Computational mechanics, an approach to structural complexity, defines a process’s causal states and gives a procedure for finding them. We show that the causal-state representation—an E-machine—is the minimal one consistent with

Lecture given at the workshop "Mathematical aspects of theories of gravitation", Stefan Banach International Mathematical Centre, 7 March 1996. A mini-introduction to critical phenomena in gravitational collapse is combined with a more detailed discussion of the regularity of the "critical spacetimes" dominating these phenomena. 1 Critical phenomena in gravitational collapse Most of the material here is contained also in the paper [1], to which I refer the reader for details. Here I'll give detail only in the calculation of the critical exponent, and on how gravity regularizes self-similar solutions, where I hope it is of more than technical interest. Initial data for general relativity (GR) may or may not form a black hole. In a hand-waving way one might compare this to a phase transition, and there is a "critical surface" in superspace (the phase space of GR) separating the two kinds of initial data. Choptuik [2] has explored this surface in a systematic way. For simplicity he...

In this paper, which is based on the important recent work of Yedidia, Freeman, and Weiss, we present a generalized form of belief propagation, viz. belie) e propagation on a partially ordered set (PBP). PBP is an iterative message-passing algorithm for solving, either exactly or approximately, the marginalized product density problem, which is a general computational problem of wide applicability. We will show that PBP can be thought of as an algorithm for minimizing a certain "free energy" function, and by exploiting this interpretation, we will exhibit a one-to-one correspondence between the fixed points of PBP and the stationary points of the free energy.

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Multiagent models of the emergence of social conventions have demonstrated that global conventions can arise from local coordination processes without a central authority. We further develop and extend previous work to address how and under what conditions emerging conventions are also socially efficient, i.e. better for all agents than potential alternative conventions. We show with computational experiments that the clustering coefficient of the networks within which agents interact is an important condition for efficiency. We also develop an analytical approximation of the simulation model that sheds some light to the original model behavior. Finally, we combine two decision mechanisms, local optimization and imitation, to study the competition between efficient and attractive actions. Our main result is that in clustered networks a society converges to an efficient convention and is stable against invasion of sub-optimal conventions under a much larger range of conditions than in a non-clustered network. On the contrary, in non-clustered networks the convention finally established heavily depends on its initial support. 1

this paper, we will provide an overview of the renormalization-group '5-4 many researchers use to understand crackling noise. Briefly, the renormalization group discusses how the effective evolution laws of our system change as we measure on longer and longer length scales. (It works by generating a coarse-graining mapping in system space, the abstract space of all possible evolution laws.) The broad range of event sizes will be attributed to a self-similarity, where the evolution laws look the same under different length scales. Using this self-similarity, we are led to a method for scaling experimental data. In the simplest case this yields power laws and fractal structures, but more generally it leads to universal scaling functions - where we argue the real predictive power lies. We will only touch upon the dauntingly complex analytical methods used in this field, but we believe we can explain faithfully and fully both what our tools are useful for, and how to apply them in practice. The renormalization group is perhaps the most impressive use of abstraction in science

The paper extends the Brock-Durlauf model of interactive discrete choice, where individuals’ decisions are influenced by the decisions of others, to richer social structures. Social structure is modelled by a graph, with individuals as vertices and interaction between individuals as edges. The paper extends the mean field case to stylized interaction topologies like the Walrasian star, the cycle and the one-dimensional lattice (or path) and compares the properties of Nash equilibria when agents act on the basis of expectations over their neighbors ’ decisions versus actual knowledge of neighbors ’ decisions. It links links social interactions theory with the econometric theory of systems of simultaneous equations modelling discrete decisions. The paper obtains general results for the dynamics of adjustment towards steady states and shows that they combine spectral properties of the adjacency matrix with those associated with the nonlinearity of the reaction functions that lead to multiplicity of steady states. When all agents have the same number of neighbors the dynamics of adjustment exhibit relative persistence. Cyclical interaction is associated with endogenous and generally transient spatial oscillations that take the form of islands of conformity, but multiplicity of equilibria leads to permanent effects of initial conditions. The paper also analyzes stochastic dynamics for general interaction topologies, when agents acts with knowledge of their neighbors ’ actual decisions, which involve networked Markov chains in sample spaces of very high dimensionality.

We analyze the population dynamics of a broad class of fitness functions that exhibit epochal evolution -- a dynamical behavior, commonly observed in both natural and artificial evolutionary processes, in which long periods of stasis in an evolving population are punctuated by sudden bursts of change. Our approach -- statistical dynamics -- combines methods from both statistical mechanics and dynamical systems theory in a way that offers an alternative to current "landscape" models of evolutionary optimization. We describe the population dynamics on the macroscopic level of fitness classes or phenotype subbasins, while averaging out the genotypic variation that is consistent with a macroscopic state. Metastability in epochal evolution occurs solely at the macroscopic level of the fitness distribution. While a balance between selection and mutation maintains a quasistationary distribution of fitness, individuals diffuse randomly through selectively neutral subbasins in genotype space. Sudden innovations occur when, through this diffusion, a genotypic portal is discovered that connects to a new subbasin of higher tness genotypes. In this way, we identify innovations with the unfolding and stabilization of a new dimension in the macroscopic state space. The architectural view of subbasins and portals in genotype space clarifies how frozen accidents and the resulting phenotypic constraints guide the evolution to higher complexity.

We review what we have learned about the "Renormalization Group peculiarities" which were discovered about twenty years ago by Griffiths and Pearce, and which questions they asked are still widely open. We also mention some related developments.

We compare phase transition(-like) phenomena in small model systems for both microcanonical and canonical ensembles. The model systems correspond to a few classical (non-quantum) point particles confined in a one-dimensional box and interacting via Lennard-Jones-type pair potentials. By means of these simple examples it can be shown already that the microcanonical thermodynamic functions of a small system may exhibit rich oscillatory behavior and, in particular, singularities (nonanalyticities) separating different microscopic phases. These microscopic phases may be identified as different microphysical dissociation states of the small system. The microscopic oscillations of microcanonical thermodynamic quantities (e.g. temperature, heat capacity, or pressure) should in principle be observable in suitably designed evaporation/dissociation experiments (which must realize the physical preconditions of the microcanonical ensemble). By contrast, singular phase transitions cannot occur, if a small system is embedded into an infinite heat bath (thermostat), corresponding to the canonical ensemble. For the simple model systems under consideration, it is nevertheless possible to identify a smooth canonical phase transition by studying the distribution of complex zeros of the canonical partition function. Key words: microscopic phase transitions, small systems, Lennard-Jones chains