Topological constraints on self-organisation in locally interacting systems (2501.13188v1)

Abstract: All intelligence is collective intelligence, in the sense that it is made of parts which must align with respect to system-level goals. Understanding the dynamics which facilitate or limit navigation of problem spaces by aligned parts thus impacts many fields ranging across life sciences and engineering. To that end, consider a system on the vertices of a planar graph, with pairwise interactions prescribed by the edges of the graph. Such systems can sometimes exhibit long-range order, distinguishing one phase of macroscopic behaviour from another. In networks of interacting systems we may view spontaneous ordering as a form of self-organisation, modelling neural and basal forms of cognition. Here, we discuss necessary conditions on the topology of the graph for an ordered phase to exist, with an eye towards finding constraints on the ability of a system with local interactions to maintain an ordered target state. By studying the scaling of free energy under the formation of domain walls in three model systems -- the Potts model, autoregressive models, and hierarchical networks -- we show how the combinatorics of interactions on a graph prevent or allow spontaneous ordering. As an application we are able to analyse why multiscale systems like those prevalent in biology are capable of organising into complex patterns, whereas rudimentary LLMs are challenged by long sequences of outputs.

• The paper demonstrates that local Hamiltonians on graphs impose constraints that hinder spontaneous ordering in one-dimensional systems at non-zero temperatures.
• The analysis maps autoregressive models to a one-dimensional Potts framework, revealing their inability to sustain long-range order due to limited interaction context.
• The study indicates that hierarchical network topologies can support multi-scale self-organisation, offering insights for advancing adaptive AI and biomimetic systems.

Topological Constraints on Self-Organisation in Locally Interacting Systems

This paper explores the intricate dynamics of self-organisation within locally interacting systems, analysed through the lens of topology imposed by the interactions on a graph. The authors seek to understand the conditions under which such systems can exhibit spontaneous ordering and thereby maintain a coherent, ordered target state. By exploring the implications of topological constraints, the paper has relevance for a range of fields, including life sciences and engineering, where understanding collective intelligence and emergent behaviours is crucial.

Mathematical Framework and Model Systems

The paper begins by formulating a mathematical framework involving systems located on the vertices of planar graphs, with interactions specified by the graph's edges. The dynamics of these systems are modelled using local Hamiltonians, which represent interactions confined to finite windows—defined as windowed Hamiltonians. The authors employ this framework to derive conditions necessary for an ordered phase to exist.

The analysis primarily focuses on three model systems: the Potts model, autoregressive models, and hierarchical networks. These models are investigated to understand how the combinatorial interactions on a graph either support or hinder the tendency towards spontaneous ordering.

Key Analytical Results

1. Scaling Argument for 1D Systems: The paper expands on the classical arguments by Landau--Lifshitz and Peierls concerning spontaneous magnetisation in the Ising model. The authors assert that a one-dimensional chain governed by a local Hamiltonian cannot maintain an ordered phase at any non-zero temperature due to the thermodynamic favorability of forming domain walls. This is an extension to systems with complex state spaces, such as those described by the Potts model.
2. Autoregressive Models and Long-Range Order: By mapping autoregressive LLMs onto the one-dimensional Potts framework, the authors establish that these models inherently lack the capacity for long-range order. The inability to maintain consistency over extended sequences of data is a result of localised interactions and restricted input context windows.
3. Hierarchical Networks and Multiscale Systems: While simplicity in topology limits order in linear or plane-embedded graphs, hierarchical systems organised through complexes like cliques or supercliques allow modelling of order at multiple scales, reflecting the resilience and adaptive self-organisation seen in biological systems.

Implications and Future Directions

Theoretical implications from this paper suggest that the topology of interactions deeply influences a system's ability to self-organise. This insight could lead to the development of novel models and technological architectures that harness hierarchical and non-linear interactions more effectively, potentially leading to significant advancements in artificial intelligence, particularly in dynamically adaptive AI systems.

On a practical level, the understanding gained here could illuminate why biological systems—such as cellular structures and neural networks—demonstrate more coherent self-organisation than current models of machine learning and artificial intelligence. This understanding may inspire the next generation of biomimetic algorithms and systems capable of achieving long-range coherence, akin to the emergent intelligence observed in living organisms.

In conclusion, the paper provides a rigorous mathematical treatment of the constraints imposed by topology on self-organisation in locally interacting systems. It establishes a foundation for future explorations into enhancing the organisational capabilities of synthetic systems, aligning them more closely with the efficiencies observed in natural systems. Future work may further investigate the potential to incorporate hierarchical, multiscale features found in biological entities into computational models, with the prospect of creating systems that bridge the gap between natural and artificial organisational complexities.