Iddo Eliazar photo


Iddo Eliazar (Personal webpage)

Holon Institute of Technology

In this talk we amalgamate ideas and concepts from various scientific disciplines – economics, mathematics, physics, probability, and statistics – to explore a topic of wide scientific interest: the omnipresence of power-laws in the distributions of sizes, commonly referred to as "Zipf’s law" and as "Pareto’s law". The talk is based on an ongoing collaboration with Morrel Cohen (Princeton & Rutgers), and is split into two parts which are outlined as follows.

Part I. Prolog: Rank distributions and Zipf’s Law:

  • Lorenz’s curve, Pietra’s formula, and Gini’s index: measuring the distribution of wealth and social inequality
  • Pareto’s Law: from absolute monarchy to pure communism
  • Lorenzian analysis of rank distributions
  • Regular variation
  • Lorenzian limit law for rank distributions: the universality classes of absolute monarchy, Pareto’s law, and pure communism
  • Network's macroscopic topologies: the universality classes of total connectedness ('solid state'), fractal connectedness ('liquid range'), and total disconnectedness (gas state)
  • Oligarchic limit law for rank distributions: the universality classes of totalitarianism, criticality, and egalitarianism
  • Interlaced universal macroscopic classification of rank distributions and their phase transitions
  • Zipfian epilog: egalitarianism, totalitarianism, and criticality

Part II. Prolog: from the single-exponent Zipf Law to the double-exponent composite Zipf Law:

  • Lorenzian analysis of rankdistributions
  • Macroscopic structures of rank distributions:absolute monarchy and versatility
  • Mapping between rankdistributions and probability laws, power-law connections
  • Oligarchic analysis of rank distributions: the universality classesof totalitarianism, criticality, and egalitarianism
  • Totalitarianism: absolute monarchy and monarchic clans
  • Heapsian analysis of rank distributions: information streams andinnovations
  • The Heaps process and the Heaps curve: a Functional Central LimitTheorem
  • The Heaps curve and Laplace transforms, power-lawconnections
  • Composite Zipfian epilog: Pareto and Inverse-Paretolimits; egalitarianism, monarchic-clan totalitarianism, andcriticality; composite Heapsian structure of innovations.