finding

active

finding: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

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

Source paper

extracted_from

Topological constraints on self-organisation in locally interacting systems

(2025) · Francesco Sacco · Dalton A R Sakthivadivel · Michael Levin

Neighborhood — ranked by edge-count

Claims (1)

claim

Hierarchical structures in biological systems enable local order while globally disordered, explaining complex patterning.
supports
Claim that multiscale organisation produces complex patterns via clique-based local coherence

Communities (4)

community

Causal emergence in biological systems
members_of
Examines how macro-scale causal power exceeds micro-scale in living and learning systems.
Hierarchical structure and multiscale coherence in physical systems
members_of
How graph topology and hierarchical interaction patterns enable or prevent phase transitions and ordered states, from statistical mechanics to biological organization.
Hierarchical network ordering & thermodynamics
members_of
Statistical mechanics of clique-structured graphs linking domain walls, free energy, and biological multiscale coherence.
Critical temperature thresholds in hierarchical magnetic systems
members_of
Analyzes phase transitions and magnetization dynamics in clique-structured graphs using coupling strength and temperature-dependent bounds.

Findings (1)

finding

There exists a non-empty critical temperature range of hierarchical behaviour (Proposition 3)
supports
Proof that the conditions of Theorem 4 are realisable in a range of temperatures

Related by similarity (8)

cosine ≥ 0.65 · no typed edge

Entities in the same semantic neighborhood but without a typed relation to this one — candidates for new edges or unrecognized duplicates.

In hierarchical systems with independent cliques, there exist parameter regimes where individual cliques maintain uniform magnetisation while others flip.finding0.842
Shows how hierarchical topology enables local order within global flexibility; explains biological multiscale organization
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Core result demonstrating topological constraints on self-organization
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Sanity check for concept independence.
Local symmetries act as a kind of glue which holds space together; the more glue there is, the more the space is one, solid, unified, coherent—and the symmetrical segments must overlap for the glue to be effectiveclaim0.726
Interpretation of the experimental finding: overlapping local symmetries are the hidden structural feature that creates perceived wholeness
Deep interlock and ambiguity appear in systems that maximize surface area or allow dual belonging, e.g., cerebellum, magnetic domains, molecular bonds.claim0.717
EI of ER random networks converges to -log2(p) with increasing size, with a phase transition at average degree ≈ log2(N).finding0.716
From Klein & Hoel (2020) analysis of artificial complex networks.
spontaneous magnetisationconcept0.712
Nonzero net magnetisation in the absence of external field; hallmark of an ordered phase
The nearly identical detailed structure on all six arms of a single snow crystal is not explained by any present diffusion-aggregation model, implying non-local coordination.claim0.710
Alexander's use of snowflake arm symmetry as evidence that something beyond local mechanics is required in morphogenesis