Topological constraints on self-organization in locally interacting systems

Area of Science:

• Complex Systems
• Network Science
• Theoretical Neuroscience

Background:

• Intelligence, in all its forms, relies on the collective behavior of its constituent parts aligning with system-level objectives.
• Understanding the factors that enable or hinder this alignment is crucial for advancing fields such as life sciences and engineering.
• Spontaneous ordering in networks of interacting systems serves as a model for self-organization, relevant to both neural and basic cognitive processes.

Purpose of the Study:

• To identify necessary topological conditions within planar graphs for the existence of an ordered phase.
• To constrain the capacity of systems with local interactions to sustain a desired ordered state.
• To elucidate how graph combinatorics influence spontaneous ordering in interacting systems.

Main Methods:

• Analysis of free energy scaling during domain wall formation in three distinct model systems: the Potts model, autoregressive models, and hierarchical networks.
• Investigation of necessary conditions on graph topology for the emergence of an ordered phase.
• Comparative study of interaction combinatorics across different network structures.

Main Results:

• Demonstrated how the specific arrangement of interactions on a graph, dictated by its combinatorics, can either facilitate or impede spontaneous ordering.
• Identified key topological constraints that limit a system's ability to maintain an ordered target state.
• Provided insights into why biological multiscale systems excel at complex pattern organization, unlike rudimentary language models struggling with long sequences.

Conclusions:

• The topology and interaction structure of a network fundamentally govern its potential for self-organization and the maintenance of collective order.
• This framework explains the organizational differences observed between complex biological systems and simpler artificial intelligence models.
• Findings offer a theoretical basis for designing systems capable of robust self-organization and complex pattern formation.