Topological constraints on self-organisation in locally interacting systems

TL;DR

This work investigates whether locally interacting systems on graphs can sustain an ordered phase and how topology controls this possibility, using windowed local Hamiltonians and a topological Peierls-type scaling argument with $\Delta F=\Delta E - T\Delta S$ and $E=O(P)$. They prove a topological equivalence theorem: all local Hamiltonians on lattices with the same combinatorial structure have asymptotically equivalent free energies, so order is determined by perimeter-based scaling on the interaction graph. Applied to three model families—1D Potts, autoregressive models, and hierarchical graphs—the results show that 1D Potts and autoregressive models do not support long-range order, while hierarchical topologies can exhibit hierarchical order (local coherence with global disorder) under certain temperature ranges. The findings explain why biological multiscale coordination can emerge while current language models struggle with long-range coherence and suggest topology-informed architectures or embodiment as potential remedies.

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 language models are challenged by long sequences of outputs.

Figures (8)

• Figure 1: A local Hamiltonian in dimension one. The Hamiltonian of this chain is a sum of windowed Hamiltonians of length $\omega = 5$. Pictured is the second window; the first begins at $s_{-1}$.
• Figure 2: Illustration of a graph Hamiltonian. Left: the graph $G$ represents the adjacency matrix of an undirected graph with six vertices, where edges indicate connections between vertices $s_i$. Right: the graph $H$ represents the graph Hamiltonian, where the coupling strengths $e_i$ have come from $\bar{H}$.
• Figure 3: Two different domains in a two-dimensional grid. The perimeter that separates the two domains is drawn in orange.
• Figure 4: Text generation in analogy to a morphogenetic process.
• Figure 5: An autoregressive model with $\omega = 5$.

Theorems & Definitions (27)

• Definition 1
• Remark
• Definition 2
• Definition 3
• Proposition 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof