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Classification of Random Boolean Networks
, 2002
"... We provide the first classification of different types of RandomBoolean Networks (RBNs). We study the differences of RBNs depending on the degree of synchronicity and determinism of their updating scheme. For doing so, we first define three new types of RBNs. We note some similarities and difference ..."
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We provide the first classification of different types of RandomBoolean Networks (RBNs). We study the differences of RBNs depending on the degree of synchronicity and determinism of their updating scheme. For doing so, we first define three new types of RBNs. We note some similarities and differences between different types of RBNs with the aid of a public software laboratory we developed. Particularly, we find that the point attractors are independent of the updating scheme, and that RBNs are more different depending on their determinism or nondeterminism rather than depending on their synchronicity or asynchronicity. We also show a way of mapping nonsynchronous deterministic RBNs into synchronous RBNs. Our results are important for justifying the use of specific types of RBNs for modelling natural phenomena.
Boolean Dynamics with Random Couplings
, 2002
"... This paper reviews a class of generic dissipative dynamical systems called NK models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends ..."
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Cited by 36 (0 self)
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This paper reviews a class of generic dissipative dynamical systems called NK models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends upon K elements in the previous time step. We review the work of many authors on the behavior of the models, looking particularly at the structure and lengths of their cycles, the sizes of their basins of attraction, and the flow of information through the systems. In the limit of infinite N, there is a phase transition between a chaotic and an ordered phase, with a critical phase in between. We argue that the behavior of this system depends significantly on the topology of the network connections. If the elements are placed upon a lattice with dimension d, the system shows correlations related to the standard percolation or directed percolation phase transition on such a lattice. On the other hand, a very different behavior is seen in the Kauffman net in which all spins are equally likely to be coupled to a given spin. In this situation, coupling loops are mostly suppressed, and the behavior of the system is much more like that of a mean field theory. We also describe possible applications of the models to, for example, genetic networks, cell differentiation, evolution, democracy in social systems and neural networks.
Closing probabilities in the Kauffman model: an annealed computation
, 2008
"... We define a probabilistic scheme to compute the distributions of periods, transients and weigths of attraction basins in Kauffman networks. These quantities are obtained in the framework of the annealed approximation, first introduced by Derrida and Pomeau. Numerical results are in good agreement wi ..."
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Cited by 14 (2 self)
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We define a probabilistic scheme to compute the distributions of periods, transients and weigths of attraction basins in Kauffman networks. These quantities are obtained in the framework of the annealed approximation, first introduced by Derrida and Pomeau. Numerical results are in good agreement with the computed values of the exponents of average periods, but show also some interesting features which can
Structure and dynamics of a gene network model incorporating small RNAs
 In Sarker et al. [107
"... this paper is twofold: firstly, to present our GRN model and describe its relevance to the biological systems we aim to simulate; secondly, to demonstrate the ability of this model to increase the range of phenomena that can be simulated by using it to model the role that small RNAs may play in gene ..."
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Cited by 8 (3 self)
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this paper is twofold: firstly, to present our GRN model and describe its relevance to the biological systems we aim to simulate; secondly, to demonstrate the ability of this model to increase the range of phenomena that can be simulated by using it to model the role that small RNAs may play in gene regulation. Initially, some background is provided on the traditional view of gene regulation in biology and the way in which this is reflected in current models. Recent discoveries concerning the role of small RNA molecules that are challenging this view are then described. A new model is proposed that generates networks with multiple levels of regulatory control. Initial results illustrating the structural and dynamic properties of this model are then presented and discussed. Finally the future directions of this model with respect to studies of evolvability are discussed
Kauffman networks: Analysis and applications
 in Proceedings of the IEEE/ACM International Conference on ComputerAided Design
, 2005
"... Abstract — A Kauffman network is an abstract model of gene regulatory networks. Each gene is represented by a vertex. An edge from one vertex to another implies that the former gene regulates the latter. Statistical features of Kauffman networks match the characteristics of living cells. The number ..."
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Abstract — A Kauffman network is an abstract model of gene regulatory networks. Each gene is represented by a vertex. An edge from one vertex to another implies that the former gene regulates the latter. Statistical features of Kauffman networks match the characteristics of living cells. The number of cycles in the network’s state space, called attractors, corresponds to the number of different cell types. The attractor’s length corresponds to the cell cycle time. The sensitivity of attractors to different kinds of disturbances, modeled by changing a network connection, the state of a vertex, or the associated function, reflects the stability of the cell to damage, mutations and virus attacks. In order to evaluate attractors, their number and lengths have to be computed. This problem is the major open problem related to Kauffman networks. Available algorithms can only handle networks with less than a hundred vertices. The number of genes in a cell is often larger. In this paper, we present a set of efficient algorithms for computing attractors in large Kauffman networks. The resulting software package is hoped to be of assistance in understanding the principles of gene interactions and discovering a computing scheme operating on these principles. I.
Reversible Boolean Networks I: Distribution of Cycle Lengths
, 2000
"... We consider a class of models describing the dynamics of N Boolean variables, where the time evolution of each depends on the values of K of the other variables. Previous work has considered models with dissipative dynamics. Here we consider timereversible models, which necessarily have the propert ..."
