Error threshold in optimal coding, numerical criteria and classes of universalities for complexity

TL;DR

This paper investigates the error threshold in optimal coding through the lens of free energy in the Random Energy Model, classifies universality classes of complexity, and explores their implications across various systems.

Contribution

It introduces a classification of complexity universality classes and connects the error threshold in coding to broader complex systems and critical phenomena.

Findings

Free energy at the transition point has finite size corrections proportional to sqrt(N).

Identifies multiple universality classes of complexity, from random graphs to complex systems like markets.

Error threshold in coding is linked to critical phenomena and universality classes.

Abstract

The free energy of the Random Energy Model at the transition point between ferromagnetic and spin glass phases is calculated. At this point, equivalent to the decoding error threshold in optimal codes, free energy has finite size corrections proportional to the square root of the number of degrees. The response of the magnetization to the ferromagnetic couplings is maximal at the values of magnetization equal to half. We give several criteria of complexity and define different universality classes. According to our classification, at the lowest class of complexity are random graph, Markov Models and Hidden Markov Models. At the next level is Sherrington-Kirkpatrick spin glass, connected with neuron-network models. On a higher level are critical theories, spin glass phase of Random Energy Model, percolation, self organized criticality (SOC). The top level class involves HOT design, error threshold in optimal coding, language, and, maybe, financial market. Alive systems are also related with the last class. A concept of anti-resonance is suggested for the complex systems.