Wandering in the state space

We analyse the topology of the state space of two systems: i) N Ising spins +/-1 with the antiferromagnetic interactions on a triangular lattice, with the condition of minimum of energy, ii) a roundabout of three access roads and three exit roads, with up to 2 cars on each road. The state space is represented by a network, and states - as nodes; two nodes are linked if an elementary process (spin flip or car shift) transforms the respective states one into another. Information is collected on the number of neighbours of states, what allows to distinguish classes and subclasses of states, and on the cluster structure of the state space. In the Ising systems, the clusters are characterized by anisotropy of the spin-spin correlation functions. In the case of a roundabout, the clusters differ by the number of empty or full roads. The method is general and it provides a basis for applications of the random walk theory

Communities and classes in symmetric fractals

Two aspects of fractal networks are considered: the community structure and the class structure, where classes of nodes appear as a consequence of a local symmetry of nodes. The analysed systems are the networks constructed for two selected symmetric fractals: the Sierpinski triangle and the Koch curve. Communities are searched for by means of a set of differential equations. Overlapping nodes which belong to two different communities are identified by adding some noise to the initial connectivity matrix. Then, a node can be characterized by a spectrum of probabilities of belonging to different communities. Our main goal is that the overlapping nodes with the same spectra belong to the same class

Combinatorial aspect of fashion

Simulations are performed according to the Axelrod model of culture dissemination, with modified mechanism of repulsion. Previously, repulsion was considered by Radillo-Diaz et al (Phys. Rev. E 80 (2009) 066107) as dependent on a predefined threshold. Here the probabilities of attraction and repulsion are calculated from the number of cells in the same states. We also investigate the influence of some homogeneity, introduced to the initial state. As the result of the probabilistic definition of repulsion, the ordered state vanishes. A small cluster of a few percent of population is retained only if in the initial state a set of agents is prepared in the same state. We conclude that the modelled imitation is successful only with respect to agents, and not only their features