On rando Boolean threshold networs Reinhard Hecel, Steffen Schober and Martin Bossert Institute of Telecounications and Applied Inforation Theory Ul University Albert-Einstein-Allee 43, 89081Ul, Gerany Eail: {reinhard.hecel, steffen.schober, artin.bossert}@uni-ul.de Abstract Ensebles of Boolean networs using linear rando threshold functions with eory are considered. Such ensebles have been studied previously by Szeja et al. [1]. They obtained analytical results for the order paraeter which can be used to predict the expected behavior of a networ randoly drawn fro the enseble. Using nuerical siulations of randoly drawn networs, Szeja et al. [1] found ared deviations fro the predicted behavior. In this wor iproved analytical results are provided that better atch up the nuerical results. Furtherore, the critical point in their analysis is identified. In the odel studied, each node is not only dependent on the K regular inputs, but also on the previous state of the node. The results show that this feedbac loop accounts for the low order paraeter and tolerance on rando errors, even for networs with high in-degree. I. INTRODUCTION A (synchronous Boolean networ (BN can be viewed as a collection of N nodes V = {v 1, v 2,..., v N } with eory. The state of a node v denoted by s v (t {0, 1} for t N and is deterined by s v (t = f v ( sv1 (t 1,..., s vv (t 1. (1 where {v 1,..., v v }, 1 v N are the controlling nodes and f v : {0, 1} v {0, 1} is a Boolean function. 1 Boolean networs serve as an abstract odel of interacting agents. For exaple, BNs have been used to odel (sall scale genetic regulatory networs, see [3] [5]. In the late 1960 s Stuart Kauffan proposed to study Boolean networs chosen at rando fro well defined ensebles of networs to understand large scale genetic regulatory networs [6], [7]. He was interested in finding general properties that possibly underlie all genetic networs and that could explain features of living organiss. Such rando Boolean networs (RBNs are constructed as follows: First for each node a function is chosen according to a well defined probability distribution fro a predefined set of Boolean functions. Second for each node, the controlling nodes are chosen fro V according soe probability distribution. Finally, a rando initial state is chosen and the networ is evolved according Equation (1. Kauffan discovered that depending on the networs topology and the choice of functions, a RBN operate in different 1 It should be noted that the state of all nodes is coputed in parallel in each tie step, hence the nae synchronous Boolean networ. In fact other updating schees are used, for exaple the updating schees can be paraeters of the networ by using different response ties of nodes on regulative actions [2]. dynaic regies. In the so-called ordered regie, ost of its coponents are frozen, i.e. eep their state when being updated. Further, single transient errors that change the state of a randoly chosen node fro 0 to 1 or vice versa tend to vanish. Contrary, in the disordered regie, only a few frozen nodes exist and single transient errors propagate to any other nodes. In this wor rando networs using linear threshold functions are considered. Let 0 < K N and h R be fixed paraeters. To each node v, a function is assigned, defined by 1, if r > h f v (x 1,..., x K, x K+1 = x K+1 if r = h (2 0, if r < h where r := r(x 1,..., x K = w i x i. (3 The weights w i are chosen uniforly at rando fro { 1, 1}, which defines a probability distribution on the set of possible functions. Then for 1 i K the controlling nodes v i are chosen uniforly at rando fro V. The state s v (t is obtained by setting x i = s vi (t 1 for 1 i K and x K+1 = s v (t 1. In other words, K inputs are chosen randoly, whereas the input K + 1 is always connected to the function s output. Hence the state of any node v at tie t depends on its own state at tie t 1. These networs were studied by Szeja et al. [1], otivated by the fact that such networs have successfully been used to study sall scale gene regulatory networs, see for exaple [3], [5] for odeling the cell cycle in yeast. Using the so-called annealed approxiation Szeja et al. [1] derived expressions in dependence of the paraeters h and K, for the tie evolution of the proportion of nodes being 1, and for the expected sensitivity of a rando Boolean function. The expected sensitivity can be used as an order paraeter for the enseble to predict its dynaical regie. Szeja et al. [1] found deviations between their analytical results for integer valued thresholds and nuerical siulations, which they attributed to the annealed approxiation. In this wor iproved analytical results are given, that better atch up the nuerical results. Further the critical point in their analysis is identified. The outline is as following: In Section II, a brief introduction in rando Boolean networs and their order paraeter i=1
