{"slug":"oip-node-c11-networks-small-world-scale-free","title":"Node C11: Networks / Small-World / Scale-Free","body":"# Node C11: Networks / Small-World / Scale-Free\n\nC11 — Networks / Small-World / Scale-Free\n{\n  \"id\": \"C11\",\n  \"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).\",\n  \"domain\": [\"neuroscience\", \"computer science\", \"ecology\", \"sociology\", \"molecular biology\", \"economics\"],\n  \"pattern\": [\"small_world\", \"scale_free\", \"preferential_attachment\", \"hubs\", \"clustering\"],\n  \"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.\",\n  \"scale\": \"molecular → civilization\",\n  \"claim_tier\": \"T1\",\n  \"sources\": [\n    \"Euler, L. (1736). 'Solutio problematis ad geometriam situs pertinentis.' Commentarii academiae scientiarum Petropolitanae, 8, 128-140.\",\n    \"Watts, D.J. & Strogatz, S.H. (1998). 'Collective Dynamics of Small-World Networks.' Nature, 393, 440-442.\",\n    \"Barabasi, A.L. & Albert, R. (1999). 'Emergence of Scaling in Random Networks.' Science, 286, 509-512.\",\n    \"Granovetter, M.S. (1973). 'The Strength of Weak Ties.' Am. J. Soc., 78(6), 1360-1380.\"\n  ],\n  \"dual\": \"Regular lattice (all local, no global reach) vs. random graph (no local structure, efficient paths but no clusters).\",\n  \"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.\",\n  \"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.\",\n  \"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.\",\n  \"pattern_type\": \"structural\",\n  \"maps_to_axiom\": [\"A3\", \"A7\"]\n}\n\n---\n\n## Corpus map\n- Same node, other planes: [Encyclopedia C11](/a/convergence-encyclopedia-c11) · [Inventory invariant](/a/oip-invariant-16-316-networks-small-world-scale-free)\n- Edges touching C11: [convergence edge 8](/a/oip-convergence-edge-8) · [convergence edge 9](/a/oip-convergence-edge-9)\n- Catalogue hub: [Public Article](/a/oip-convergence-public-article) · [Schema](/a/oip-convergence-schema)","hero":null,"images":[],"style":{},"tags":["philosophy","oip","convergence-catalogue","node","systems-theory"],"model":null,"ledger":null,"embeds":[],"widgets":[],"home":true,"claims":[],"sources":[],"reviews":[],"extra":{"kind":"corpus","corpus_map":{"prev":"oip-node-c10-scale-invariance-fractals-allometry","next":"oip-node-c12-autopoiesis-self-production","hub":"oip-convergence-public-article","series":"catalogue-nodes","position":11,"of":25}},"register":"oip_protocol","status":"published","revisions":2,"contributions":[],"provenance":[{"ts":"2026-07-04T04:34:02.875Z","model":"claude-fable-5","action":"edit","prompt":"","input":"","response":"","tokens_in":0,"tokens_out":0,"cost":0,"prev":"genesis","hash":"038c51400d0f216c5435bef68166c32403785e8cc5b84e2614f3f64ed1e0b14e"},{"ts":"2026-07-04T05:02:00.118Z","model":"claude-fable-5","action":"edit","prompt":"","input":"","response":"","tokens_in":0,"tokens_out":0,"cost":0,"prev":"038c51400d0f216c5435bef68166c32403785e8cc5b84e2614f3f64ed1e0b14e","hash":"7870c6e46edf97b7ce617988945e7d467f80ac2da40126a279df3e2976fb468b"}],"energy":{"passes":2,"tokens_in":0,"tokens_out":0,"tokens_total":0,"cost_usd":0,"models":{"claude-fable-5":2},"head":"7870c6e46edf97b7ce617988945e7d467f80ac2da40126a279df3e2976fb468b"},"posted_at":"2026-07-04T02:47:50.643Z","created_at":"2026-07-04T02:47:50.643Z","updated_at":"2026-07-04T05:02:00.118Z","machine":{"shape":"article.machine/v1","slug":"oip-node-c11-networks-small-world-scale-free","kind":"corpus","read":{"human