{"slug":"oip-invariant-16-316-networks-small-world-scale-free","title":"Invariant 3.16 — Networks / small-world / scale-free","body":"## 3.16 Networks / small-world / scale-free\nConnectivity converges on a few topologies. **Sources:** Euler (graph theory); Watts–Strogatz (small-world); Barabási (scale-free, preferential attachment); Granovetter (weak ties). **Domains:** neurons, internet, food webs, metabolism, society. **Dual:** regular lattice vs random graph (small-world sits between — a criticality echo). **Tier:** T1. **Falsifier:** large adaptive networks that are neither small-world nor scale-free. **Maps:** A₃, A₇. *(Your voxel graph is itself an instance.)*\n\n---\n\n## Corpus map\n- Same invariant, other planes: [Catalogue node C11](/a/oip-node-c11-networks-small-world-scale-free) · [Encyclopedia C11](/a/convergence-encyclopedia-c11)\n- Inventory hub: [Convergence Catalogue — Public Article](/a/oip-convergence-public-article)","hero":null,"images":[],"style":{},"tags":["philosophy","oip","catalogue","invariant","systems-theory"],"model":null,"ledger":null,"embeds":[],"widgets":[],"home":true,"claims":[],"sources":[],"reviews":[],"extra":{"kind":"corpus","corpus_map":{"prev":"oip-invariant-15-315-optimization-under-constraint-pareto-fronts","next":"oip-invariant-17-317-attractors-dynamical-systems-chaos","hub":"oip-convergence-public-article","series":"catalogue-inventory-invariants","position":16,"of":22}},"register":"oip_protocol","status":"published","revisions":2,"contributions":[],"provenance":[{"ts":"2026-07-04T04:33:48.635Z","model":"claude-fable-5","action":"edit","prompt":"","input":"","response":"","tokens_in":0,"tokens_out":0,"cost":0,"prev":"genesis","hash":"3edad9149470993ffa2228344e6d4f9f98877bf11cfdad7e80f2faf207dabfb7"},{"ts":"2026-07-04T05:01:46.987Z","model":"claude-fable-5","action":"edit","prompt":"","input":"","response":"","tokens_in":0,"tokens_out":0,"cost":0,"prev":"3edad9149470993ffa2228344e6d4f9f98877bf11cfdad7e80f2faf207dabfb7","hash":"5de8752996997b52794b8a12fc58d2d1415c3d3a36f3b082762d55607914b1a6"}],"energy":{"passes":2,"tokens_in":0,"tokens_out":0,"tokens_total":0,"cost_usd":0,"models":{"claude-fable-5":2},"head":"5de8752996997b52794b8a12fc58d2d1415c3d3a36f3b082762d55607914b1a6"},"posted_at":"2026-07-04T02:40:34.621Z","created_at":"2026-07-04T02:40:34.621Z","updated_at":"2026-07-04T05:01:46.987Z","machine":{"shape":"article.machine/v1","slug":"oip-invariant-16-316-networks-small-world-scale-free","kind":"corpus","read":{"human":"https://miscsubjects.com/a/oip-invariant-16-316-networks-small-world-scale-free","json":"https://miscsubjects.com/api/articles/oip-invariant-16-316-networks-small-world-scale-free","bundle":"https://miscsubjects.com/api/articles/oip-invariant-16-316-networks-small-world-scale-free/bundle?format=markdown"},"traversal":{"prev":{"slug":"oip-invariant-15-315-optimization-under-constraint-pareto-fronts","human":"https://miscsubjects.com/a/oip-invariant-15-315-optimization-under-constraint-pareto-fronts","json":"https://miscsubjects.com/api/articles/oip-invariant-15-315-optimization-under-constraint-pareto-fronts"},"next":{"slug":"oip-invariant-17-317-attractors-dynamical-systems-chaos","human":"https://miscsubjects.com/a/oip-invariant-17-317-attractors-dynamical-systems-chaos","json":"https://miscsubjects.com/api/articles/oip-invariant-17-317-attractors-dynamical-systems-chaos"},"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-inventory-invariants","position":16,"of":22},"ledger":{"claims":0,"sources":0,"contributions":0,"revisions":2,"objections_url":"https://miscsubjects.com/api/articles/oip-invariant-16-316-networks-small-world-scale-free/objections","thread_state_url":"https://miscsubjects.com/api/protocol/thread-state?target=oip-invariant-16-316-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-invariant-16-316-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-invariant-16-316-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-invariant-16-316-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-invariant-16-316-networks-small-world-scale-free\",\"raw_text\":\"<material delta>\"}'  # open intake, no key","read_back":"curl -s https://miscsubjects.com/api/articles/oip-invariant-16-316-networks-small-world-scale-free | python3 -c 'import json,sys; d=json.load(sys.stdin); print(json.dumps(d[\"claims\"][-3:], indent=1))'"}}}