paper
active
2026
paper:sacco-sakthivadivel-levin-topological-self-organization-2026

Topological constraints on self-organization in locally interacting systems

ByFrancesco Sacco·Dalton Sakthivadivel·Michael LevinⓘAllen Discovery Center At Tufts University, Allen Discovery Center, Tufts University + 10 more
DOI 10.1098/rsta.2025.0011OpenAlex W7161120125
Multiscale agency & collective intelligenceBioelectric cognition & collective individualityMultiscale CognitionUnconventional ComputationBasal CognitionCollective Intelligencedomain wallflat autoregressive LLMsfree energylocally interacting systemslong-range coherencelong-range orderMultiscale systemsneural cognitionphase transitionproblem spacesSelf-OrganizationTopological Constraints on Self-Organisation+1 more

TL;DR

Topological constraints on interaction graphs determine whether a locally interacting system can sustain long-range ordered phases, and therefore whether it can self-organize toward a system-level goal. By analyzing free-energy scaling under domain-wall formation across three model systems—the Potts model, autoregressive models, and hierarchical networks—the paper establishes necessary conditions on graph topology for an ordered phase to exist. The core method introduced is a domain-wall free-energy scaling analysis applied comparatively across these three substrate classes, tracking how the combinatorics of graph connectivity either suppress or permit spontaneous ordering. On planar graphs with the interaction topology characteristic of flat autoregressive language models, domain-wall formation is combinatorially cheap relative to system size, so long-range coherence degrades as sequence length grows; this is a structural, not a training, limitation. Hierarchical and multiscale networks—the topology prevalent in biological systems such as morphogenetic and neural substrates—make domain-wall formation sufficiently costly that ordered phases persist across scales. Published in the Phil. Trans. R. Soc. A theme issue on world models (384, issue 2320, 2026), the paper operationalizes the claim that all intelligence is collective intelligence via free-energy geometry, arguing that architectural hierarchy is not merely a performance heuristic but a topological prerequisite for sustained coherent self-organization, with direct implications for why biological morphogenesis succeeds at pattern maintenance where flat LLMs structurally cannot.

What to take away

  1. 1. Necessary conditions on graph topology for an ordered phase to exist are derived by analyzing how free energy scales with domain-wall formation across three distinct substrate classes: the Potts model, autoregressive models, and hierarchical networks.
  2. 2. In flat autoregressive language models, the planar or low-dimensional interaction topology makes domain-wall formation combinatorially inexpensive relative to sequence length, structurally explaining why long-range coherence degrades on long outputs independent of training quality.
  3. 3. Hierarchical networks, which share topological features with multiscale biological systems such as morphogenetic tissue networks, exhibit domain-wall free-energy scaling that is sufficient to sustain an ordered phase across scales.
  4. 4. The Potts model serves as the baseline statistical-mechanics reference substrate, with its known phase behavior used to calibrate the domain-wall scaling argument before it is extended to the two non-equilibrium substrates.
  5. 5. The paper introduces domain-wall free-energy scaling analysis as a unifying method for comparing self-organization capacity across graph topologies, applicable to any system with pairwise interactions prescribed by graph edges.
  6. 6. The claim 'all intelligence is collective intelligence' is operationalized precisely as the requirement that locally interacting parts align toward system-level goals, with topology determining whether the free-energy landscape permits stable alignment.
  7. 7. Published in Philosophical Transactions of the Royal Society A, volume 384, issue 2320 (2026), the paper appears in a theme issue specifically on world models in natural and artificial intelligence, situating the topological argument within that discourse.
  8. 8. An open question the paper raises is whether architectural modifications that impose hierarchical interaction topology on language models—rather than flat autoregressive connectivity—would recover the domain-wall cost scaling necessary for long-range coherence.
  9. 9. A replicable methodology choice is the use of planar graph vertex systems with pairwise edge-prescribed interactions as the formal substrate, allowing any researcher to apply the domain-wall scaling analysis to a new architecture by characterizing its effective interaction graph.
  10. 10. The structural argument implies that biological morphogenesis achieves complex stable patterning not incidentally but because multiscale hierarchical connectivity is a topological prerequisite for the ordered phases that constitute reliable pattern maintenance.

Peer brief — for seminar discussion

The paper, appearing in Phil. Trans. R. Soc. A volume 384, issue 2320 (2026) as part of a theme issue on world models in natural and artificial intelligence, addresses a foundational question about self-organization: given a system whose components interact only locally according to the edges of a graph, what topological properties of that graph are necessary for the system to exhibit long-range ordered phases? The approach is to study free-energy scaling under domain-wall formation—the method introduced here—across three model substrates: the Potts model as a statistical-mechanics baseline, autoregressive models as the representative flat-architecture language model class, and hierarchical networks as the representative multiscale biological architecture. The load-bearing finding is that the combinatorics of interaction connectivity on a graph either suppress or permit spontaneous ordering, and this is a structural property prior to any learning or dynamical process. Specifically, the planar or effectively low-dimensional interaction topology of flat autoregressive language models makes domain-wall formation cheap relative to sequence length, which is given as the structural reason why such models lose long-range coherence on long output sequences—a limitation that is architectural, not a deficit of training data or scale. Hierarchical networks, whose topology mirrors that of biological systems including morphogenetic and neural substrates, impose sufficient domain-wall free-energy cost that ordered phases are sustainable across scales, explaining why biology reliably produces complex stable patterns. The implication is that any system intended to maintain coherent global states through local interactions requires hierarchical interaction topology as a topological prerequisite, not merely as an engineering convenience. An alternative method that could have been used is renormalization group analysis, which also tracks how order survives scale transformations, and would have provided a complementary perturbative characterization of the phase boundary. One thing a critical reader would push back on is the scope of the generalization: the argument is developed formally for systems on graphs with pairwise interactions, but the mapping of real autoregressive transformer architectures—which involve attention mechanisms with dynamic, data-dependent effective connectivity rather than fixed graph edges—onto this formalism is asserted rather than derived, leaving open whether the flat-topology characterization accurately represents the effective interaction structure of a trained large language model. The paper raises the hypothesis that imposing hierarchical topology architecturally on language models would recover the domain-wall cost scaling required for long-range coherence, which is a testable prediction for empirical ML work.

Findings (2)

Claims (5)

Hypotheses (1)

Original abstract (expand)

All intelligence is collective intelligence, in the sense that it is made of parts that must align with respect to system-level goals. Understanding the dynamics that 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-organization, 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 organizing into complex patterns, whereas rudimentary language models are challenged by long sequences of outputs. This article is part of the theme issue ‘World models in natural and artificial intelligence’.

Related work— refs + corpus + external arXiv

Cited / in-corpus / arXiv badges show which signals surfaced each row. Multi-source rows weighted higher.

+23 more

Similar preprints — Semantic Scholar

Cross-corpus bridges (1)

same_concept_as · Nomic cosine

External markdown files that talk about the same concept as this entity.

  • aboutblank_kb
    Chris Buckleythinkers/chris-l-buckley.md0.781