On the Lyapunov Exponent of Monotone Boolean Networks

We derive asymptotic formulas for the Lyapunov exponent of almost all monotone Boolean networks. The formulas are different depending on whether the number of variables of the constituent Boolean functions, or equivalently, the connectivity of the Boolean network, is even or odd.


Introduction
Boolean networks are complex dynamical systems that were proposed as models of genetic regulatory networks [13,14] and have since then been used to model a range of complex phenomena. Random Boolean networks (RBNs) are ensembles of randomly generated Boolean networks with random topology and random updating functions. For a bias p of the random Boolean functions, which is the probability that the function takes on the value 1, the critical connectivity is equal to K c = [2p (1 − p)] −1 (ref. [8]). Under a synchronous updating scheme, for K < K c , perturbations decay while for K > K c , perturbations spread throughout the network. This relationship is valid when the number of nodes in the network is infinite and coincides with the Lyapunov exponent [20,30]. Networks that are known to operate in the dynamically critical regime, meaning that their Lyapunov exponent is 0, are widely known to have many optimal properties and are thought to be a hallmark of living systems [14,24].
We consider updating rules that belong to the class of monotone Boolean functions. This class of functions is one of the most widely studied classes of Boolean functions [16]. Monotone Boolean networks have been primarily studied in the asynchronous updating scheme setting, whereby only one node is updated at a time. Some work has focused on long-term dynamics, such as fixed points and limit cycles [2,3,1]. It is also known that certain classes of fully asynchronous Boolean networks can be simulated by monotone Boolean networks [21]. However, in the context of the Lyapunov exponent, asynchronous updating schemes appear to be less relevant than synchronous updating schemes [22].
In the remainder, we will use n, rather than the conventional K in the Boolean network literature, to denote the number of variables of the Boolean functions. Our results concerning almost all monotone Boolean networks can be understood in a probabilistic manner, meaning that the asymptotic formulas are valid with probability almost 1 if a monotone Boolean network is chosen at random from the set of all such networks.
As the formulas are asymptotic, they should not be interpreted for small n. At the same time, they are quite accurate even for n = 7, which follows from the results originally published in [15]. Absent additional constraints on the Boolean functions, such as the classes of canalizing functions [11,25,19,30] or Post classes [31], random monotone Boolean networks quickly enter the disordered regime relative to n, but slower than RBNs [8]. Our main results here are asymptotic formulas, depending on whether n is even or odd, for the so-called expected average sensitivity of a monotone Boolean function, first given in [28]. Given the results in [30], the logarithm of the sensitivities,ŝ f , in Theorems 4 and 5, can be directly interpreted as the Lyapunov exponent.

