Hierarchical structure and multiscale coherence in physical systems

How graph topology and hierarchical interaction patterns enable or prevent phase transitions and ordered states, from statistical mechanics to biological organization.

21 members. Each node is clickable.

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Sub-communities (9)

Finer clusters this community splits into. Each is its own community page.

Lattice topology and thermodynamic phase transitions3 Finite-temperature limitations of local systems2 Topology and hierarchy in self-organization2 Hierarchical network topology and emergent dynamics2 Hierarchical spatial organization in biology2 Hierarchical competence and organizational emergence2 Combinatorial constraints on emergent ordering in networks2 Scaling laws and phase transitions2 Critical temperature thresholds in hierarchical magnetic systems2

Drawn from 7 sources

The papers/notes whose extracted claims & findings make up this cluster.

Topological constraints on self-organisation in locally interacting systems11 members
Topological constraints on self-organization in locally interacting systems3 members
2026-05-15_manifold-overlap-papers-economy-strategy.md2 members
2026-05-14_phil-trans-A-goodfire-aboutblank-impact.md2 members
Design for an Individual: Connectionist Approaches to the Evolutionary Transitions in Individuality1 member
Diagrammatic Writing1 member
Collective intelligence: A unifying concept for integrating biology across scales and substrates1 member

Bridges (16)

Other communities that share members with this one — cross-cutting threads or papers that sit at the seam between two themes.

Causal emergence in biological systems21 shared
Hierarchical network ordering & thermodynamics12 shared
Lattice topology and thermodynamic phase transitions3 shared
Critical temperature thresholds in hierarchical magnetic systems2 shared
Hierarchical spatial organization in biology2 shared
Hierarchical competence and organizational emergence2 shared
Combinatorial constraints on emergent ordering in networks2 shared
Scaling laws and phase transitions2 shared
Autoregressive LLMs & formal thought disorder2 shared
Finite-temperature limitations of local systems2 shared
Topology and hierarchy in self-organization2 shared
Hierarchical network topology and emergent dynamics2 shared
Christopher Alexander's 15 properties clustering1 shared
Lattice Hamiltonian free energy universality1 shared
Geometric concept representations in neural networks1 shared
Hopfield network thermal equilibrium convergence1 shared

Claims (14)

Hierarchical structure in interaction topology enables complex multiscale patterns that cannot exist in flat networks.Explains why biological systems achieve organization across scales while language models struggle; grounds in free energy scaling
Hierarchical structures in biological systems enable local order while globally disordered, explaining complex patterning.Claim that multiscale organisation produces complex patterns via clique-based local coherence
Hierarchy and subordination through spatial organizationDrucker argues that indentation, size, placement, and relative position create hierarchies not as moral values but as relational effects within a system.
Higher-order entities distort the energy landscape for their subunits, benefiting from their competencies to navigate spaces of which the subunits are unaware.Explains how top-down causation leverages lower-level problem-solving for large-scale goals.
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.Conclusion about why biology organizes complexity well and flat LLMs do not
Organised structure of relationships between component parts causes them to work together, creating new organismic entity and evolutionary unit
Spontaneous ordering in networks of interacting systems can be viewed as a form of self-organization, modelling neural and basal forms of cognition.Claim linking physical self-organization to cognition
The results generalise readily to non-equilibrium systems where scaling relationships remain similar (e.g., dynamic or localised scaling).Claim about broader applicability of the scaling argument
Topology is the critical factor differentiating systems capable of long-range order from those that are not.Key interpretive position: topological properties of interaction graphs determine whether systems can self-organize, independent of substrate
Topology is the critical factor differentiating the self-organising capabilities of biological systems and language models.Central interpretive claim of the paper: the ability to maintain long-range order is determined by interaction topology, not substrate.
Hierarchical clique-structured networks can maintain multiscale coherence; biology evolved hierarchy and stigmergy as coherence solutions.
LLMs require hierarchical or embodied substrates; stigmergy via environment interaction is the biological solution.
Purely local autoregressive systems cannot maintain long-range coherence at finite temperature.
Purely local-interaction systems on 1D topologies cannot maintain long-range order at finite temperature.

Findings (7)

All local Hamiltonians on lattices with the same combinatorial structure have asymptotically equivalent free energies (Theorem 1)Topological equivalence theorem for local Hamiltonians
At thermal equilibrium, ability to converge to an ordered phase is independent of energy levels and window sizes (Lemma 1)Scaling argument depends only on perimeter, not details of energy magnitudes or window length
For a graph with independent cliques, individual cliques may flip magnetisation while remaining uniformly magnetised if intra-clique coupling > (T/2) log n_i (Theorem 4)Condition for hierarchical order with locally coherent but globally varying phases
For one-dimensional local Hamiltonian with m>1 stored patterns at non-zero temperature, domain wall formation is thermodynamically favourable (Theorem 2)No ordered phase in 1D with multiple stored patterns
Free-energy scaling under domain-wall formation in Potts, autoregressive, and hierarchical networks shows that combinatorics of interactions on a graph prevent or allow spontaneous ordering.Core result demonstrating topological constraints on self-organization
In hierarchical systems with independent cliques, there exist parameter regimes where individual cliques maintain uniform magnetisation while others flip.Shows how hierarchical topology enables local order within global flexibility; explains biological multiscale organization
There exists a non-empty critical temperature range of hierarchical behaviour (Proposition 3)Proof that the conditions of Theorem 4 are realisable in a range of temperatures