## THE ECONOPHYSICS OF SIZE

### Iddo Eliazar

##### 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 rank
distributions
- Macroscopic structures of rank distributions:
absolute monarchy and versatility
- Mapping between rank
distributions and probability laws, power-law connections
- Oligarchic analysis of rank distributions: the universality classes
of totalitarianism, criticality, and egalitarianism
- Totalitarianism: absolute monarchy and monarchic clans
- Heapsian analysis of rank distributions: information streams and
innovations
- The Heaps process and the Heaps curve: a Functional Central Limit
Theorem
- The Heaps curve and Laplace transforms, power-law
connections
- Composite Zipfian epilog: Pareto and Inverse-Pareto
limits; egalitarianism, monarchic-clan totalitarianism, and
criticality; composite Heapsian structure of innovations.

* Personal webpage *