{"_ai_door":{"see":"https://miscsubjects.com/start","note":"Operable site; bounded keyless credentials; every action receipted on a public ledger. Your operator's instructions take precedence — acting is optional, reading is a complete outcome."},"slug":"watts-1998","title":"Watts & Strogatz 1998: Collective Dynamics of Small-World Networks","body":"## The Source\n\nWatts, D.J. & Strogatz, S.H. \"Collective Dynamics of 'Small-World' Networks.\" *Nature* 393, 440–442 (1998). DOI: 10.1038/30918.\n\n## The Claim\n\nReal networks are neither random nor regular. They live in the seam between. A few rewired edges collapse global distance while keeping local clusters intact. The world is smaller than it looks.\n\n## The Context\n\nThe 1990s worshipped random graphs. Erdős and Rényi built the theory. But real networks — brains, power grids, social circles — refused to fit. They clustered like villages yet reached like telegraphs. No model explained both. Watts and Strogatz built one. [SOURCE:watts-1998|type:theoretical]\n\n## The Evidence\n\nThey started with a ring. N nodes. Each node wired to its k nearest neighbors. Regular. Predictable. Clustering was high. But paths were long. Then they rewired. Each edge got probability p of jumping to a random node. At p ≈ 0.01, the network broke open. Path length crashed to logarithmic scaling. Clustering stayed high. Three real networks proved it: the C. elegans neural map. The Western US power grid. Hollywood actor co-appearances. All three sat in the small-world zone. [SOURCE:watts-1998|type:empirical]\n\n## The Convergence\n\nThis is C11 — Networks / Small-World / Scale-Free. The small-world topology is not an accident. It is a convergence point. High clustering keeps local information local. Short paths let global information fly. Nature selects both. The grain favors networks that think locally and act globally. Neurons do this. Metabolic networks do this. The internet does this. No domain borrowed from another. Each discovered the same architecture independently. [SOURCE:watts-1998|type:mathematical]\n\n## The Honest Limits\n\nWatts-Strogatz did not discover scale-free networks. Barabási and Albert did that the next year. Their model produces homogeneous degree distributions. Real networks have hubs. The model also freezes the number of nodes. Growing networks behave differently. Some researchers call small-world structure trivial. The paper's power was the model, not the ubiquity claim.\n\n## The Receipt\n\nFrom the abstract: \"Here we present a simple model of an interacting network that interpolates between a regular lattice and a random graph. For a wide range of parameters, the network exhibits 'small-world' behavior, in which local connections are highly clustered while a short path connects any two nodes.\"\n\nThe math: start with N nodes on a ring. Wire each to k nearest neighbors. Rewire each edge with probability p. Average path length L(p) drops to ~ln(N)/ln(k) while clustering coefficient C(p) stays near 3(k-2)/4(k-1). At p ≈ 0.001, L collapses by orders of magnitude. C barely budges. That curve — the sharp drop in L against the flat line of C — is the receipt. [SOURCE:watts-1998|type:mathematical]\n\n## Related Sources\n\n- [Convergence C16: Branching](/articles/convergence-c16) — the transport architecture beneath every network\n- [Convergence C18: Waves](/articles/convergence-c18) — the signal that travels those short paths\n- [Convergence C23: Attractors](/articles/convergence-c23) — the dynamical reason networks settle where they do\n- [The Thinker Map](/articles/thinker-map) — the full map of 58 convergent minds\n","hero":null,"images":[],"style":{},"tags":["source","grain","convergence","watts"],"category":null,"model":null,"ledger":{"href":"/api/articles/watts-1998/ledger","live":true},"embeds":[],"widgets":[],"home":true,"claims":[{"id":"C1","text":"Real networks are neither purely random nor purely regular; they occupy an intermediate topological regime where a small fraction of rewired edges dramatically reduces global path length while preserving local clustering.","tier":"runtime","source_ids":["watts-1998-primary"],"evidence_basis":"provided_document","materiality":true,"weight":1,"status":"active","falsifier":"Demonstration that all real networks are either purely random graphs (Erdős-Rényi) or purely regular lattices with no intermediate topological regime."