is given. In the following sections our ain results are derived. In Section III-A the so-called bias ap is derived for rando threshold networs and soe results that will be needed for further calculations are given. In Section III-B the order paraeter is derived, by first calculating the average sensitivity of threshold functions, where the distinctions is ade between integer and non integer thresholds. In section III-C the average sensitivity of functions whose input x K+1 is randoly chosen is considered, instead of being set to the output. In Section III-D, the previously derived results are used to obtain the phase diagra. The phase diagra visualizes for which cobination of h and K, rando threshold networs operate in the chaotic regie. Furtherore rando threshold function with h = 0 are discussed in ore detail. II. RANDOM BOOLEAN NETWORKS AND THEIR ORDER PARAMETERS An iportant question for rando Boolean networs is their expected dynaical behavior and the expected robustness against single transient errors. Let us concentrate on the last point. If a rando chosen node is disturbed, i.e. its state is changed fro one to zero or vice versa, one is interested in the evolution of this disturbance. Will it tend to spread through the whole networ, possibly affecting all nodes? Or will the disturbance die out, indicating the so-called ordered phase? In soe case, depending on the paraeters of the rando construction process, there exists a single order paraeter that can be used to answer this question. Soe well nown exaple are the so-called N K-networs studied by Kauffan [6]. These functions are chosen uniforly at rando fro the set of all Boolean functions with K arguents. The controlling ( nodes are chosen uniforly at rando fro all N K possibilities. This enseble can be described by the single paraeter K. It is well nown that ordered behavior is only found if and only if K 2. Often the order paraeter is obtained by the so-called annealed approxiation [8]. It provides the probabilistic fraewor to deterine the dynaical regie of a rando networ. The annealed approxiation is a ean-field theory that neglects correlations between nodes. It is assued that at each tie step the functions and the controlling nodes are drawn at rando again. If N is large, this procedure allows for quite accurate predictions [8]. Also for ensebles lie those studied here the predictions for the annealed odel coincides the non-annealed odel if N [9]. Suppose the networ operates in a stationary state. Under the assuptions of the annealed analysis we ay assue that a node chosen uniforly at rando has probability b of being in state 1. Let p(f denote the probability of choosing the function f. Then the order paraeter is defined as λ = p(f s(f, xb x (1 b K x (4 f x Ω K where denotes the Haing weight and s(f, x denotes the sensitivity of f at x defined by s(f, x = #{y Ω K x y = 1 and f(x f(y} where denotes the coponent wise addition od 2. It is well now that if N and λ 1 (5 any single perturbation introduced at a randoly chosen node will vanish with probability one [10]. The paraeter b can be obtained as follows (cnf. [11]. Suppose that at t the probability of a rando chosen node to be 1 is equal to b t. Then the so-called bias ap is defined by b t+1 = p(f f(xb x t (1 b t K x. (6 f x Ω K Let b 0 = 0.5. The fix point of the bias ap (if it exists, denoted by b, is the expected nuber of nodes that are one in the stationary state. Soe coents on (5 and (4 are necessary. For convience let us introduce the average sensitivity with respect to b as(f = s(f, xb x (1 b K x. x Ω K By x i we denote a vector that is obtained fro x by flipping its ith position. Then P e := Pr [f(x f(x i] = as(f K where i is chosen uniforly at rando fro {1,..., K}. Hence (4 ay be written as λ = E f [as(f] = K E f [P e ] where the expectation is taen with respect to the whole function enseble. Now assue a rando perturbation at the input of soe rando function. We expect the state of the node to be changed with E[P e ] = λ/k which in turn will spread to K other nodes 2. Hence after t tie steps λ t nodes are affected by the perturbation on average. Therefore if λ 1 the perturbation will vanish with high probability. A rigorous treatent of this topic is given in [10]. A. The bias ap III. RANDOM THRESHOLD NETWORKS In this section, we ai at deriving an expression for the bias ap in order to find fixed points b. Denote the nuber of positive weights of a function f by = (f = #{i i 1 K and w i = 1}, where #{ } denotes the cardinality of the set. For convinience we write f for a function with = (f. The corresonding function r(x 1,..., x K is denoted by r. Proposition 1: Consider a rando threshold networ where all functions f = f(x 1,..., x K, x K+1 depend on K + 1 variables. The threshold h is a fixed constant and P (w i = 1 = P (w i = 1 = 1/2 for all i independently. Any 2 As the in-degree of any node is K, the average out-degree of any node is also K.
