EMERGENCE OF SCALING: FROM INFORMATION-THEORETIC CONSTRAINTS TO SAMPLE SPACE REDUCING PROCESSES

Bernat Corominas-Murtra (Personal webpage)

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 so-called Zipf's Law [2]. Insuch probability distribution, the probability of observing the k-thmost 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 self-organized 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'rich-gets-richer' class of models or the critical exponentsappearing at the percolation threshold of a random graph [5].

The main focus will be on two non-standard frameworks, having,however, a huge explanatory potential: Information-theoreticconstraints for the evolution of complex codes. This leads to amathematical formalisation of the so-called "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 information-theoretic 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 power-law 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.