Hierarchical network ordering & thermodynamics

Statistical mechanics of clique-structured graphs linking domain walls, free energy, and biological multiscale coherence.

13 members. Each node is clickable.

Loading graph…

Drawn from 3 sources

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

Topological constraints on self-organisation in locally interacting systems10 members
Topological constraints on self-organization in locally interacting systems3 members
2026-05-14_phil-trans-A-goodfire-aboutblank-impact.md1 member

Bridges (11)

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 systems12 shared
Hierarchical structure and multiscale coherence in physical systems12 shared
Combinatorial constraints on emergent ordering in networks2 shared
Lattice topology and thermodynamic phase transitions2 shared
Topology and hierarchy in self-organization2 shared
Hierarchical network topology and emergent dynamics2 shared
Scale-free goal-directed agency1 shared
Hierarchical spatial organization in biology1 shared
Scaling laws and phase transitions1 shared
Critical temperature thresholds in hierarchical magnetic systems1 shared
Relational self, care & aliveness1 shared

Claims (9)

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
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
Self-organisation can be viewed as a form of autopoietic cognition navigating problem spaces toward target morphologies.Linking self-organisation to cognition and navigation of configuration space
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.

Findings (4)

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