Influence and Dynamic Behavior in Random Boolean Networks C. Seshadhri, Yevgeniy Vorobeychik, Jackson R. Mayo, Robert C. Armstrong, and Joseph R. Ruthruff Sandia National Laboratories, P.O. Box 969, Livermore, California 94551-0969, USA We present a rigorous mathematical framework for analyzing dynamics of a broad class of Boolean network models. We use this framework to provide the first formal proof of many of the standard critical transition results in Boolean network analysis, and offer analogous characterizations for novel classes of random Boolean networks. We precisely connect the short-run dynamic behavior of a Boolean network to the average influence of the transfer functions. We show that some of the assumptions traditionally made in the more common mean-field analysis of Boolean networks do not hold in general. For example, we offer some evidence that imbalance, or expected internal inhomogeneity, of transfer functions is a crucial feature that tends to drive quiescent behavior far more strongly than previously observed. Introduction. Complex systems can usually be repre- tween short-run and long-run sensitivity is not a foregone con- sented as a network of interdependent functional units. clusion [10] and remains an open question. Boolean networks were proposed by Kauffman as models of We provide a formal mathematical framework to analyze genetic regulatory networks [1, 2] and have received consider- the behavior of Booleam networks over a logarithmic (in the able attention across several scientific disciplines. They model size of the graph) number of discrete time steps, and give a variety of complex phenomena, particularly in theoretical conditions for exponential divergence in Hamming distance biology and physics [3–8]. in terms of the indegree distribution and influence of transfer A Boolean network N with n nodes can be described by functions in T . a directed graph G = (V;E) and a set of transfer functions. Assumptions. We assume that the Boolean network N is We use V and E to denote the sets of nodes and edges respec- constructed as follows. First, we specify an indegree distribu- tively, and denote the indegree of node i by Ki. Each node tion D with a maximum possible indegree Kmax, and for each Ki i is assigned a Ki-ary Boolean function fi : {−1;+1g ! node i independently draw its indegree Ki ∼ D. We then con- {−1;+1g, termed transfer function. If the state of node i at struct G by choosing each of the Ki neighbors of every node i time t is xi(t), its state at time t + 1 is described by uniformly at random from all n nodes. Next, for each node i we independently choose a Ki-input transfer function accord- x (t + 1) = f (x (t);:::;x (t)): i i i1 iKi ing to T . We assume that the family T has either of the following properties: The state of N at time t is just the vector (x1(t);x2(t);:::;xn(t)). • Full independence: Each entry in the truth table of a Boolean networks are studied by positing a distribution of transfer function is i.i.d., or graph topologies and Boolean functions from which indepen- dent random draws are made. We denote the distribution of • Balanced on average: Transfer functions drawn from transfer functions by T . An early observation was that when T have, on average, an equal number of +1 and −1 the indegree of a network is fixed at K and each transfer func- output entries in the truth table. Formally, Pr f ;x[ f (x) = tion is chosen uniformly randomly from the set of all K-input +1] = 1=2, where Pr f ;x denotes the probability of an possibilities, the network dynamics undergo a critical transi- event when f is drawn from T , and input x for f is tion at K = 2, such that for K < 2 the network behavior is chosen uniformly at random. quiescent and small perturbations die out, while for K > 2 it exhibits chaotic features [2]. This result has been generalized Influence. The notion of influence of variables on Boolean to non-homogeneous distributions of transfer functions, when functions was defined by Kahn et al. [11] and introduced to the output bit is set to 1 with probability p (called bias) in- the study of Boolean networks by Shmulevich and Kauff- dependently for every possible input string [9]. The resulting man [4]. The influence of input i on a Boolean function f , critical boundary is described by the equation 2K p(1− p) = 1. denoted by Infi( f ), is All analysis of Boolean networks to date uses mean-field (i) approximations, an annealed approximation [9], simulation Infi( f ) = Prx[ f (x) 6= f (x )]; studies [1, 7], or combinations of these, to understand the dy- namic behavior. Many previous studies