We study the random growth of surfaces from within the perspective of a single column, namely, the fluctuation of the column height around the mean value, y͑t͒ϵh͑t͒ − ͗h͑t͒͘, which is depicted as being subordinated to a standard fluctuation-dissipation process with friction ␥. We argue that the main properties of Kardar-Parisi-Zhang theory, in one dimension, are derived by identifying the distribution of return times to y͑0͒ = 0, which is a truncated inverse power law, with the distribution of subordination times. The agreement of the theoretical prediction with the numerical treatment of the ͑1+1͒-dimensional model of ballistic deposition is remarkably good, in spite of the finite-size effects affecting this model. is an example of self-organization: As pointed out by Family [7], a growing surface spontaneously evolving into a steady state with universal fractal properties is similar to the mechanism of self-organized criticality [8]. The columns of the material growing due to the deposition of particles can be thought of as the individuals of a society. The joint action of the randomness driving the particle deposition and the interaction among columns results in the emergence of anomalous scaling coefficients, which can be interpreted as the signature of cooperation. However, only a little attention has been devoted so far to studying the dynamics of the single individuals of this society, namely, the single growing columns of the sample under study. Usually the authors of this field of research study the correlation among distinct columns [9] without paying attention to the dynamics of an individual. Yet, a single column is expected to carry information about cooperation.The single column perspective was recently adopted by Merikoski et al.[10] to study combustion fronts in paper. The individual property under observation iswhere h͑t͒ denotes the height of a single column at time t and ͗h͑t͒͘ the average over the heights of the columns of the whole sample. The authors of Ref.[10] record the times at which the variable y͑t͒ changes sign and builds up the corresponding time series t i so as to create the new time series The coefficient ␤ refers to the interface growth prior to saturation, a physical condition where the standard deviation of all L columns, the interface widthgrows as w͑L , t͒ ϰ t ␤ . Equation (2) establishes a connection between a single column property, D , and a collective property, ␤, thereby playing an important role for the perspective adopted in this paper. The theoretical foundation for this important relation is given in earlier papers [11][12][13][14] and has been more recently discussed by Majumdar [15].In this paper we prove that the KPZ condition emerges from the identification of D ͑͒ with the distribution function S ͑͒, the essential ingredient of the subordination theory [16][17][18] stemming from the original work of Montroll and Weiss [19]. In the subdiffusion case, anomalous diffusion is derived from the ordinary diffusion process by assuming that the time distance between on...

Extensive empirical evidence suggests that there is a maximal number of people with whom an individual can maintain stable social relationships (the Dunbar number). We argue that this arises as a consequence of a natural phase transition in the dynamic self-organization among N individuals within a social system. We present the calculated size dependence of the scaling properties of complex social network models to argue that this collective behavior is an enhanced form of collective intelligence. Direct calculation establishes that the complexity of social networks as measured by their scaling behavior is nonmonotonic, peaking around 150, thereby providing a theoretical basis for the value of the Dunbar number. Thus, we establish a theory-based bridge spanning the gap between sociology and psychology.

We illustrate a technique of analysis of time series, aiming at revealing crucial events, as a theoretical tool for the physical foundation of the subordination approach.

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We apply the method of Diffusion Entropy (DE) to the study of stromatolites by means of a two-dimensional procedure that makes it possible for us to compare the DE analysis to the results of a compression method. As done with the compression method, we analyze two pairs of samples, one biotic and the other a-biotic. Each pair consists of a target, the putative stromatolite sample, and of its surrounding matrix. We use two different procedures, referring to single colors and to a color combination, respectively. We apply the DE method to both procedures and we find the same result, this being that the scaling index of the time series stemming from the biotic target yields a scaling index larger than the scaling indices of the other three curves. We argue that the DE analysis confirms the results of the compression method.

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