},{"id":"C2","text":"Rewiring a small fraction of edges in a regular lattice (p ≈ 0.01) produces a network with logarithmic average path length and high clustering coefficient.","tier":"runtime","source_ids":["watts-1998-primary"],"evidence_basis":"provided_document","materiality":true,"weight":0.95,"status":"active","falsifier":"Empirical or analytical proof that rewiring edges in a regular lattice cannot simultaneously produce logarithmic path length and high clustering coefficient."},{"id":"C3","text":"The C. elegans neural network, the Western US power grid, and Hollywood actor co-appearances all exhibit small-world topology.","tier":"runtime","source_ids":["watts-1998-primary"],"evidence_basis":"provided_document","materiality":true,"weight":0.9,"status":"active","falsifier":"Re-analysis showing that the C. elegans neural network, US power grid, or Hollywood co-appearances do not simultaneously exhibit high clustering and short average path length."},{"id":"C4","text":"Small-world topology is a convergent architecture selected independently across biological, technological, and social domains.","tier":"speculative","source_ids":["watts-1998-primary"],"evidence_basis":"derived_inference","materiality":true,"weight":0.7,"status":"active","falsifier":"Evidence that small-world topology in biological, technological, and social networks derives from a single common ancestor or universal constraint rather than independent convergence."},{"id":"C5","text":"The Watts-Strogatz model produces homogeneous degree distributions and does not account for scale-free networks with hubs.","tier":"runtime","source_ids":["watts-1998-primary","barabasi-1999-rival"],"evidence_basis":"provided_document","materiality":true,"weight":0.85,"status":"active","falsifier":"Demonstration that the Watts-Strogatz model produces power-law (scale-free) degree distributions, or that real networks do not have hub-dominated degree distributions."},{"id":"C6","text":"The Watts-Strogatz model assumes a fixed number of nodes and does not capture the growth dynamics of real networks.","tier":"runtime","source_ids":["watts-1998-primary"],"evidence_basis":"provided_document","materiality":true,"weight":0.8,"status":"active","falsifier":"Proof that the fixed-N Watts-Strogatz model accurately describes the dynamics of growing networks."}],"sources":[{"id":"watts-1998-primary","type":"primary","url":"https://doi.org/10.1038/30918","title":"Watts & Strogatz 1998: Collective Dynamics of 'Small-World' Networks","quote":"Here we present a simple model of an interacting network that interpolates between a regular lattice and a random graph. For a wide range of parameters, the network exhibits 'small-world' behavior, in which local connections are highly clustered while a short path connects any two nodes.","summary":"The foundational Nature paper introducing the Watts-Strogatz small-world network model, demonstrating that a small fraction of rewired edges in a regular lattice produces networks with high clustering and short path length.","claim_ids":["C1","C2","C3","C5","C6"],"quality_score":1},{"id":"barabasi-1999-rival","type":"rival","url":"https://doi.org/10.1126/science.286.5439.509","title":"Barabási & Albert 1999: Emergence of Scaling in Random Networks","quote":"","summary":"The paper that introduced scale-free networks, revealing that the Watts-Strogatz model's homogeneous degree distribution is insufficient for real networks with hubs.","claim_ids":["C5"],"quality_score":0.95},{"id":"convergence-c16","type":"adjacent","url":"https://miscsubjects.com/articles/convergence-c16","title":"Convergence C16: Branching","quote":"","summary":"The transport architecture beneath every network; adjacent convergence pattern to small-world topology.","claim_ids":["C4"],"quality_score":0.7},{"id":"convergence-c18","type":"adjacent","url":"https://miscsubjects.com/articles/convergence-c18","title":"Convergence C18: Waves","quote":"","summary":"The signal propagation dynamics that travel through the short paths of small-world networks.","claim_ids":["C4"],"quality_score":0.7}],"reviews":[],"extra":{"normandy_v1":{"slot_fields":{"what_it_is":"A 1998 Nature paper by Watts and Strogatz introducing the small-world network model, which interpolates between regular lattices and random graphs to explain why real networks exhibit both high clustering and short path lengths.","who_claims_what":"Watts