External field and critical exponents in controlling dynamics on complex networks

TL;DR

This paper investigates how controlling a fraction of nodes in complex networks influences phase transitions, revealing critical exponents and universality classes that govern the dynamics.

Contribution

It introduces a framework linking external control to phase transitions in network dynamics, identifying critical exponents and universality classes.

Findings

Controlling nodes acts as an external field in phase transitions.

Critical exponents are derived for a general class of dynamics.

Distinct universality classes are identified through examples.

Abstract

Dynamical processes on complex networks, ranging from biological, technological and social systems, show phase transitions between distinct global states of the system. Often, such transitions rely upon the interplay between the structure and dynamics that takes place on it, such that weak connectivity, either sparse network or frail interactions, might lead to global activity collapse, while strong connectivity leads to high activity. Here, we show that controlling dynamics of a fraction of the nodes in such systems acts as an external field in a continuous phase transition. As such, it defines corresponding critical exponents, both at equilibrium and of the transient time. We find the critical exponents for a general class of dynamics using the leading orders of the dynamic functions. By applying this framework to three examples, we reveal distinct universality classes.