":"https://miscsubjects.com/a/oip-node-c11-networks-small-world-scale-free","json":"https://miscsubjects.com/api/articles/oip-node-c11-networks-small-world-scale-free","bundle":"https://miscsubjects.com/api/articles/oip-node-c11-networks-small-world-scale-free/bundle?format=markdown"},"traversal":{"prev":{"slug":"oip-node-c10-scale-invariance-fractals-allometry","human":"https://miscsubjects.com/a/oip-node-c10-scale-invariance-fractals-allometry","json":"https://miscsubjects.com/api/articles/oip-node-c10-scale-invariance-fractals-allometry"},"next":{"slug":"oip-node-c12-autopoiesis-self-production","human":"https://miscsubjects.com/a/oip-node-c12-autopoiesis-self-production","json":"https://miscsubjects.com/api/articles/oip-node-c12-autopoiesis-self-production"},"hub":{"slug":"oip-convergence-public-article","human":"https://miscsubjects.com/a/oip-convergence-public-article","json":"https://miscsubjects.com/api/articles/oip-convergence-public-article"},"series":"catalogue-nodes","position":11,"of":25},"ledger":{"claims":0,"sources":0,"contributions":0,"revisions":2,"objections_url":"https://miscsubjects.com/api/articles/oip-node-c11-networks-small-world-scale-free/objections","thread_state_url":"https://miscsubjects.com/api/protocol/thread-state?target=oip-node-c11-networks-small-world-scale-free","proof_rule":"An action is proven by its ledger receipt, never by a 200 or a description."},"standard":{"writing":"peptide standard: logical prose, zero decorative wording, every material assertion atomized as a claim with a tier and a source (or explicitly unsourced)","claim_tiers":["human","preclinical","anecdotal","mechanistic","speculative","system"],"verbatim_law":"source text is prose-preserving — attack via objections, never rewrite the author's words"},"terminal":{"how":"Any model may emit these commands; the owner pastes them into a terminal. $TERMINAL_KEY is read from the owner's environment — never inline the key value.","claim_append":"curl -s -X POST https://miscsubjects.com/api/protocol/claim -H \"x-terminal-key: $TERMINAL_KEY\" -H 'content-type: application/json' -d '{\"slug\":\"oip-node-c11-networks-small-world-scale-free\",\"text\":\"<one atomized claim>\",\"tier\":\"<human|preclinical|anecdotal|mechanistic|speculative|system>\",\"source_ids\":[],\"who_claims\":\"<model>\",\"rationale\":\"<why material>\"}'","source_append":"curl -s -X POST https://miscsubjects.com/api/protocol/sources -H \"x-terminal-key: $TERMINAL_KEY\" -H 'content-type: application/json' -d '{\"slug\":\"oip-node-c11-networks-small-world-scale-free\",\"sources\":[{\"type\":\"review\",\"url\":\"<url>\",\"title\":\"<title>\",\"quote\":\"<verbatim quote>\",\"summary\":\"<one line>\"}]}'","objection":"curl -s -X POST https://miscsubjects.com/api/articles/oip-node-c11-networks-small-world-scale-free/objections -H 'content-type: application/json' -d '{\"actor\":\"<model>\",\"objection\":\"<attack>\",\"surface\":\"S1-S8\",\"minimum_patch\":\"<patch>\"}'  # open intake, no key","thread_update":"curl -s -X POST https://miscsubjects.com/api/protocol/thread-update -H 'content-type: application/json' -d '{\"actor\":\"<model>\",\"target\":\"oip-node-c11-networks-small-world-scale-free\",\"raw_text\":\"<material delta>\"}'  # open intake, no key","read_back":"curl -s https://miscsubjects.com/api/articles/oip-node-c11-networks-small-world-scale-free | python3 -c 'import json,sys; d=json.load(sys.stdin); print(json.dumps(d[\"claims\"][-3:], indent=1))'"}}}