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We consider a class of models describing the dynamics of N Boolean variables, where the time evolution of each depends on the values of K of the other variables. Previous work has considered models with dissipative dynamics. Here we consider timereversible models, which necessarily have the property that every possible point in the statespace is an element of one and only one cycle. As in the dissipative case, when K is large, typical orbit lengths grow exponentially with N, whereas for small enough K, typical orbit lengths grow much more slowly with N. The numerical data are consistent with the existence of a phase transition at which the average orbit length grows as a power of N at a value of K between 1.4 and 1.7. However, in the reversible models the interplay between the discrete symmetry and quenched randomness can lead to enormous fluctuations of orbit lengths and other interesting features that are unique to the reversible case. The orbits can be classified by their behavior under time reversal. The orbits that transform into themselves under time reversal have properties quite different from those that do not; in particular, a significant fraction of lattertype orbits have lengths enormously longer than orbits that are time reversalsymmetric. For large K and moderate N, the vast majority of points in the statespace are on one of the time reversal singlet orbits, and a random hopping model gives an accurate description of orbit lengths. However, for any finite K, the random hopping approximation fails qualitatively when N is large enough
Structural Circuits and Attractors in Kauffman Networks
"... There has been some ambiguity about the growth of attractors in Kauffman networks with network size. Some recent work has linked this to the role and growth of circuits or loops of boolean variables. Using numerical methods we have investigated the growth of structural circuits in Kauffman networks ..."
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Cited by 2 (0 self)
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There has been some ambiguity about the growth of attractors in Kauffman networks with network size. Some recent work has linked this to the role and growth of circuits or loops of boolean variables. Using numerical methods we have investigated the growth of structural circuits in Kauffman networks and suggest that the exponential growth in the number of structural circuits places a lower bound on the complexity of the growth of boolean dependency loops and hence of the number of attractors. We use a fast and exact circuit enumeration method that does not rely on sampling trajectories. We also explore the role of structural selfedges, or selfinputs in the NKmodel, and how they affect the number of structural circuits and hence of attractors.
Regulatory Dynamics on Random Networks: Asymptotic Periodicity and Modularity”, submitted (2007). Available at http://lanl.arxiv.org/abs/0707.1551
"... Abstract. We study the dynamics of discrete–time regulatory networks on random digraphs. For this we define ensembles of deterministic orbits of random regulatory networks, and introduce some statistical indicators related to the long–term dynamics of the system. We prove that, in a random regulator ..."
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Abstract. We study the dynamics of discrete–time regulatory networks on random digraphs. For this we define ensembles of deterministic orbits of random regulatory networks, and introduce some statistical indicators related to the long–term dynamics of the system. We prove that, in a random regulatory network, initial conditions converge almost surely to a periodic attractor. We study the subnetworks, which we call modules, where the periodic asymptotic oscillations are concentrated. We proof that those modules are dynamically equivalent to independent regulatory networks.
Selforganized criticality and adaptation in discrete dynamical networks (arXiv:0811.0980
, 2008
"... Abstract It has been proposed that adaptation in complex systems is optimized at the critical boundary between ordered and disordered dynamical regimes. Here, we review models of evolving dynamical networks that lead to selforganization of network topology based on a local coupling between a dynami ..."
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Abstract It has been proposed that adaptation in complex systems is optimized at the critical boundary between ordered and disordered dynamical regimes. Here, we review models of evolving dynamical networks that lead to selforganization of network topology based on a local coupling between a dynamical order parameter and rewiring of network connectivity, with convergence towards criticality in the limit of large network size N. In particular, two adaptive schemes are discussed and compared in the context of Boolean Networks and Threshold Networks: 1) Active nodes loose links, frozen nodes aquire new links, 2) Nodes with correlated activity connect, decorrelated nodes disconnect. These simple local adaptive rules lead to coevolution of network topology anddynamics. Adaptive networks are strikingly different from random networks: They evolve inhomogeneous topologies and broad plateaus of homeostatic regulation, dynamical activity exhibits 1 / f noise and attractor periods obey a scalefree distribution. The proposed coevolutionary mechanism of topological selforganization is robust against noise and does not depend on the details of dynamical transition rules. Using finitesize scaling, it is shown that networks converge to a selforganized critical state in the thermodynamic limit. Finally, we discuss open questions and directions for future research, and outline possible applications of these models to adaptive systems in diverse areas.
Circuits, Attractors and Reachability in MixedK Kauffman Networks
, 2007
"... The growth in number and nature of dynamical attractors in Kauffman NK network models are still not well understood properties of these important random boolean networks. Structural circuits in the underpinning graph give insights into the number and length distribution of attractors in the NK model ..."
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Cited by 1 (1 self)
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The growth in number and nature of dynamical attractors in Kauffman NK network models are still not well understood properties of these important random boolean networks. Structural circuits in the underpinning graph give insights into the number and length distribution of attractors in the NK model. We use a fast direct circuit enumeration algorithm to study the NK model and determine the growth behaviour of structural circuits. This leads to an explanation and lower bound on the growth properties and the number of attractor loops and a possible Krelationship for circuit number growth with network size N. We also introduce a mixedK model that allows us to explore N 〈K 〉 between pairs of integer K values in Kauffmanlike systems. We find that the circuits ’ behaviour is a useful metric in identifying phase transitional behaviour around the critical connectivity in that model too. We identify an intermediate phase transition in circuit growth behaviour at K = KS ≈ 1.5, that is distinct from both the percolation transition at KP ≡ 1 and the Kauffman transition at KC ≡ 2. We relate this transition to mutual node reachability within the giant component of nodes.