Nino Boccara

Cellular automaton rules conserving the number of active sites

This paper shows how to determine all of the unidimensional two-state cellular automaton rules of a given number of inputs which conserve the number of active sites. These rules have to satisfy a necessary and sufficient condition. If the active sites are viewed as cells occupied by identical particles, these cellular automaton rules represent evolution operators of systems of identical interacting particles whose total number is conserved. Some of these rules, which allow motion in both directions, mimic ensembles of one-dimensional pseudorandom walkers. Numerical evidence indicates that the corresponding stochastic processes might be non-Gaussian.

Car accidents and number of stopped cars due to road blockage on a one-lane highway

Within the framework of a simple model of car traffic on a one-lane highway, we study the probability of the occurrence of car accidents when drivers do not respect the safety distance between cars, and, as a result of the blockage during the time T necessary to clear the road, we determine the number of stopped cars as a function of car density. We give a simple theory in good agreement with our numerical simulations.

Cellular Automata and Cooperative Systems

Complexity of Infinite Sequences and the Ising Transducer.- Renormalization Group Analysis of Directed Models.- Maximal Lyapunov Exponent for 1D Boolean Cellular Automata.- Automata Network Epidemic Models.- Structure Evolution of Neural Networks.- Lower Bounds on the Memory Capacity of the Dilute Hopfield Model.- Schrodinger Operators with Substitution Potentials.- An Interplay Between Local and Global Dynamics in Biological Networks: the Case of Genetic Sequences.- How to Fire almost any Arbitrary Pattern on a Cellular Automaton.- Dynamics of Random Neural Nets.- A Matrix Method of Solving an Asymmetric Exclusion Model with Open Boundaries.- A Formula of full Semiinvariants.- Non-Gibbsian States for Renormalization-Group Transformations and Beyond.- Fluctuations in the Asymmetric Simple Exclusion Process.- Analysis of a Network Model.- Drift and Diffusion in Phase Space. An Application to Celestial Mechanics.- Real Computation with Cellular Automata.- Lyapunov Functionals Associated to Automata.- Sand-Pile Dynamics in a One-Dimensional Bounded Lattice.- Differential Inequalities for Potts and Random-Cluster Processes.- Cryptography with Dynamical Systems.- On Entropic Repulsion in Low Temperature Ising Models.- The Distribution of Lyapunov Exponents for Large Random Matrices.- Transience and Dislocations in One-Dimensional Cellular Automata.- Properties of Limit Sets of Cellular Automata.- Stochastic Equations on Random Trees.- The Branching Diffusion, Stochastic Equations and Travelling Wave Solutions to the Equation of Kolmogorov - Petrovskii - Piskunov.- Spin Models with Random Interactions: Some Rigorous Results.- Low Temperature Phase Transitions on Quasiperiodic Lattices.- Stability of Interfaces in Stochastic Media.- Hydrodynamic Equation for Mean Zero Asymmetric Zero Range Processes.- Clustering and Coexistence in Threshold Voter Models.- Analytical Results for the Maximal Lyapunov Exponent.- Some Remarks on Almost Gibbs States.- Some Coded Systems that are Not Unstable Limit Sets of Cellular Automata.- Constructive Criteria for the Ergodicity of Interacting Particle Systems.- Self-Similar Fractals Can be Generated by Cellular Automata.- Finite Volume Mixing Conditions for Lattice Spin Systems and Exponential Approach to Equilibrium of Glauber Dynamics.- Quasi-Stationary Distributions for Birth-Death Chains. Convergence Radii and Yaglom Limit.- Cooperative Phenomena in Swarms.- Comparison of Semi-Oriented Bootstrap Percolation Models with Modified Bootstrap Percolation.- Chaotic Size Dependence in Spin Glasses.- On the Gibbs States for One-Dimensional Lattice Boson Systems with a Long-Range Interaction.- Hydrodynamic Limits and Ergodicity for Hamiltonian System with Small Noise.- Relaxation Times for Stochastic Ising Models in the Limit of Vanishing External Field at Fixed Low Temperatures.