function receives the previous state of the corresponding node as arguent y K+1. Then b = 1 ( K 2 K b, (7 with =0 b = P (r (b > h 1 P (r (b = h. (8 Above [ ] P (r (b = h = Pr w i x i = h weights are +1, i where P (x i = 1 = b independently for all i. Also P (r (b > h = P (r (b = h. h+1 For the proof, the following lea is needed. Lea 1: Let f 1 and f 2 be threshold functions with r 1 and r 2 respectively and (f 1 = (f 2 =. Then P (r 1 = u = P (r 2 = u. Proof: Lets consider the probability distribution for r, w, P (r = u w. As each x i follows the sae distribution, r = x i x i, i {j w j=1} i {j w j= 1} and the lea follows. Proof: First Equation 8 is proved. Let x t i be the state of an input variable at tie t. f(x t 1,..., x t K, xt K+1 = 1 if r(xt > 0. Also f(x t 1,..., x t K, xt K+1 = 1 if r(xt = 0 and x t K+1 = 1. Then, fro Lea 1 and Equation 1 follows b t+1 = P (r (b t > h + b t 1 P (r (b t = h. (9 If a fixed point b is reached, also a fixed point for each b is reached. Then b t+1 = b t 1 = b and Equation 8 follows fro 9. Fro Lea 1 and the fact that w i is unifor distributed, Equation 7 follows. Lea 2: Clearly P (r > h = = h 1 h l=0 ( b (1 b ( K l b l (1 b K l. Proof: For y { 1, 1} define the rando variable w (y = #{i x i = 1, w i = y}. P (w (1 = = ( b (1 b, (10 and ( K P (w ( 1 = = b (1 b K. (11 P (r > h = P (r > h as r is integer valued. r > h holds true, if at least h + 1 ore of the positive weighted inputs are 1, than there are negative weighted inputs that are 1. Considering all possible constellations gives: P (r > h = P (w (1 = P (w ( 1 +h 1. =ax( h,0 Note that, if 1 h 0, the second su in equation 2 is 0. Lea 3: Let the threshold h be integer valued, then P (r = h = =0 ( K h ( b 2 h (1 b 2+K+h and let h be non integer valued, h R\N then P (r = h = 0. Proof: The proof follows the notation of Lea 2. For integer valued h N, P (r = h can be calculated as following. When denotes the nuber of positive inputs that are on, r = h holds true, if there are also h of the negative inputs on. Considering all possible constellations gives: P (r = h = P (w (1 = P (w ( 1 = h. =0 The result follows fro (10 and (11. If h isn t integer valued, then P (r = h = 0 because r is integer valued and h not. 1 Results for b: In order to find fixed points b, Equation 7 can be solved nuerically. As entioned in the introduction, in the context of the annealed approxiation, fixed points are found and interpreted (stable or instable with the bias ap. There, a fixed point is stable if the gradient is saller then 0. For threshold functions the apping b t+1 = 1 2 K =0 ( K P (r (b t > h+b t 1 P (r (b t = h, is found, with the difference that b t+1 is not only dependent on b t but also on b t 1. Fixed points can be found nuerically, by starting at b 0 = 0.5 and b 0 = 0.5, {1, K}. An exaple iteration is shown in Figure 1. The fixed points found by solving nuerically Preposition 1 are discussed now. The fixed point for h = 0, b = 0.5 is independent of K. For each each negative threshold ( K h < 0, exactly one stable fixed point between 0.5 and 1 could be found. For h = 1 and K 10 the b = 0. For K > 11, fixed points unequal to zero are found. For larger positive thresholds h, K has to be very large for the existence of fixed points unequal to 0. To validate the results, rando threshold networs of the size N = 10000 with K inputs per function have been generated and initialized with an bias of b 0 = 0.5. The fixed points for h 0 agree with less than 0.5% deviation with the predicted ones. For a threshold of h = 1 the siulation doesn t