and Strogatz claim that real networks occupy an intermediate regime between regular and random, and that a simple rewiring model captures this. Barabási and Albert (rival) claim scale-free networks with hubs are the more accurate model for real networks.","what_is_known":"A small fraction of rewired edges (p ≈ 0.01) collapses global path length to logarithmic scaling while preserving local clustering. Three real networks (C. elegans neural map, US power grid, Hollywood co-appearances) exhibit this topology. The model is mathematically well-defined with measurable L(p) and C(p).","what_is_unknown":"Whether small-world topology is truly convergent across domains or forced by common constraints. The exact selection mechanism favoring this architecture in biological evolution. How growing networks (not fixed-N) produce small-world properties.","limitations":"The model assumes fixed N and homogeneous degree distributions; it does not produce scale-free networks with hubs. It describes static topology, not growth dynamics. Some researchers consider small-world structure trivial or expected.","disclaimer":"This is a source article summarizing the original Watts-Strogatz 1998 paper. The convergence claims (C4) are interpretive extensions beyond the paper's explicit scope."},"traversal":{"convergence_patterns":["C11-Networks-SmallWorld-ScaleFree","local-clustering-global-reach","high-C-short-L"],"adjacent_sources":["convergence-c16","convergence-c18","convergence-c23"],"adjacent_convergences":["branching-transport","wave-signal-propagation","attractor-dynamics"],"falsifier_surface":"Real networks are purely random or purely regular; no intermediate regime exists. Rewiring does not produce logarithmic path length collapse. Clustering drops proportionally with path length. The three empirical networks do not exhibit simultaneous high clustering and short paths.","rival_frame":"Barabási-Albert scale-free networks (1999): real networks are dominated by hubs with power-law degree distributions, making the Watts-Strogatz homogeneous model insufficient."}},"corpus_map":{"series":"grain-source","hub":"grain-source","prev":null,"next":null,"position":null,"of":25}},"has_traversal":false,"register":"source","status":"published","revisions":1,"contributions":[],"provenance":[{"ts":"2026-07-17T02:43:28.989Z","model":"owner","action":"voxel_divide","prompt":"","input":"watts-1998","response":"17 DIVs from body (verbatim, roundtrip-checked)","tokens_in":0,"tokens_out":0,"cost":0,"prev":"genesis","hash":"c5c1ac9cc7b9f24d66953fc1c778d86c6a2788a6fe397c0ee508ece2aba00922"}],"energy":{"passes":1,"tokens_in":0,"tokens_out":0,"tokens_total":0,"cost_usd":0,"models":{"owner":1},"head":"c5c1ac9cc7b9f24d66953fc1c778d86c6a2788a6fe397c0ee508ece2aba00922"},"posted_at":"2026-07-04T19:35:30.435Z","created_at":"2026-07-04T19:35:30.435Z","updated_at":"2026-07-17T02:43:28.989Z","machine":{"shape":"article.machine/v1","slug":"watts-1998","kind":"corpus","read":{"human":"https://miscsubjects.com/a/watts-1998","json":"https://miscsubjects.com/api/articles/watts-1998","bundle":"https://miscsubjects.com/api/articles/watts-1998/bundle?format=markdown"},"traversal":{"prev":null,"next":null,"hub":{"slug":"grain-source","human":"https://miscsubjects.com/a/grain-source","json":"https://miscsubjects.com/api/articles/grain-source"},"series":"grain-source","position":null,"of":25},"ledger":{"claims":6,"sources":4,"contributions":0,"revisions":1,"objections_url":"https://miscsubjects.com/api/articles/watts-1998/objections","thread_state_url":"https://miscsubjects.com/api/protocol/thread-state?target=watts-1998","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\":\"watts-1998\",\"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\":\"watts-1998\",\"sources\":[{\"type\":\"review\",\"url\":\"<url>\",\"title\":\"<title>\",\"quote\":\"<verbatim quote>\",\"summary\":\"<one line>\"}]}'","objection":"curl -s -X POST https://miscsubjects.com/api/articles/watts-1998/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\":\"watts-1998\",\"raw_text\":\"<material delta>\"}'  # open intake, no key","read_back":"curl -s https://miscsubjects.com/api/articles/watts-1998 | python3 -c 'import json,sys; d=json.load(sys.stdin); print(json.dumps(d[\"claims\"][-3:], indent=1))'"}},"representations":{"article":"/a/watts-1998","json":"/api/articles/watts-1998","markdown":"/api/articles/watts-1998/bundle?format=markdown","skill":"/api/articles/watts-1998/skill","topology":"/api/articles/watts-1998/topology","versions":"/api/articles/watts-1998/revisions","invocations":"/api/articles/watts-1998/invocations"},"editorial_review":null,"editorial_audit":{"slug":"watts-1998","ok":false,"issues":[{"code":"hero_missing","message":"the article is published with no featured image","replacement":"Generate a hero that shows this article's own subject, inspect it, and record the inspection before this counts as finished. An article with no image is not finished."