[Paper Review] Nesting of dynamic systems and mode-dependent networks

David I. Spivak, Joshua Tan|arXiv (Cornell University)|Feb 25, 2015

Cellular Automata and Applications17 references3 citations

TL;DR

This paper introduces a category-theoretic framework using operads and their algebras to model mode-dependent networks—dynamic systems where network topology evolves based on internal states. It formalizes nesting of such systems through a semantics that ensures consistent composition across scales, enabling hierarchical modeling of complex, adaptive networks like neural systems.

ABSTRACT

For many networks, the connection pattern (often called the topology) can vary in time, depending on the changing state, or mode, of the modules within the network. For example, "airplane mode" is the name for one communicative mode of a modern cellphone, in which it will not connect with any cellphone towers; thus the topology of the cellular network is dependent on the modes of its modules. This paper addresses the issue of nesting such mode-dependent networks, in which a local network can be abstracted as a single module in a larger network. Each module in the network represents a dynamic system, whose behavior includes repeatedly updating its communicative mode. It is in this way that the dynamics of the modules controls the topology of the networks at all levels. This paper provides a formal semantics, using the category-theoretic framework of operads and their algebras, to capture the nesting property and dynamics of mode-dependent networks. We provide a detailed running example to ground the mathematics.

Motivation & Objective

• To formalize the nesting of dynamic, mode-dependent networks where topology evolves with internal states.
• To provide a consistent, compositional semantics for networks of networks with time-varying topologies.
• To model complex biological systems—such as the visual pathway—using hierarchical, state-dependent network dynamics.
• To extend wiring diagram theory to include state-dependent connectivity, enabling recursive system composition.
• To establish a formal foundation for building and reasoning about large-scale, adaptive dynamical systems using category theory.

Proposed method

• Uses operads and their algebras to formalize the hierarchical composition of networks with dynamic topologies.
• Introduces the operad of mode-dependent networks (OMDN) to model systems where node states dictate connection patterns.
• Defines modal dynamical systems via state sets, mode functions, update functions, and readout maps that depend on communicative modes.
• Applies coherence maps and composition rules to ensure consistency when composing networks at different levels.
• Employs tensor products and morphisms to build layered structures, such as feedforward neural networks.
• Uses a plug-and-play composition strategy to assemble complex systems from atomic components like neurons and reflex modules.

Experimental results

Research questions

• RQ1 How can networks with time-varying topologies be formally composed in a way that preserves consistency across scales?
• RQ2 What categorical structure enables the nesting of dynamical systems where connectivity depends on internal states?
• RQ3 How can a unified semantics model both local dynamics and global network topology in adaptive systems?
• RQ4 Can operads provide a formal foundation for modeling hierarchical, heterogeneous networks like the brain’s visual pathway?
• RQ5 What conditions ensure that mode-dependent network composition is independent of the zoom level or decomposition choice?

Key findings

• The operad of mode-dependent networks (OMDN) provides a consistent, compositional framework for systems where topology evolves with internal states.
• Modal dynamical systems are formally defined with state sets, mode functions, update rules, and readout functions that depend on communicative modes.
• The framework supports hierarchical composition: individual neurons can be abstracted as modules within larger networks, such as the visual cortex or blink reflex.
• A feedforward neural network can be constructed as a suboperad (OFFN) of OMDN, enabling modular design and composition.
• The visual system model VS is formally composed as a nested sequence of operations, reducing to basic components like nerves and neurons.
• The framework ensures that rewiring and composition are independent of the choice of decomposition, guaranteeing consistency across scales.