Henryk Fuks

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.

Individual-based lattice model for spatial spread of epidemics

We present a lattice gas cellular automaton (LGCA) to study spatial and temporal dynamics of an epidemic of SIR (susceptible-infected-removed) type. The automaton is fully discrete, i.e., space, time and number of individuals are discrete variables. The automaton can be applied to study spread of epidemics in both human and animal populations. We investigate effects of spatial inhomogeneities in initial distribution of infected and vaccinated populations on the dynamics of epidemic of SIR type. We discuss vaccination strategies which differ only in spatial distribution of vaccinated individuals. Also, we derive an approximate, mean-field type description of the automaton, and discuss differences between the mean-field dynamics and the results ofLGCA simulation.

Response Curves for Cellular Automata in One and Two Dimensions: An Example of Rigorous Calculations

In this paper, the authors consider the problem of computing a response curve for binary cellular automata, that is, the curve describing the dependence of the density of ones after many iterations of the rule on the initial density of ones. The authors demonstrate how this problem could be approached using rule 130 as an example. For this rule, preimage sets of finite strings exhibit recognizable patterns; therefore, it is possible to compute both cardinalities of preimages of certain finite strings and probabilities of occurrence of these strings in a configuration obtained by iterating a random initial configuration n times. Response curves can be rigorously calculated in both one- and two-dimensional versions of CA rule 130. The authors also discuss a special case of totally disordered initial configurations, that is, random configurations where the density of ones and zeros are equal to 1/2.

Inflection system of a language as a complex network

We investigate inflection structure of a synthetic language using Latin as an example. We construct a bipartite graph in which one group of vertices correspond to dictionary headwords and the other group to inflected forms encountered in a given text. Each inflected form is connected to its corresponding headword, which in some cases in non-unique. The resulting sparse graph decomposes into a large number of connected components, to be called word groups. We then show how the concept of the word group can be used to construct coverage curves of selected Latin texts. We also investigate a version of the inflection graph in which all theoretically possible inflected forms are included. Distribution of sizes of connected components of this graphs resembles cluster distribution in a lattice percolation near the critical point.

Lattice gas cellular automaton modeling of surface roughening in homoepitaxial growth in nanowires

Our research addresses the problem of bridging large time and length scale gaps in simulating atomistic processes during thin film deposition. We introduce a new simulation approach based on a discrete description of atoms so that the unit length scale coincides with the atomic diameter. The interaction between atoms is defined using a coarse-grained approach to boost the computation speed. This approach does not heavily sacrifice the atomistic details in order to study structural evolution of a growing thin film on time scales in the order of seconds and even minutes. Our approach is inspired by lattice gas cellular automata models for chemically reacting systems, where individual particles interact with surrounding through assumed local driving forces. For homoepitaxial thin film deposition, the local driving force is the propensity of an atom to establish as many chemical bonds as possible to the underlying substrate atoms when it executes surface diffusion. Simulation results of Si layers deposited on a flat Si(001) substrate are presented.

Dynamics of large scale networks following a merger

We studied the dynamic network of relationships among avatars in the massively multiplayer online game Planetside 2. In the spring of 2014, two separate servers of this game were merged, and as a result, two previously distinct networks were combined into one. We observed the evolution of this network in the seven month period following the merger. We found that some structures of original networks persist in the combined network for a long time after the merger. As the original avatars are gradually removed, these structures slowly dissolve, but they remain observable for a surprisingly long time. We present a number of visualizations illustrating the post-merger dynamics and discuss time evolution of selected quantities characterizing the topology of the network.

Cellular Automata Simulations - Tools and Techniques

The purpose of this chapter is to provide a concise introduction to cellular automata simulations, and to serve as a starting point for those who wish to use cellular automata in modelling and applications. No previous exposure to cellular automata is assumed, beyond a standard mathematical background expected from a science or engineering researcher. Cellular automata (CA) are dynamical systems characterized by discreteness in space, time, and in state variables. In general, they can be viewed as cells in a regular lattice updated synchronously according to a local interaction rule, where the state of each cell is restricted to a finite set of allowed values. Unlike other dynamical systems, the idea of cellular automaton can be explained without using any advanced mathematical apparatus. Consider, for instance, a well-known example of the so-called majority voting rule. Imagine a group of people arranged in a line line who vote by raising their right hand. Initially some of them vote “yes”, others vote “no”. Suppose that at each time step, each individual looks at three people in his direct neighbourhood (himself and two nearest neighbours), and updates his vote as dictated by the majority in the neighbourhood. If the variable si(t) represents the vote of the i-th individual at the time t (assumed to be an integer variable), we can write the CA rule representing the voting process as