A unifying approach to self-organizing systems interacting via conservation laws

TL;DR

This paper introduces a unified framework based on conservation laws and projection operators for analyzing and designing self-organizing systems across various physical, biological, and engineered networks.

Contribution

It develops a formalism that captures diverse systems using graph-based representations and extends to collective dynamics via the PrEDS method, enabling analysis and design of self-organizing behaviors.

Findings

PrEDS effectively models diverse networked systems.

The framework aligns with non-equilibrium thermodynamics principles.

New insights into swarm dynamics and optimization.

Abstract

We present a unified framework for embedding and analyzing dynamical systems using generalized projection operators rooted in local conservation laws. By representing physical, biological, and engineered systems as graphs with incidence and cycle matrices, we derive dual projection operators that decompose network fluxes and potentials. This formalism aligns with principles of non-equilibrium thermodynamics and captures a broad class of systems governed by flux-forcing relationships and local constraints. We extend this approach to collective dynamics through the PRojective Embedding of Dynamical Systems (PrEDS), which lifts low-dimensional dynamics into a high-dimensional space, enabling both replication and recovery of the original dynamics. When systems fall within the PrEDS class, their collective behavior can be effectively approximated through projection onto a mean-field space. We demonstrate the versatility of PrEDS across diverse domains, including resistive and memristive circuits, adaptive flow networks (e.g., slime molds), elastic string networks, and particle swarms. Notably, we establish a direct correspondence between PrEDS and swarm dynamics, revealing new insights into optimization and self-organization. Our results offer a general theoretical foundation for analyzing complex networked systems and for designing systems that self-organize through local interactions.