Definitions and Preliminaries
Let f : {0, 1} n → {0, 1} be a Boolean function of n variables x 1 , . . . , x n . Let be the partial derivative of f with respect to x j , where ⊕ is addition modulo 2 (exclusive OR) andx (j,k) = (x 1 , . . . , x j−1 , k, x j+1 , . . . x n ), k = 0, 1. Clearly, the partial derivative is a Boolean function itself that specifies whether a change in the jth input causes a change in the original function f . Now, the activity of variable x j in function f can be defined as Note that although the vectorx consists of n components (variables), the jth variable is fictitious in 1) . For a n-variable Boolean function f , we can form its activity vector α f = [α f 1 , . . . , α f n ]. It is easy to see that 0 ≤ α f j ≤ 1, for any j = 1, . . . , n. In fact, we can consider α f j to be a probability that toggling the jth input bit changes the function value, when the input vectors x are distributed uniformly over {0, 1} n . Since we're in the binary setting, the activity is also the expectation of the partial derivative with respect to the uniform distribution: Under an arbitrary distribution, α f j is referred to as the influence of variable x j on the function f [12]. The influence of variables was used in the context of genetic regulatory network modeling in [29]. Another important quantity is the sensitivity of a Boolean function f , which measures how sensitive the output of the function is to changes in the inputs.
The sensitivity s f (x) of f on vectorx is defined as the number of Hamming neighbors ofx on which the function value is different than onx (two vectors are Hamming neighbors if they differ in only one component). That is, where e i is the unit vector with 1 in the ith position and 0s everywhere else and χ [A] is an indicator function that is equal to 1 if and only if A is true. The average sensitivity s f is defined by taking the expectation of s f (x) with respect to the distribution ofx. It is easy to see that under the uniform distribution, the average sensitivity is equal to the sum of the activities: Therefore, s f is a number between 0 and n. Letα = (α 1 , · · · , α n ) andβ = (β 1 , · · · , β n ) be two different n-element binary vectors. We say thatα precedesβ, denoted asα ≺β, if α i ≤ β i for every i, 1 ≤ i ≤ n. Ifα ⊀β andβ ⊀α, thenα andβ are said to be incomparable. Relative to the predicate ≺, the set of all binary vectors of a given length is a partially ordered set. A Boolean function f (x 1 , · · · , x n ) is called monotone if for any two vectorsα andβ such thatα ≺β, we have f (α) ≤ f (β).
We denote by M (n) the set of all monotone Boolean functions of n variables. Let E n denote the Boolean n-cube, that is, a graph with 2 n vertices each of which is labeled by an n-element binary vector. Two verticesα = (α 1 , · · · , α n ) andβ = (β 1 , · · · , β n ) are connected by an edge if and only if the Hamming The set of those vectors from E n in which there are exactly k units, 0 ≤ k ≤ n, is called the kth layer of E n and is denoted by E n,k .
A vectorα ∈ E n is called a minimal one or minimal unit of monotone Boolean function f (x 1 , . . . , x n ) if f (α) = 1 and f (β) = 0 for anyβ ≺α. A vector α ∈ E n is called an maximal zero of monotone Boolean function f (x 1 , . . . , x n ) if f (α) = 0 and f (β) = 1 for anyβ ≻α. The minimal ones correspond directly to the terms in the minimal disjunctive normal form (DNF) representation of the monotone Boolean function. In [18], asymptotic formulae for the number of monotone Boolean functions of n variables with a most probable number of minimal ones were derived, confirming the conjecture in [27] that the number of monotone Boolean functions relative to the number of minimal ones asymptotically follows a normal distribution, with the assumption of all monotone Boolean functions being equiprobable.
The average sensitivity has been studied intensively by a number of authors [10,5,4,6,9,30,26,23,33,7]. For example, it was shown by Friedgut [9] that if the average sensitivity of f is k then f can be approximated by a function depending on only c k variables where c is a constant depending only on the accuracy of the approximation but not on n. Shmulevich and Kauffman [30] have shown that the average sensitivity determines the critical phase transition curve in random Boolean networks. Shi [26] showed that the average sensitivity can serve as a lower bound of quantum query complexity. Average sensitivity was used to characterize the noise sensitivity of monotone Boolean functions by Mossel and O'Donnell [23]. Zhang [33] gives lower and upper bounds of the average sensitivity of a monotone Boolean function. The upper bound is asymptotic to √ n, which has been shown by Bshouty and Tamon [7]. Our main results are given in Theorems 4 and 5.

The structure of special monotone Boolean functions
We now briefly review some known results concerning the structure of so-called special monotone Boolean functions. Let M 0 (n) denote the set of functions in M (n) possessing the following properties. If n is even, then M 0 (n) contains only functions f ∈ M (n) such that all minimal ones of f are situated in E n,n/2−1 , E n,n/2 , and E n,n/2+1 while function f is equal to 1 on all vectors in E n,n/2+2 , · · · , E n,n . For odd n, M 0 (n) contains only functions f ∈ M (n) such that all minimal ones of f are situated in either E n,(n−3)/2 , E n,(n−1)/2 , and E n,(n+1)/2 or E n,(n−1)/2 , E n,(n+1)/2 , and E n,(n+3)/2 . In the first case, f (α) = 1 for allα in E n,(n+3)/2 , · · · , E n,n while in the second case, f (α) = 1 for allα in E n,(n+5)/2 , · · · , E n,n . Then, as shown in [15], which we denote by |M 0 (n)| ∼ |M (n)| . In [17], asymptotic formulae for the number of special functions from M 0 (n) were established and subsequently used to characterize statistical properties of a popular class of nonlinear digital filters called stack filters [32]. The set of these special functions is denoted by M 1 0 (n) and, depending on whether n is even or odd, is defined differently. While we shall omit the rather lengthy definitions of special functions, the result from [17] that will be important to us is that M 1 0 (n) ∼ |M (n)|. In other words, almost all monotone Boolean functions are special. We shall also need the following results.
Let us start with the case of even n. Let z 0 = 1 2 n n/2 .
Let M 1 0 (n, r, z, v) denote the set of functions f ∈ M 1 0 (n) such that f has r minimal ones in E n,n/2−1 , v maximal zeros in E n,n/2+1 , and f is equal to 1 on z vertices in E n,n/2 . In [17], the following result was proved.