0.8 1 b 3 b 2 any arguent will not affect the output. This is also true for r < h with the sae type of arguents. Hence consider r = i=1 x i w i = h. For (a, b (0, 1 ( 1, 1 define the sets b t+1 ; b t+1 0.6 0.4 0.2 0 b 1 b 0 b A(a, b = {x i (x i, w i = (a, b} Assue arguent x i A(0, 1 is flipped fro 0 to 1. The function will change its output if and only if it was 0 before the flip. But the later event has probability 1 b. Consider all other cases and assue that an x i is chosen uniforly at rando yields 0 0.2 0.4 0.6 0.8 1 b t ; b t K E [P e ] = Pr [r = h] { A(0, 1 (1 b + A(0, 1 b + A(1, 1 b + A(1, 1 (1 b }. Fig. 1. The cobweb diagra for a threshold function with K = 3 and h = 0. It can be seen how each b evolve to its fixed point, when starting with b 0 = 0.5, {1, K} and b 0 = 0.5. agree with the predicted results, e.g. for K = 10 a fixed point of 0 is expected, whereas in the siulation, a average b of 0.23 is found. B. Sensitivity and order paraeter Let us consider a threshold function f attached to soe node v at tie t. Reeber that f = f(x 1,..., x K, x K+1 where x K+1 = s v (t 1, i.e. it is set to previous state of the attached node. To derive λ it is assued that the networ operates in its stationary state. Hence for all 1 i K the arguents x i are independently chosen at rando with Pr [x i = 1] = b. Furtherore it is assued that x K+1 is chosen randoly with Pr [x K+1 = 1] = E[f] =: b f. As b f only depends on (f, b is used 3. Fro Lea 1 and P (w i = 1 = 1/2 it follows that λ = KE f [P e ] = 1 2 K =0 ( K K E [P e]. f:w(f= For convenience E[P e ] is written instead of E [P e]. f:w(f= Here P e ust be coputed under the assuption entioned above, i.e. the first K arguents are binoial distributed with paraeter b whereas x K+1 is one with probability b and otherwise zero. It will be shown that if h is an integer then K E [P e ] = p (h {b (2h 2 + K h + }, (12 and if h is not integer valued K E [P e ] = p ( h (K +h+p ( h ( h. (13 In both cases p ( = P (r = as given by Lea 3. Proof of Equation (12: First let r > h which iplies that f = 1. If r would be lowered by one the function will still output its previous state naely 1. Hence a flip of 3 Follows fro Lea 1. Now by definition A(0, 1 = A(1, 1 and A(1, 1 = A[1, 1] h. Also the su of the cardinalities of all sets is equal to K. Substituting the constraints into the equation copletes the proof. Proof of Equation (13: A threshold function with a non integer threshold can only change its output by changing one input when r is directly above r = h or directly below r = h the threshold. Suppose r = h, and j of the positive inputs are 1. Then also j h of the negative inputs ust be 1. The function s output can change, by lowering r which can be done by changing one of the j positive inputs that have value 1 to 0 and one of the K (j h negative inputs fro 0 to 1. Suppose r = h, and j is again the nuber of positive inputs that have value 1. To change the output of a function, r has to be increased by one. This can be done by changing one of the j positive inputs which have value 0 to 1 or by changing one of the j h negative inputs that have the value 1 to 0. That gives the lea. Corollary 1: The average sensitivity is equal for all threshold values in between two consecutive integers, i < h < i + 1 where i is an integer. Further, the function s output is unabiguously defined by the K regular inputs of the function, and independent fro the input x K+1 and therfore also fro the previous output. This result is derived differently, but in accordance with [1]. C. A changed decision rule for r = h The dependency on the previous state is an iportant property of the threshold functions discussed here, and it is closely related to the functions low average sensitivity. To deonstrate what ipact this decision rule has on the average sensitivity and order paraeter, another decision rule will be discussed: the input x K+1 is 1 with probability b, P (x K+1 = 1 = b, instead of being set to the output. Deriving an equation for fixed points b is analogous as in section?? and sipped because of liited space. Lea 4: Let f be a threshold function, where
