Phase transition in a class of non-linear random networks

arXiv:1003.0871v3 [nlin.AO] 30 Jul 2010 Phase transition in a class of non-linear random networks M. Andrecut1 and S. A. Kauffman2 November 5, 2018 1Institute for Space Imaging Science 2Institute for Biocomplexity and Informatics University of Calgary, Alberta, T2N 1N4, Canada Abstract We discuss the complex dynamics of a non-linear random networks model, as a function of the connectivity k between the elements of the network. We show that this class of networks exhibit an order-chaos phase transition for a critical connectivity kc = 2. Also, we show that both, pairwise correlation and complexity measures are maximized in dy- namically critical networks. These results are in good agreement with the previously reported studies on random Boolean networks and random threshold networks, and show once again that critical networks provide an optimal coordination of diverse behavior. 1 Introduction Random Boolean networks (RBNs) are a class of complex systems, that show a well-studied transition between ordered and disordered phases. The RBN model was initially introduced as an idealization of genetic regulatory networks. Since then, the RBN model has attracted much interest in a wide variety of fields, ranging from cell differentiation and evolution to social and physical spin sys- tems (for a review of the RBN model see [1] and [2], and the references within). The dynamics of RBNs can be classified as ordered, disordered, or critical, as a function of the average connectivity k, between the elements of the network, and the bias p in the choice of Boolean functions. For equiprobable Boolean functions, p = 1/2, the critical connectivity is kc = 2. The RBNs operating in the ordered regime (k < kc) exhibit simple dynamics, and are intrinsically robust under structural and transient perturbations. In contrast, the RBNs in the disordered regime (k > kc) are extremely sensitive to small perturbations, which rapidly propagate throughout the entire system. Recently, it has been shown that the pairwise mutual information exhibits a jump discontinuity at the critical value kc of the RBN model [3]. More recently, similar results have 1 been reported for a related class of discrete dynamical networks, called random threshold networks (RTNs) [4]. In this paper we consider a non-linear random networks (NLRNs) model, which represents a departure from the discrete valued state representation, cor- responding to the RBN and RTN models, to a continuous valued state represen- tation. We discuss the complex dynamics of the NLRN model, as a function of the average connectivity (in-degree) k. We show that the NLRN model exhibits an order-chaos phase transition, for the same critical connectivity value kc = 2, as the RBN and RTN models. Also, we show that both, pairwise correlation and complexity measures are maximized in dynamically critical networks. These re- sults are in very good agreement with the previously reported studies on the RBN and RTN models, and show once again that critical networks provide an optimal coordination of diverse behavior. 2 NLRN model The NLRN model consists of N randomly interconnected variables, with con- tinuously valued states −1 ≤xn ≤+1, n = 1, ..., N. At time t the state of the network is described by an N dimensional vector x(t) = [x1(t), ..., xN(t)]T , (1) which is updated at time t + 1 using the following map: x(t + 1) = f (w, x(t)) , (2) where f (w, x(t)) = [f1 (w, x(t)) , ..., fN (w, x(t))]T , (3) and fn (w, x(t)) = tanh N X m=1 wnmxm(t) + x0 ! , n = 1, ..., N. (4) Here, w is an N × N interaction matrix, with the following randomly assigned elements: wnm =    −1 with probability k 2N 0 with probability N−k N +1 with probability k 2N , (5) and k is the average in-degree of the network. The interaction weights can be interpreted as excitatory, if wnm = 1, and respectively inhibitory, if wnm = −1. Also, we have wnm = 0, if xm is not an input to xn. Obviously, the threshold x0 can be considered as a constant input, with a fixed weight wn0 = 1, to each variable xn. Therefore, in the following discussion we do not lose generality by assuming that the threshold parameter is always set to x0 = 0. 2 3 Phase transition In order to illustrate the complex dynamics of the NLRN system, we consider the results of the simulation of three networks, each containing N = 128 variables, and having different average in-degrees: k = 1, k = 2 and respectively k = 4. Also, the continuous values of the variables xn(t) are encoded in shades of gray, with black and white corresponding to the extreme values ±1. In Figure 1, one can easily see the three qualitatively different types of behavior: ordered (k = 1), critical (k = 2), and respectively chaotic (k = 4). A quantitative characterization of the transition from the ordered phase to the chaotic phase is given by the Lyapunov exponents [5], which measure the rate of separation of infinitesimally close trajectories of a dynamical system. The linearized dynamics in tangent s

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