Fractional calculus ties the microscopic and macroscopic scales of complex network dynamics

TL;DR

This paper demonstrates how fractional calculus can connect microscopic decision-making processes with macroscopic complex network behaviors, revealing phase transitions and temporal complexity through a fractional master equation approach.

Contribution

It introduces a fractional master equation framework that accurately models the inverse power-law switching times in complex networks, linking microscopic and macroscopic dynamics.

Findings

Fractional master equation describes network element behavior.

Inverse power-law distribution of switching times observed.

Model captures phase transitions and temporal complexity.

Abstract

A two-state master equation based decision making model has been shown to generate phase transitions, to be topologically complex and to manifest temporal complexity through an inverse power-law probability distribution function in the switching times between the two critical states of consensus. These properties are entailed by the fundamental assumption that the network elements in the decision making model imperfectly imitate one another. The process of subordination establishes that a single network element can be described by a fractional master equation whose analytic solution yields the observed inverse power-law probability distribution obtained by numerical integration of the two-state master equation to a high degree of accuracy.