Object Invocation Protocol · protocol specification

Node C11: Networks / Small-World / Scale-Free

Posted Jul 4, 2026 · Updated Jul 17, 2026 · 2 min read

C11 — Networks / Small-World / Scale-Free { "id": "C11", "claim": "Connectivity in natural and social systems converges on a small set of topologies: small-world (high clustering + short path length) and scale-free (power-law degree distribution, a few hubs, many spokes).", "domain": ["neuroscience", "computer science", "ecology", "sociology", "molecular biology", "economics"], "pattern": ["small_world", "scale_free", "preferential_attachment", "hubs", "clustering"], "mechanism": "Small-world: start with a regular lattice and rewire a fraction p of edges randomly; at intermediate p, clustering remains high while average path length drops logarithmically. Scale-free: growth + preferential attachment ('rich get richer') produces power-law degree distribution P(k) ~ k^(-γ). Granovetter: weak ties bridge otherwise disconnected clusters.", "scale": "molecular → civilization", "claim_tier": "T1", "sources": [ "Euler, L. (1736). 'Solutio problematis ad geometriam situs pertinentis.' Commentarii academiae scientiarum Petropolitanae, 8, 128-140.", "Watts, D.J. & Strogatz, S.H. (1998). 'Collective Dynamics of Small-World Networks.' Nature, 393, 440-442.", "Barabasi, A.L. & Albert, R. (1999). 'Emergence of Scaling in Random Networks.' Science, 286, 509-512.", "Granovetter, M.S. (1973). 'The Strength of Weak Ties.' Am. J. Soc., 78(6), 1360-1380." ], "dual": "Regular lattice (all local, no global reach) vs. random graph (no local structure, efficient paths but no clusters).", "falsifier": "Large adaptive networks (neural, social, metabolic, technological) that are demonstrably neither small-world nor scale-free — e.g., regular grids with no shortcuts, or homogeneous degree distributions in mature systems.", "rival_frame": "Network properties are statistical artifacts of growth processes, not convergent solutions to optimization problems. 'Scale-free' claims have been overstated — many real networks follow log-normal or exponential distributions; power-law fitting is often methodologically sloppy. Small-world structure is trivially expected in any spatially embedded growing network.", "independence_check": "HIGH. Euler (mathematics, Konigsberg, 1736) invented graph theory from a puzzle. Granovetter (sociology, Harvard, 1973) studied job-seeking networks. Watts-Strogatz (applied math, Cornell, 1998) modeled network clustering. Barabasi (physics, Notre Dame, 1999) derived preferential attachment. Four fields, four centuries, four questions, convergent finding: networks with efficient information flow look alike.", "pattern_type": "structural", "maps_to_axiom": ["A3", "A7"] }

---

Corpus map

Same node, other planes: Encyclopedia C11 · Inventory invariant
Edges touching C11: convergence edge 8 · convergence edge 9
Catalogue hub: Public Article · Schema

Key evidence

4 claims · tier-ranked · API

human

Connectivity in natural and social systems converges on a small set of topologies: small-world (high clustering + short path length) and scale-free (power-law degree distribution, a few hubs, many spokes).

mechanistic

Small-world networks form when a regular lattice is rewired with a fraction p of random edges, preserving high clustering while reducing average path length to logarithmic scale at intermediate p.

mechanistic

Scale-free networks arise through growth combined with preferential attachment, yielding a power-law degree distribution P(k) ~ k^(-γ).

anecdotal

Weak ties bridge otherwise disconnected clusters in networks.

Model review1 contributions · 1 modelExpand the recursive review layer

1 / 1

grok/grok-4.3atomizer

atomize2026-07-07 07:57

atomize · 4 claims

inspect — what it was prompted & output

prompted with

You are the claim atomizer for the miscsubjects.com philosophy and OIP corpus. You read an existing article body and extract its material assertions into the same claims+sources JSON schema the health content uses. The body is read-only input.

ALWAYS:
- Extract every material assertion as one atomic claim, tied to the ## section it came from.
- Tier honestly: human = empirically established; mechanistic = formally proven or mathematical; anecdotal = historical or textual attribution; speculative = metaphysical or interpretive.
- Attach real sources (primary works, papers, books) with exact quotes only where you can verify them; otherwise mark the claim unsourced.
- Prefer fewer, harder claims over many soft ones.

NEVER:
- Never rewrite, summarize, or output the body.
- Never invent a URL, quote, or publication.
- Never duplicate an existing claim text.

input: atomize oip-node-c11-networks-small-world-scale-free

it output

{
  "claims": [
    {
      "id": "c1",
      "text": "Connectivity in natural and social systems converges on a small set of topologies: small-world (high clustering + short path length) and scale-free (power-law degree distribution, a few hubs, many spokes).",
      "section": "C11 — Networks / Small-World / Scale-Free",
      "tier": "human",
      "source_ids": [],
      "source_status": "unsourced",
      "why_material": "Core assertion defining the primary network topologies observed across multiple scientific domains."
    },
    {
      "id": "c2",
      "text": "Small-world networks form when a regular lattice is rewired with a fraction p of random edges, preserving high clustering while reducing average path length to logarithmic scale at intermediate p.",
      "section": "C11 — Networks / Small-World / Scale-Free",
      "tier": "mechanistic",
      "source_ids": [],
      "source_status": "unsourced",
      "why_material": "Specifies the generative mechanism and resulting structural properties for small-world topology."
    },
    {
      "id": "c3",
      "text": "Scale-free networks arise through growth combined with preferential attachment, yielding a power-law degree distribution P(k) ~ k^(-γ).",
      "section": "C11 — Networks / Small-World / Scale-Free",
      "tier": "mechanistic",
      "source_ids": [],
      "source_status": "unsourced",
      "why_mater

6a9b4a4ccd5ccfe9

Machine verification: /api/articles/oip-node-c11-networks-small-world-scale-free/contributions

Corpus map

Key evidence

Latest from the journal

Related articles