) ∈ F N .Suppose that a vertex i has K i in-edges from vertices v i, , . . . , v i,K i . For y ∈ F N we define B ( y ) = (cid:16) f (y v , , . . . , y v ,K ) , . . . , f N (y v N, , . . . , y v N,KN ) (cid:17) . The state of B at time 0 is called the initial state x , so we define B ( x ) = x . Fortime t ≥ B t ( x ) = B ( B t − ( x )). Hence wecan in interpret V as set of gates, E and ˜ F describes their functional dependenceand x is the networks initial state.Assume some ordering f , f , ... on the set of all Boolean functions F , whereeach function f i depends on K i arguments. Further a random variable F : Ω →F with probabilities p i = P ( F = f i ) such that P ∞ i = i p i = 1 and P ∞ i =1 p i K i < ∞ . Now a random Boolean network consisting of N gates is constructed asfollows: For each gate a Boolean function is chosen independently, where theprobability of choosing f i is given by p i . Suppose a function f was chosen thathas K arguments, these arguments are chosen at random from all (cid:0) NK (cid:1) equallylikely possibilities. At last an initial state is chosen at random from the set onall equally likely states. If the Boolean functions are chosen according to ourpreviously defined random variable F K we will call this networks N K -Networkswith connectivity K . If the functions are chosen according to F fixed K,p or F K,p we will call this networks biased random Boolean networks with connectivity K and fixed bias p respectively mean bias p .Let us now state Lynch’s results. His analysis depends on a parameter R ∋ λ ≥ λ later in Definition 3. First we have to state Lynch’s definition of freezing and ineffective gates: Definition 2 (Lynch [4] Definition 1 Item 2 and 5) . Let x ∈ F N and v ∈ V .1. Gate v freezes to y ∈ F N in t steps on input x if B t ′ v ( x ) = y for all t ′ ≥ t .2. Let u ( i ) ∈ F n .A gate v is t -ineffective at input x ∈ F K if B t ( x ) = B t ( x ⊕ u ( v ) ) . Now we will state the main result.
Theorem 4 (Lynch [4] Theorem 4 and 6) . Let α , β be positive constants satisfying α log δ + 2 β < and α log δ < β where δ = E ( K i ) .1. There is a constant r such that for all x ∈ F N lim n →∞ P ( v is ineffective in α log N steps ) = r When λ ≤ , r = 1 and when λ > , r < . RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
2. There is a constant r such that for all x ∈ F N lim n →∞ P ( v is freezing in α log N steps ) = r When λ ≤ , r = 1 and when λ > , r < . The above theorem shows that if λ ≤ λ > Corollary 1 (Lynch [4] Corollary 3 and Corollary 6) . Let λ > . For almostall random Boolean networks1. if gate v is not α log N -ineffective, there is a positive constant W such thatfor t ≤ α log N , the number of gates affected by v at time t is asymptoticto W λ t ,2. if gate v is not freezing in α log N steps , there is a positive constant W such that for t ≤ α log N , the number of gates that affect v at time t isasymptotic to W λ t . Now we will state the definition of λ for Boolean networks: Definition 3 (Lynch [4], Definition 4) . Let f be a Boolean function of K ar-guments. For i ∈ { , . . . , K } , we say that argument i directly affects f on input w ∈ F K if f ( w ) = f ( w ⊕ u ( i ) ) . Now put γ ( f, w ) as the number of i ’s thatdirectly affect f on input w . Given a constant a ∈ [0 , , we define λ = ∞ X i =1 p i X w ∈ F Ki γ ( f i , w ) a w H ( w ) (1 − a ) K i − w H ( w ) . Obviously γ ( f, w ) is identical to s f ( w ) which will be used instead in thefurther discussion. The constant a is the probability that a random gate is one(at infinite time) given that all gates at time 0 have probability 0 . F which should be either F K , F fixed K,p or F K,p . The functions are chosenout the set F K , we denote a function’s probability with p f . It follows that λ = X w ∈ F K a w H ( w ) (1 − a ) K − w H ( w ) X f p f s f ( w ) (5)= X w ∈ F K a w H ( w ) (1 − a ) K − w H ( w ) E ( s F ( w )) (6)= E ( s F ( w )) K X i =0 (cid:18) Ki (cid:19) a i (1 − a ) K − i (7)= E ( s F ( w )) = E F ( s f ) (8) Please note that we here state a slightly weaker result than in the original analysis. RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- E ( s F ( w )) denotes the expectation of the sensitivity for a fixed w , Equation (7)follows from Lemma 1. Therefore, together with Theorem 1 and Theorem 3 weproved the following: Theorem 5 (Biased random Boolean networks) . For random Boolean networks,if 1. the functions are chosen according random variable F K,p , it follows that λ = 2 Kp (1 − p ) ,
2. the functions are chosen according random variable F fixed K,p , it follows that λ = 2 K +1 Kp (1 − p )2 K − . As a special case of the above theorem we get (or by using Theorem 1)
Theorem 6 ( N K -Networks) . In random Boolean networks, where the functionsare chosen according to the random variable F K λ = K . The results about
N K -Networks are consistent with experimental results. Infact if K ≤ λ = 1. The resulting phase diagram for biased random Boolean networks,where the functions are chosen according to F fixed K,p and F K,p is shown in Figure1. It it interesting to note that if the functions are chosen with fixed bias, thenalso Boolean networks with connectivity K = 2 can become unstable. Thisconclusion can be drawn from Lynch’s original result already. As mentioned inthe introduction, he showed for K = 2, that λ > .
5, the probability of choosing a constant function iszero, whereas both XOR and inverted XOR function have probability greaterzero, hence λ > annealed approximation (see [3]).In their annealed model the functions and connections are chosen at randomat each time step. Considering two instances of the same annealed networkstarting in two randomly chosen initial states s (0) , s (0) they show thatlim N →∞ lim t →∞ d H ( s ( t ) , s ( t )) N = c RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- o r d e r e d ph a s e d i s o r d e r e d ph a s e p K Figure 1: Phase diagram for biased random networks: Functions chosen accord-ing F K,p (dashed) and F fixed K,p (solid)where c = 1 if 2 Kp (1 − p ) ≤ c ≤ We would like to thank our colleges Georg Schmidt and Stephan Stiglmayr forproofreading and Uwe Schoening for useful hints.
References [1] S. Kauffman, Metabolic stability and epigenesis in randomly constructednets, Journal of Theoretical Biology 22 (1969) 437–467.[2] S. Kauffman, The large scale structure and dynamics of genetic controlcircuits: an ensemble approach, Journal of Theoretical Biology 44 (1974)167–190.[3] B. Derrida, Y. Pomeau, Random networks of automata - a simple annealedapproximation, Europhysics Letters 2 (1986) 45–49.[4] J. F. Lynch, Dynamics of random boolean networks, in: Conference onMathematical Biology and Dynamical Systems, University of Texas atTyler, 2005. 10