P (x K+1 = 1 = b. If h is an integer then K E [P e ] =P (r = h 1 ( h + 1 b+ P (r = h ( + h 2b 2bh + bk+ P (r = h + 1 (K + h + 1 (b 1, (14 and if h is non integer valued, 13 holds. Proof: For the constellations r = h 1, r = h and r = h + 1, the output can change by changing one input. If r = h 1 the output changes if r is increased by one with probability b. Suppose j = #{i x i = 1, w i = 1}, then also j (h 1 = #{x i = 1, w i = 1} holds. r can be increased by 1 if one of the j positive inputs that are 0 changes, or if one of the j (h 1 negative inputs that are 1 changes. The output changes with probability b because this is the probability for the output to be 1 if r = h. Now the case r = h is considered. Then P (f = 1 = b by definition. Suppose j = #{i x i = 1, w i = 1} then j h = #{x i = 1, w i = 1}. If one of the j positive inputs that are 1, or one of the K (j +h negative that are 0 is flipped, then the output is f t+1 = 1, if f t = 0. Changing one of the j positive inputs that are 0, or one of the i + h negative inputs that are 1, changes the output, f t+1, if f t = 1. If r = h + 1 the output changes if r is decreased by one with probability 1 b. Changing one of the j positive inputs that are 1, or one of the K (j h negative inputs that are 0 will decrease h. Putting all together and shortening j for each constellation gives Equation 14. For non-integer thresholds, r = h is not possible, and therefore the input x K+1 doesn t have an influence, which gives the rest of the Lea. D. Nuerical results In this section, the analytical results fro the previous sections are evaluated. After choosing paraeters for an enseble, deterining the order paraeter is a two step process: First a stable fixed point is calculated using the results fro Section III-A. Then the order paraeter can be obtained with the results fro Section III-B. 1 Order paraeter for h = 0: In Table I are soe results of the order paraeter λ: λ org refers to the functions whose input x K+1 is set to the output, and λ rdd refers to the functions whose input x K+1 is 1 with probability b. Further, λ approx org and are obtained by siulation and will be explained later. λ approx rdd K λ org λ approx org λ rdd λ approx rdd 3 0.3375 0.3515 0.9375 0.9371 4 0.4270 0.4374 1.0937 1.0937 15 0.9789 0.9875 2.1670 2.1670 16 1.0155 1.0240 2.2391 2.2391 TABLE I ANALYTICAL AND ESTIMATED RESULTS FOR THE ORDER PARAMETER λ OF BOOLEAN THRESHOLD FUNCTIONS. Rando Boolean networs are expected to operate in the chaotic regie for λ > 2. In contrast (see Table I, rando threshold networs with K regular inputs per function and the arguent x K+1 set to the function s output are expected to operate in the chaotic regie, for K 16. But if a rando decision is ade for the case r = 0, instead of eeping the previous state, the networs are expected to operate in the chaotic regie already for K 4. Therefore if x K+1 is set to the function s previous output aes a huge difference for the stability and spread or rando transient errors, for a syste odeled as a threshold networ. The dependency on the previous output of the function results in a low average sensitivity and has a highly stabilizing effect on the threshold networs. Szeja et. al. [1] where also deriving an expression for λ, however, but they neglected that each function has its own expectancy value. Substituting b with b in equation 12 leads to the results of Szeja et. al. for λ. Consequently they concluded that for h = 0 the networs are in the chaotic regie for K > 12, obviously this results differs significantly fro the one derived here. By using siulations to validate their results, Szeja et. al. found that in all siulations of networs with connectivities up to K = 16, we find only fixed point attractors, which eans that these networs are in the frozen phase. Therefore their siulations are in accordance