rely solely on short- where x(i) is the same as x in all coordinates except i. Given run characteristics (e.g., Derrida plots that consider only a a distribution T of transfer functions, let Td denote the very short, often only a single-step, horizon [4, 5, 7]) and ex- induced distribution over d-input transfer functions. The trapolate to understand long-term dynamics. Hamming dis- expected total influence under Td, denoted by I(Td), is tance between Boolean network states that diverges exponen- E f ∼Td [∑i Infi( f )]. When Td is clear from the context we tially over time for small perturbations to initial state sug- write this simply as I(d). Suppose that we have an indegree gests sensitivity to initial conditions typically associated with distribution where p(d) is the probability that indegree is d. chaotic dynamical systems. Nonetheless, the connection be- We show that the quantity that characterizes the dynamic be- 2 havior of Boolean networks is In this model, the probability that an edge is f -bichromatic d K is exactly 1=2. Hence, I(d) = (total number of edges)=2 . max d d−1 I = ∑ p(d)I(d): Since the total number of edges (of B ) is d2 , we obtain d=1 I(d) = d=2. Notice that I(d) is linear in this case, and, conse- quently, considering I(K) = K=2 suffices for any distribution Main Result. We present our main result that character- p(d). Applying Theorem 1 then gives us the well-known crit- izes dynamic behavior of Boolean networks under the as- ∗ ical transition at K = 2. sumptions stated above. Define t = logn=(4logKmax). The following theorem tracks the evolution of Hamming distance Transfer functions with a bias p. A simple generalization up to time t∗, starting with a small (single-bit) perturbation. of uniform random transfer functions is to introduce a bias, We note that our theorem applies for any distribution of inde- that is, a probability p that an entry in the truth table is +1 (but still filling in the truth table with i.i.d. entries) [2]. In this grees with a maximum bounded by Kmax, though increasing ∗ case, the probability that an edge is f -bichromatic is 2p(1− p) density (Kmax) shortens the effective horizon t . and therefore I(d) = 2dp(1 − p). Since I(d) is linear, we can Theorem 1 Choose a random Boolean network N having a characterize the critical transition in this case at 2K p(1− p) = random graph G with n nodes and a distribution of trans- 1 for any indegree distribution with mean K. fer functions T . Evolve N in parallel from a uniform ran- Canalizing functions. Kauffman [2] and others have ob- (i) dom starting state x and its flip perturbation x (with a uni- served that since uniform random transfer functions are typ- form random i). The expected Hamming distance between ically chaotic, they are unlikely to represent a distribution ∗ the respective states of N at time t ≤ t lies in the range of transfer functions that accurately models real phenomena, t 1=4 I ± 1=n . such as genetic regulatory networks. Biased transfer functions The proof of this theorem is non-trivial and is provided in only partially resolve this, as they still tend to fall easily into the supplement. It shows that the effects of flip perturbations a chaotic regime for a rather broad range of p [6]. Empirical vanish when I < 1 while perturbations diverge exponentially studies of genetic networks suggest another class of transfer when I > 1. Thus, criticality of the system is equivalent to functions called canalizing. A canalizing function has at least I = 1. one input, i, such that there is some value of that input, vi, that Much of the past work assumed (or explicitly stated) that determines the value of the Boolean function. Shmulevich it suffices to consider the expected influence value I(K) for and Kauffman [4] show heuristically that canalizing functions the mean indegree K. A direct consequence of Theorem 1 is have I(K) = (K + 1)=4 and thus exhibit a critical transition at that I(K) characterizes a critical transition iff I(d) is affine. K = 3. We now show that this is a corollary of our theorem, using Proposition 2 to obtain I(d). To see this, observe that I(K) = I iff I(K) = I (∑d dp(d)) = ∑d p(d)I(d): This is true if and only if I(d) is affine. To compute I(d), fix (without loss of generality) the canal- Applications. In this section we use Theorem 1 to recover izing input index to be 1 and the canalizing input and output most of the characterizations of critical indegree thresholds values to +1. Consider the distribution of functions condi- to date and prove results for new natural classes of transfer tional on these properties.