
EMERGENCE OF SCALING: FROM INFORMATIONTHEORETIC CONSTRAINTS TO SAMPLE SPACE REDUCING PROCESSESBernat CorominasMurtra (Personal webpage)Medical University of Vienna
The comprehension of the mechanisms behind scaling patterns hasbecome on of the hot topics of modern statistical physics [1]. Fromcomplex networks to critical phenomena, scaling laws emerge insomewhat regular way. In these two lectures we will revise parts ofcurrent explanatory proposals for the emergence of such behaviours.We will put a special emphasis on the socalled Zipf's Law [2]. Insuch probability distribution, the probability of observing the kthmost common event scales as 1/k. Its remarkable ubiquity spans fromword frequencies in written texts to the distribution of city sizes,family names, wealth distribution or the size of avalanches insystems exhibiting selforganized criticality. Its origin and theconsequences one can extract from its observation in real systems isa matter of an intense debate. We will start with a short critical review of the scope of theresults one can derive from the statistical study of scalingbehaviours, both from the fundamental and practical side [3,4]. Wewill then briefly revise some of the standard approaches for theemergence of such statistical behaviours, such as the'richgetsricher' class of models or the critical exponentsappearing at the percolation threshold of a random graph [5]. The main focus will be on two nonstandard frameworks, having,however, a huge explanatory potential: Informationtheoreticconstraints for the evolution of complex codes. This leads to amathematical formalisation of the socalled "least effort"hypothesis, informally proposed by G.K. Zipf as the origin of thescaling behaviour observed in complex codes [2,6,7]. The increase ofcomplexity under informationtheoretic constraints has, however,larger ranges of application than the communication systems alone[8]. The "Sample Space Reducing" (SSR) processes, a recently introducedfamily of stochastic processes displaying a minimal degree ofhistory dependence [9]. SSR process are a totally new route toscaling which can explain a huge range of powerlaw exponents thanksto the unique assumption that the sampling space is reduced as longas the process unfolds. The intuitive rationale behind the SSRprocesses and the surprisingly simple mathematical apparatus neededto understand them makes the SSR process approach a new researcharea with promising applications.
References:
[1] M.E.J. Newman. Power laws, Paretodistributions and Zipf's law. Contemp. Phys. 46 (2005)323–351. 