Critical behaviour of a cellular automaton highway traffic model

We derive the critical behaviour of a cellular automaton traffic flow model using an order parameter breaking the symmetry of the jam-free phase. Random braking appears to be the symmetry-breaking field conjugate to the order parameter. For vmax = 2, we determine the values of the critical exponents β, γ and δ using an order-3 cluster approximation and computer simulations. These critical exponents satisfy a scaling relation, which can be derived assuming that the order parameter is a generalized homogeneous function of |ρ-ρc| and p in the vicinity of the phase transition point.

MODELS OF OPINION FORMATION: INFLUENCE OF OPINION LEADERS

This paper studies the evolution of the distribution of opinions in a population of individuals in which there exist two distinct subgroups of highly-committed, well-connected opinion leaders endowed with a strong convincing power. Each individual, located at a vertex of a directed graph, is characterized by her name, the list of people she is interacting with, her level of awareness, and her opinion. Various temporal evolutions according to different local rules are compared in order to find under which conditions the formation of strongly polarized subgroups, each adopting the opinion of one of the two groups of opinion leaders, is favored.

The Dynamics of Street Gang Growth and Policy Response

Street gangs have emerged as tremendously powerful institutions in many communities. In urban ghettos, they may very well be the most important institutions in the lives of a large proportion of adolescent and young adult males. In this paper, we will examine the dynamics of street gang growth and the effects of efforts to control it. Our most basic hypothesis is that there is an intrinsic dynamic that drives the growth of street gangs. Thus, the growth of the gang population is a function of its size. The most important reason why size and growth rate are related is probably that gang participation is contagious. Ethnographic work emphasizes the importance of peer influence and social bonding both in the initial decision to join a gang and in the maintenance of membership, often through a lengthy prison term (e.g., Cohen, 1955; Crane, 1989; Hagedorn, 1988; Padilla, 1992; Moore, 1978, 1991; Short and Strodtbeck, 1965; Taylor, 1990). Empirical tests of differential association theory, while not addressing the issue of gang participation per se, have consistently

Power-Law Distributions

The essential points of this chapter are The Pareto law The log-normal distribution The two regimes of income distribution and how it is explained The Zipf’s law of word frequency The behavior of stock prices as a function of time according to Louis Bachelier and Benoit Mandelbrot The definition of econophysics The universal behavior of rank as a function size for cities The distribution of family names (in Japan) The distribution of votes (in Sao Paulo) The explanation of the power-law behavior The log-normal behavior The definition of self-organized criticality The sandpile model and the avalanches

ON THE EXISTENCE OF A VARIATIONAL PRINCIPLE FOR DETERMINISTIC CELLULAR AUTOMATON MODELS OF HIGHWAY TRAFFIC FLOW

It is shown that a variety of deterministic cellular automaton models of highway traffic flow obey a variational principle which states that, for a given car density, the average car flow is a nondecreasing function of time. This result is established for systems whose configurations exhibits local jams of a given structure. If local jams have a different structure, it is shown that either the variational principle may still apply to systems evolving according to some particular rules, or it could apply under a weaker form to systems whose asymptotic average car flow is a well-defined function of car density. To establish these results, it has been necessary to characterize among all number-conserving cellular automaton rules which ones may reasonably be considered as acceptable traffic rules. Various notions such as free-moving phase, perfect and defective tiles, and local jam play an important role in the discussion. Many illustrative examples are given.

Modeling Diffusion of Innovations with Probabilistic Cellular Automata

We present a family of one-dimensional cellular automata modeling the diffusion of an innovation in a population. Starting from simple deterministic rules, we construct models parameterized by the interaction range and exhibiting a second-order phase transition. We show that the number of individuals who eventually keep adopting the innovation strongly depends on connectivity between individuals.

Transformations of one-dimensional cellular automation rules by translation-invariant local surjective mappings

Abstract If M is a noninvertible translation-invariant local surjective mapping, it is shown that some local one-dimensional deterministic cellular automaton rules F have a transform Φ by M defined by Φ ∘ M = M ∘ F . When it exists Φ is local and its Wolframs class is the same as F . The evolution of a cellular automaton according to rule Φ is simply related to the evolution according to rule F . In the case of class-3 rules, the evolution to the attractor may often be viewed as particle-like structures evolving in a regular background. If the structure of these particles and their interactions for a rule F are known, then the structure and interactions of the transformed particles for rule Φ are also known. If M is a nontrivial invertible translation-invariant local surjective mapping, Φ always exists, but it is, in general, site- and time-dependent.