Relevant elments, Magnetization and Dynamical Properties in Kauffman Networks: a Numerical Study

This is the first of two papers about the structure of Kauffman networks. In this paper we define the relevant elements of random networks of automata, following previous work by Flyvbjerg and Flyvbjerg and Kjaer, and we study numerically their probability distribution in the chaotic phase and on the critical line of the model. A simple approximate argument predicts that their number scales as sqrt(N) on the critical line, while it is linear with N in the chaotic phase and independent of system size in the frozen phase. This argument is confirmed by numerical results. The study of the relevant elements gives useful information about the properties of the attractors in critical networks, where the pictures coming from either approximate computation methods or from simulations are not very clear.Comment: 22 pages, Latex, 8 figures, submitted to Physica

Relaxation, closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks

Attractors in asymmetric neural networks with deterministic parallel dynamics were shown to present a "chaotic" regime at symmetry eta < 0.5, where the average length of the cycles increases exponentially with system size, and an oscillatory regime at high symmetry, where the typical length of the cycles is 2. We show, both with analytic arguments and numerically, that there is a sharp transition, at a critical symmetry \e_c=0.33, between a phase where the typical cycles have length 2 and basins of attraction of vanishing weight and a phase where the typical cycles are exponentially long with system size, and the weights of their attraction basins are distributed as in a Random Map with reversal symmetry. The time-scale after which cycles are reached grows exponentially with system size $N$, and the exponent vanishes in the symmetric limit, where $T\propto N^{2/3}$. The transition can be related to the dynamics of the infinite system (where cycles are never reached), using the closing probabilities as a tool. We also study the relaxation of the function $E(t)=-1/N\sum_i |h_i(t)|$, where $h_i$ is the local field experienced by the neuron $i$. In the symmetric system, it plays the role of a Ljapunov function which drives the system towards its minima through steepest descent. This interpretation survives, even if only on the average, also for small asymmetry. This acts like an effective temperature: the larger is the asymmetry, the faster is the relaxation of $E$, and the higher is the asymptotic value reached. $E$ reachs very deep minima in the fixed points of the dynamics, which are reached with vanishing probability, and attains a larger value on the typical attractors, which are cycles of length 2.Comment: 24 pages, 9 figures, accepted on Journal of Physics A: Math. Ge

Attractors in fully asymmetric neural networks

The statistical properties of the length of the cycles and of the weights of the attraction basins in fully asymmetric neural networks (i.e. with completely uncorrelated synapses) are computed in the framework of the annealed approximation which we previously introduced for the study of Kauffman networks. Our results show that this model behaves essentially as a Random Map possessing a reversal symmetry. Comparison with numerical results suggests that the approximation could become exact in the infinite size limit.Comment: 23 pages, 6 figures, Latex, to appear on J. Phys.

Self-organized Networks of Competing Boolean Agents

A model of Boolean agents competing in a market is presented where each agent bases his action on information obtained from a small group of other agents. The agents play a competitive game that rewards those in the minority. After a long time interval, the poorest player's strategy is changed randomly, and the process is repeated. Eventually the network evolves to a stationary but intermittent state where random mutation of the worst strategy can change the behavior of the entire network, often causing a switch in the dynamics between attractors of vastly different lengths.Comment: 4 pages, 3 included figures. Some text revision and one new figure added. To appear in PR

Closing probabilities in the Kauffman model: an annealed computation

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 not be explained whithin the annealed approximation.Comment: latex, 36 pages, figures added in uufiles format,error in epsffile nam

Periodicity of chaotic trajectories in realizations of finite computer precisions and its implication in chaos communications

Fundamental problems of periodicity and transient process to periodicity of chaotic trajectories in computer realization with finite computation precision is investigated by taking single and coupled Logistic maps as examples. Empirical power law relations of the period and transient iterations with the computation precisions and the sizes of coupled systems are obtained. For each computation we always find, by randomly choosing initial conditions, a single dominant periodic trajectory which is realized with major portion of probability. These understandings are useful for possible applications of chaos, e.g., chaotic cryptography in secure communication.Comment: 10 pages, 3 figures, 2 table

Exact Solution of Semi-Flexible and Super-Flexible Interacting Partially Directed Walks

We provide the exact generating function for semi-flexible and super-flexible interacting partially directed walks and also analyse the solution in detail. We demonstrate that while fully flexible walks have a collapse transition that is second order and obeys tricritical scaling, once positive stiffness is introduced the collapse transition becomes first order. This confirms a recent conjecture based on numerical results. We note that the addition of an horizontal force in either case does not affect the order of the transition. In the opposite case where stiffness is discouraged by the energy potential introduced, which we denote the super-flexible case, the transition also changes, though more subtly, with the crossover exponent remaining unmoved from the neutral case but the entropic exponents changing

The computational complexity of Kauffman nets and the P versus NP problem

Complexity theory as practiced by physicists and computational complexity theory as practiced by computer scientists both characterize how difficult it is to solve complex problems. Here it is shown that the parameters of a specific model can be adjusted so that the problem of finding its global energy minimum is extremely sensitive to small changes in the problem statement. This result has implications not only for studies of the physics of random systems but may also lead to new strategies for resolving the well-known P versus NP question in computational complexity theory.Comment: 4 pages, no figure