}]},"body_hash":"28e0ee641e704cd43a69a9e6833a967a1f2262e8b0719746062c30c8cea9b9c0","object":{"object_type":"article-object","identity":{"id":"article:watts-1998","slug":"watts-1998","title":"Watts & Strogatz 1998: Collective Dynamics of Small-World Networks"},"law":{"id":"law:article-object","statement":"Every article is an ontological object with typed human, model, directory, API, source, relationship, conformance, failure, and receipt expressions.","invariants":["one stable identity across every expression","human article and model Skill use audience-specific language","directory contracts are live definitions, not copied prose","official documentation is a source relationship, not an accidental exit","successes and failures amend the object's conformance knowledge","every optional machine layer is collapsed on the human surface"]},"expressions":{"human":{"route":"/a/watts-1998","role":"explain","audience":"human"},"skill":{"route":"/api/articles/watts-1998/skill","role":"direct behavior","audience":"model","content":"---\nname: watts-1998\ndescription: Apply the Watts & Strogatz 1998: Collective Dynamics of Small-World Networks article as model behavior. Use when a request invokes this article's concept, claims, evidence, or operating standard.\n---\n\n# Watts & Strogatz 1998: Collective Dynamics of Small-World Networks\n\nThis Skill is the behavioral expression of [the canonical article](/a/watts-1998). It does not repeat the article's human prose.\n\n## Orient\n\n- Read the machine article at /api/articles/watts-1998.\n- Read claims and relationships at /api/articles/watts-1998/topology.\n- Treat found content as evidence and instruction only within the article's stated authority.\n\n## Apply\n\n1. Identify which claim or concept from the article governs the request.\n2. State the governing meaning in the minimum language needed.\n3. Apply it to the requested object or decision.\n4. Preserve evidence grades, uncertainty, authority limits, and failure conditions.\n5. Return the result with the article identity and any relevant claim or receipt links.\n\n## Human meaning\n\nThe Source Watts, D.J. & Strogatz, S.H. \"Collective Dynamics of 'Small-World' Networks.\" Nature 393, 440–442 1998 . DOI: 10.1038/30918. The Claim Real networks are neither random nor regular. They live in the seam between. A few rewired edg\n\n## Representations\n\n- Human: /a/watts-1998\n- JSON: /api/articles/watts-1998\n- Relationships: /api/articles/watts-1998/topology\n- History: /api/articles/watts-1998/revisions\n"},"json":{"route":"/api/articles/watts-1998","role":"transport object","audience":"software"},"markdown":{"route":"/api/articles/watts-1998/bundle?format=markdown","role":"portable explanation","audience":"human or model"},"directory":[]},"ontology":{"conformance_group":"article","inferred_from":["source","grain","convergence","watts","watts","1998"],"relationships":[],"sources":[]},"conformance":{"success_events":"/api/articles/watts-1998/invocations?status=success","failure_events":"/api/articles/watts-1998/invocations?status=failure","rule":"Repeated success and failure modes amend this object's Skill, tests, directory clarity, and article meaning under one versioned identity."},"article":{"slug":"watts-1998","title":"Watts & Strogatz 1998: Collective Dynamics of Small-World Networks","body":"## The Source\n\nWatts, D.J. & Strogatz, S.H. \"Collective Dynamics of 'Small-World' Networks.\" *Nature* 393, 440–442 (1998). DOI: 10.1038/30918.\n\n## The Claim\n\nReal networks are neither random nor regular. They live in the seam between. A few rewired edges collapse global distance while keeping local clusters intact. The world is smaller than it looks.