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Register a new userFrom Microscopic Heterogeneity to Macroscopic Complexity in the Contrarian Voter Model
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Sven Banisch
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An analytical treatment of a simple opinion model with contrarian behavior is presented. The focus is on the stationary dynamics of the model and in particular on the effect of inhomogeneities in the interaction topology on the stationary behavior. We start from a micro-level Markov chain description of the model. Markov chain aggregation is then used to derive a macro chain for the complete graph as well as a meso-level description for the two-community graph composed of two (weakly) coupled sub-communities. In both cases, a detailed understanding of the model behavior is possible using Markov chain tools. More importantly, however, this setting provides an analytical scenario to study the discrepancy between the homogeneous mixing case and the model on a slightly more complex topology. We show that memory effects are introduced at the macro level when we aggregate over agent attributes without sensitivity to the microscopic details and quantify these effects using concepts from information theory. In this way, the method facilitates the analysis of the relation between microscopic processes and a their aggregation to a macroscopic level of description and informs about the complexity of a system introduced by heterogeneous interaction relations.
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Starting from the microscopic description of a normal fluid in terms of any kind of local interacting many-particle theory we present a well defined step by step procedure to derive the hydrodynamic equations for the macroscopic phenomena. We specify the densities of the conserved quantities as the relevant hydrodynamic variables and apply the methods of non-equilibrium statistical mechanics with projection operator techniques. As a result we obtain time-evolution equations for the hydrodynamic variables with three kinds of terms on the right-hand sides: reversible, dissipative and fluctuating terms. In their original form these equations are completely exact and contain nonlocal terms in space and time which describe nonlocal memory effects. Applying a few approximations the nonlocal properties and the memory effects are removed. As a result we find the well known hydrodynamic equations of a normal fluid with Gaussian fluctuating forces. In the following we investigate if and how the time-inversion invariance is broken and how the second law of thermodynamics comes about. Furthermore, we show that the hydrodynamic equations with fluctuating forces are equivalent to stochastic Langevin equations and the related Fokker-Planck equation. Finally, we investigate the fluctuation theorem and find a modification by an additional term.
We give exact formulae for a wide family of complexity measures that capture the organization of hidden nonlinear processes. The spectral decomposition of operator-valued functions leads to closed-form expressions involving the full eigenvalue spectrum of the mixed-state presentation of a processs epsilon-machine causal-state dynamic. Measures include correlation functions, power spectra, past-future mutual information, transient and synchronization informations, and many others. As a result, a direct and complete analysis of intrinsic computation is now available for the temporal organization of finitary hidden Markov models and nonlinear dynamical systems with generating partitions and for the spatial organization in one-dimensional systems, including spin systems, cellular automata, and complex materials via chaotic crystallography.
The main subject of the thesis is the study of stationary nonequilibrium states trough the use of microscopic stochastic models that encode the physical interaction in the rules of Markovian dynamics for particles configurations. These models are known as interacting particles systems and are simple enough to be treated analytically but also complex enough to capture essential physical behaviours. The thesis is organized in two parts. The part 1 is devoted to the microscopic theory of the stationary states. We characterize these states developing some general structures that have an interest in themselves. In this part there is an interlude dedicated to discrete calculus on discrete manifolds with an exposition a little bit different to the one available in literature and some original definitions. The part 2 studies the problem macroscopically. In particular we consider the large deviations asymptotic behavior for a class of solvable one dimensional models of heat conduction. Both part 1 and 2 begin with an introduction of motivational character followed by an overview of the relevant results and a summary explaining the organization. Even tough the two parts are strictly connected they can be read independently after chapter 1. The material is presented in such a way to be self-consistent as much as possible.
The Fisher-Rao metric from Information Geometry is related to phase transition phenomena in classical statistical mechanics. Several studies propose to extend the use of Information Geometry to study more general phase transitions in complex systems. However, it is unclear whether the Fisher-Rao metric does indeed detect these more general transitions, especially in the absence of a statistical model. In this paper we study the transitions between patterns in the Gray-Scott reaction-diffusion model using Fisher information. We describe the system by a probability density function that represents the size distribution of blobs in the patterns and compute its Fisher information with respect to changing the two rate parameters of the underlying model. We estimate the distribution non-parametrically so that we do not assume any statistical model. The resulting Fisher map can be interpreted as a phase-map of the different patterns. Lines with high Fisher information can be considered as boundaries between regions of parameter space where patterns with similar characteristics appear. These lines of high Fisher information can be interpreted as phase transitions between complex patterns.
We study a generalization of the voter model on complex networks, focusing on the scaling of mean exit time. Previous work has defined the voter model in terms of an initially chosen node and a randomly chosen neighbor, which makes it difficult to disentangle the effects of the stochastic process itself relative to the network structure. We introduce a process with two steps, one that selects a pair of interacting nodes and one that determines the direction of interaction as a function of the degrees of the two nodes and a parameter $alpha$ which sets the likelihood of the higher degree node giving its state. Traditional voter model behavior can be recovered within the model. We find that on a complete bipartite network, the traditional voter model is the fastest process. On a random network with power law degree distribution, we observe two regimes. For modest values of $alpha$, exit time is dominated by diffusive drift of the system state, but as the high nodes become more influential, the exit time becomes becomes dominated by frustration effects. For certain selection processes, a short intermediate regime occurs where exit occurs after exponential mixing.
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