Latest Research Papers
2025-01-22
arXiv
Topological constraints on self-organisation in locally interacting systems
The paper explores the conditions under which systems with local interactions can self-organize into ordered phases, using models like the Potts model and autoregressive models. It identifies topological constraints that influence the ability of such systems to maintain order, explaining why complex biological systems can form intricate patterns while simple language models struggle with long sequences.
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.