Main Results
Since |M 0 (n)| ∼ |M (n)| , we can focus our attention on functions in M 0 (n) and derive the average sensitivity of a typical function from M 0 (n) . By 'typical' we mean the most probable Boolean function relative to the parameters k, t, and u in Theorems 1-3. It can easily be seen that the most probable special Boolean functions will have k = t = u = 0. This will imply, to take the n-even case as an example, that the most probable function f has r 0 minimal ones in E n,n/2−1 , v 0 maximal zeros in E n,n/2+1 , and f is equal to 1 on z 0 vertices in E n,n/2 , where r 0 , v 0 , and z 0 are given in (6). Our proofs are thus based on the derivation of the average sensitivity of such a function. Whenever we make probabilistic assertions using words such as 'most probable' or 'typical' or talk about expectations, we are implicitly endowing the set M 1 0 (n, r, z, v) with a uniform probability distribution for fixed parameters n, r, z, v. This should not be confused with the Gaussian-like distribution of M 1 0 (n, r, z, v) relative to its parameters n, r, z, v. We will also omit the floor notation ⌊·⌋ as the results are asymptotic.
Theorem 4 Let n be even and let f ∈ M 1 0 (n) be a typical monotone Boolean function. Then, the expected average sensitivityŝ f = E s f of f iŝ Proof. We will proceed by first focusing on determining the activity of an arbitrary variable x j of a typical function f. By simple symmetry arguments, if we were to sample randomly from the set M (n) of monotone Boolean functions, the expected activities would be equal for all the variables. It will follow by (4) that the expected average sensitivity will be equal to n multiplied by the expected activity. Since the function f is such that its minimal ones are situated in E n,n/2−1 , E n,n/2 , and E n,n/2+1 while it is equal to 1 on all vectors in E n,n/2+2 , · · · , E n,n , the only non-trivial behavior occurs between the layers E n,n/2−2 and E n,n/2+2 . Let us consider the minimal ones, and hence all of the ones, on E n,n/2−1 . Since we are considering variable x j , half of these minimal ones will have x j = 0 (i.e.x (j,0) ) and the other half will have x j = 1 (i.e.x (j,1) ). It is easy to see that ifx (j,0) ∈ E n,n/2−1 is a minimal one, then by monotonicity, f x (j,1) = 1. Consequently, the Hamming neighborsx (j,0) andx (j,1) contribute nothing to the sum in (2). On the other hand, ifx (j,1) ∈ E n,n/2−1 is a minimal one, then ∂f (x) /∂x j = 1, since f (x) = 0 for allx ∈ E n,n/2−2 . The (most probable 1 ) number of such minimal ones on E n,n/2−1 contributing to the sum in (2) is thus equal to 1 2 n n/2 − 1 2 −n/2−1 .
The number of zeros on E n,n/2−1 is equal to As above, half of these will have x j = 0 and half will have x j = 1. We need not consider vectorsx (j,1) ∈ E n,n/2−1 , since f (x) = 0 for allx ∈ E n,n/2−2 . However, we should consider the number of ones situated on the middle layer E n,n/2 . The middle layer contains ones and an equal number of zeros. Thus, half of the vectorsx (j,1) ∈ E n,n/2 will be ones and the other half will be zeros. In total, the number of vectors x ∈ E n,n/2−1 such that f (x) = 0, x j = 0, and f x (j,1) = 1, is equal to We have now examined all partial derivatives above and below the layer E n,n/2−1 .
Let us now jump to layer E n,n/2+1 , as it will be similar by duality considerations. The number of maximal zeros on that layer is equal to v 0 = r 0 . Half of these will have x j = 1 and thus ∂f (x) /∂x j = 0 due to monotonicity. The other half will have x j = 0 and since f (x) = 1 for allx ∈ E n,n/2+2 , the same total as in (14) will result. Similarly, the number of ones on E n,n/2+1 is the same as in (15). We are only concerned withx (j,1) ∈ E n,n/2+1 , such that f x (j,1) = 1 and f x (j,0) = 0. As above, because the middle layer contains the same number of ones and zeros, the total number of such pairs of vectors is the same as in (17). Thus, having accounted for all partial derivatives above and below E n,n/2+1 and having convinced ourselves that their contribution to the overall activity of variable x j is the same as in (14) and (17), we can multiply (14) and (17) Since there are 2 −n+1 Hamming neighborsx (j,0) ≺x (j,1) and since the average sensitivity is n times the activity, we must multiply (19) by n2 −n+1 , resulting in the statement of the theorem. The case of odd n is somewhat more involved because there is no "middle" layer E n,n/2 . Instead, typical functions break up into two sets: M 1 0,1 (n, r, z, v) and M 1 0,2 (n, r, z, v) . In the first case, all minimal ones are on layers E n,(n−3)/2 , E n,(n−1)/2 , and E n,(n+1)/2 while in the second case, all minimal ones are situated on E n,(n−1)/2 , E n,(n+1)/2 , and E n,(n+3)/2 . Under random sampling, these two cases will occur with equal probabilities. As we shall see, the results will be different for each case. Thus, the expected average sensitivity will be the average of the the expected average sensitivities corresponding to these two cases.
Theorem 5 Let n be odd and let f ∈ M 1 0 (n) be a typical monotone Boolean function. Then, the expected average sensitivityŝ f = E s f of f iŝ whereŝ f 1 andŝ f 2 are given in (28) and (37), respectively.
Proof. Let us first address the set M 1 0,1 (n, r, z, v) . As in Theorem 4, we only need to concern ourselves with four cases. These are: 1.