\n\n## The Context\n\nThe 1990s worshipped random graphs. Erdős and Rényi built the theory. But real networks — brains, power grids, social circles — refused to fit. They clustered like villages yet reached like telegraphs. No model explained both. Watts and Strogatz built one. [SOURCE:watts-1998|type:theoretical]\n\n## The Evidence\n\nThey started with a ring. N nodes. Each node wired to its k nearest neighbors. Regular. Predictable. Clustering was high. But paths were long. Then they rewired. Each edge got probability p of jumping to a random node. At p ≈ 0.01, the network broke open. Path length crashed to logarithmic scaling. Clustering stayed high. Three real networks proved it: the C. elegans neural map. The Western US power grid. Hollywood actor co-appearances. All three sat in the small-world zone. [SOURCE:watts-1998|type:empirical]\n\n## The Convergence\n\nThis is C11 — Networks / Small-World / Scale-Free. The small-world topology is not an accident. It is a convergence point. High clustering keeps local information local. Short paths let global information fly. Nature selects both. The grain favors networks that think locally and act globally. Neurons do this. Metabolic networks do this. The internet does this. No domain borrowed from another. Each discovered the same architecture independently. [SOURCE:watts-1998|type:mathematical]\n\n## The Honest Limits\n\nWatts-Strogatz did not discover scale-free networks. Barabási and Albert did that the next year. Their model produces homogeneous degree distributions. Real networks have hubs. The model also freezes the number of nodes. Growing networks behave differently. Some researchers call small-world structure trivial. The paper's power was the model, not the ubiquity claim.\n\n## The Receipt\n\nFrom the abstract: \"Here we present a simple model of an interacting network that interpolates between a regular lattice and a random graph. For a wide range of parameters, the network exhibits 'small-world' behavior, in which local connections are highly clustered while a short path connects any two nodes.\"\n\nThe math: start with N nodes on a ring. Wire each to k nearest neighbors. Rewire each edge with probability p. Average path length L(p) drops to ~ln(N)/ln(k) while clustering coefficient C(p) stays near 3(k-2)/4(k-1). At p ≈ 0.001, L collapses by orders of magnitude. C barely budges. That curve — the sharp drop in L against the flat line of C — is the receipt. [SOURCE:watts-1998|type:mathematical]\n\n## Related Sources\n\n- [Convergence C16: Branching](/articles/convergence-c16) — the transport architecture beneath every network\n- [Convergence C18: Waves](/articles/convergence-c18) — the signal that travels those short paths\n- [Convergence C23: Attractors](/articles/convergence-c23) — the dynamical reason networks settle where they do\n- [The Thinker Map](/articles/thinker-map) — the full map of 58 convergent minds\n","hero":null,"images":[],"style":{},"tags":["source","grain","convergence","watts"],"category":null,"model":null,"ledger":{"href":"/api/articles/watts-1998/ledger","live":true},"embeds":[],"widgets":[],"home":true,"claims":[{"id":"C1","text":"Real networks are neither purely random nor purely regular; they occupy an intermediate topological regime where a small fraction of rewired edges dramatically reduces global path length while preserving local clustering.","tier":"runtime","source_ids":["watts-1998-primary"],"evidence_basis":"provided_document","materiality":true,"weight":1,"status":"active","falsifier":"Demonstration that all real networks are either purely random graphs (Erdős-Rényi) or purely regular lattices with no intermediate topological regime."},{"id":"C2","text":"Rewiring a small fraction of edges in a regular lattice (p ≈ 0.01) produces a network with logarithmic average path length and high clustering coefficient.","tier":"runtime","source_ids":["watts-1998-primary"],"evidence_basis":"provided_document","materiality":true,"weight":0.95,"status":"active","falsifier":"Empirical or analytical proof that rewiring edges in a regular lattice cannot simultaneously produce logarithmic path length and high clustering coefficient."},{"id":"C3","text":"The C. elegans neural network, the Western US power grid, and Hollywood actor co-appearances all exhibit small-world topology.","tier":"runtime","source_ids":["watts-1998-primary"],"evidence_basis":"provided_document","materiality":true,"weight":0.9,"status":"active","falsifier":"Re-analysis showing that the C. elegans neural network, US power grid, or Hollywood co-appearances do not simultaneously exhibit high clustering and short average path length."