with the result given here. For a RBN the average sensitivity can be obtained by siulation using the Derrida plot as it has been done e.g. in [12]. This is not feasible here, due to the dependency of the functions on their previous state. Therefore, to further validate the results, Algorith 1 has been used. Algorith 1 estiates the average sensitivity for a given function, by estiating K Pr [f(x f(x i] and therefore also wors if a function has eory, e.g. when the function is dependent on its previous state. To obtain λ approx org and λ approx rdd, for each Algorith 1: Average Sensitivity estiation for a Boolean function with eory Data: Boolean Threshold function f of order K, t trials Result: estiated average sensitivity as(f begin as(f 0 for i 1 to t do generate a rando x Ω K according to the probability distribution of x generate a rando e i Ω K with wt(e i = 1 f i f(x ˆf i f(x e i if f i ˆf i then as(f as(f + 1 as(f K end as(f t function f, Algorith 1 has been used to estiate K E [P e ].
Then K E [P e ] has been averaged over all functions. It can be seen that the analytical results atch very good with the estiated ones. 2 The Phase diagra: In this part it is visualized in the so-called phase diagra, for which cobination of the paraeters K and h, threshold networs operate in the chaotic regie (λ > 1. To obtain Figure III-D2, for each pair of values K and h first a stable fixed point b was calculated with the results fro section III-A, and then the order paraeter λ was calculated with the results fro section III-B. It can be seen in Figure III-D2 that threshold networs deonstrate how very different behaviour for integer an non integer valued thresholds are. This is because for non-integer thresholds the case r = h isn t possible, therefore the functions are not dependent on their previous output. In contrast to negative valued thresholds, where a fixed point exists for each K, for positive valued thresholds, the fixed point is b = 0 until K becoes large enough, to find a b 0. Then, λ > 1 for the considered values, therefore b deterines there the edge of chaos. Threshold h 2 0 2 4 6 0 10 20 30 40 50 K where the chaotic regie starts Fig. 2. The Phase diagra for rando threshold networs (RTNs, using functions whose input x K+1 is set to its output. The bars indicate values for K for which RTNs operate in the ordered regie. Thin bars refer to the integer valued thresholds, whereas the solid ones refer to non integer valued thresholds. IV. DISCUSSION & CONCLUSION Rando threshold networs (RTNs were considered. In the odel studied here each node is not only controlled by K other nodes, but is also dependent on the previous state of the node, i.e. possess a local feedbac loop. The assuptions of the annealed approxiation [8] were used in order to deterine the dynaical regie of the RTNs. First the fixed points of the biased ap were obtained and solved nuerically. In order to obtain the order paraeter λ, an expression for the average sensitivity was derived. Evaluating the analytical results, the focus was first on a threshold of h = 0. Due to the low average sensitivity of the functions a RTN with h = 0 is expected to operate in the ordered regie for K < 16. This is a very high degree copared to the so-called NKnetwors, that are expected to operate in the chaotic regie already for K > 2. The stability even for networs with high in-degree can be traced bac to the local feedbac. These results are in agreeent with siulations done here and in [1]. To deonstrate the effect of the local feedbac loop on the networ stability, it was shown that if the local feedbac is oitted the networs enter the chaotic regie already for K 4. These results iply that, if a dynaical syste is odeled with a threshold networ, assuptions about the feedbac are crucial. 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