Towards an Algebra for Cascade Effects

TL;DR

This paper develops an algebraic framework for analyzing cascade effects in dynamical systems, focusing on fixed points, failure modes, and resilience, with applications to understanding cascading failures.

Contribution

It introduces a novel axiomatic theory for cascade systems, characterizing their behavior through fixed points, operators, and measures of resilience and fragility.

Findings

Characterization of cascade systems via fixed points and operators

Introduction of mu-rank to quantify system energy

Analysis of system resilience and fragility limits

Abstract

We introduce a new class of (dynamical) systems that inherently capture cascading effects (viewed as consequential effects) and are naturally amenable to combinations. We develop an axiomatic general theory around those systems, and guide the endeavor towards an understanding of cascading failure. The theory evolves as an interplay of lattices and fixed points, and its results may be instantiated to commonly studied models of cascade effects. We characterize the systems through their fixed points, and equip them with two operators. We uncover properties of the operators, and express global systems through combinations of local systems. We enhance the theory with a notion of failure, and understand the class of shocks inducing a system to failure. We develop a notion of mu-rank to capture the energy of a system, and understand the minimal amount of effort required to fail a system, termed resilience. We deduce a dual notion of fragility and show that the combination of systems sets a limit on the amount of fragility inherited.