Adaptive Complex Contagions and Threshold Dynamical Systems

TL;DR

This paper extends the theory of threshold dynamical systems on networks to include adaptive, dynamic thresholds, providing a complete characterization of their attractor structures and implications for modeling adaptive systems.

Contribution

It introduces and analyzes three classes of dynamic threshold systems, expanding current theory to cover adaptive thresholds and characterizing their attractor structures.

Findings

Sequential systems only have fixed points as limit sets.

Parallel systems only have period orbits of size at most two.

Characterization of attractor states for general graphs and specific cases.

Abstract

A broad range of nonlinear processes over networks are governed by threshold dynamics. So far, existing mathematical theory characterizing the behavior of such systems has largely been concerned with the case where the thresholds are static. In this paper we extend current theory of finite dynamical systems to cover dynamic thresholds. Three classes of parallel and sequential dynamic threshold systems are introduced and analyzed. Our main result, which is a complete characterization of their attractor structures, show that sequential systems may only have fixed points as limit sets whereas parallel systems may only have period orbits of size at most two as limit sets. The attractor states are characterized for general graphs and enumerated in the special case of paths and cycle graphs; a computational algorithm is outlined for determining the number of fixed points over a tree. We expect our results to be relevant for modeling a broad class of biological, behavioral and socio-technical systems where adaptive behavior is central.