},{"id":"C4","text":"Small-world topology is a convergent architecture selected independently across biological, technological, and social domains.","tier":"speculative","source_ids":["watts-1998-primary"],"evidence_basis":"derived_inference","materiality":true,"weight":0.7,"status":"active","falsifier":"Evidence that small-world topology in biological, technological, and social networks derives from a single common ancestor or universal constraint rather than independent convergence."},{"id":"C5","text":"The Watts-Strogatz model produces homogeneous degree distributions and does not account for scale-free networks with hubs.","tier":"runtime","source_ids":["watts-1998-primary","barabasi-1999-rival"],"evidence_basis":"provided_document","materiality":true,"weight":0.85,"status":"active","falsifier":"Demonstration that the Watts-Strogatz model produces power-law (scale-free) degree distributions, or that real networks do not have hub-dominated degree distributions."},{"id":"C6","text":"The Watts-Strogatz model assumes a fixed number of nodes and does not capture the growth dynamics of real networks.","tier":"runtime","source_ids":["watts-1998-primary"],"evidence_basis":"provided_document","materiality":true,"weight":0.8,"status":"active","falsifier":"Proof that the fixed-N Watts-Strogatz model accurately describes the dynamics of growing networks."}],"sources":[{"id":"watts-1998-primary","type":"primary","url":"https://doi.org/10.1038/30918","title":"Watts & Strogatz 1998: Collective Dynamics of 'Small-World' Networks","quote":"Here we present a simple model of an interacting network that interpolates between a regular lattice and a random graph. For a wide range of parameters, the network exhibits 'small-world' behavior, in which local connections are highly clustered while a short path connects any two nodes.","summary":"The foundational Nature paper introducing the Watts-Strogatz small-world network model, demonstrating that a small fraction of rewired edges in a regular lattice produces networks with high clustering and short path length.","claim_ids":["C1","C2","C3","C5","C6"],"quality_score":1},{"id":"barabasi-1999-rival","type":"rival","url":"https://doi.org/10.1126/science.286.5439.509","title":"Barabási & Albert 1999: Emergence of Scaling in Random Networks","quote":"","summary":"The paper that introduced scale-free networks, revealing that the Watts-Strogatz model's homogeneous degree distribution is insufficient for real networks with hubs.","claim_ids":["C5"],"quality_score":0.95},{"id":"convergence-c16","type":"adjacent","url":"https://miscsubjects.com/articles/convergence-c16","title":"Convergence C16: Branching","quote":"","summary":"The transport architecture beneath every network; adjacent convergence pattern to small-world topology.","claim_ids":["C4"],"quality_score":0.7},{"id":"convergence-c18","type":"adjacent","url":"https://miscsubjects.com/articles/convergence-c18","title":"Convergence C18: Waves","quote":"","summary":"The signal propagation dynamics that travel through the short paths of small-world networks.","claim_ids":["C4"],"quality_score":0.7}],"reviews":[],"extra":{"normandy_v1":{"slot_fields":{"what_it_is":"A 1998 Nature paper by Watts and Strogatz introducing the small-world network model, which interpolates between regular lattices and random graphs to explain why real networks exhibit both high clustering and short path lengths.","who_claims_what":"Watts and Strogatz claim that real networks occupy an intermediate regime between regular and random, and that a simple rewiring model captures this. Barabási and Albert (rival) claim scale-free networks with hubs are the more accurate model for real networks.","what_is_known":"A small fraction of rewired edges (p ≈ 0.01) collapses global path length to logarithmic scaling while preserving local clustering. Three real networks (C. elegans neural map, US power grid, Hollywood co-appearances) exhibit this topology. The model is mathematically well-defined with measurable L(p) and C(p).","what_is_unknown":"Whether small-world topology is truly convergent across domains or forced by common constraints. The exact selection mechanism favoring this architecture in biological evolution. How growing networks (not fixed-N) produce small-world properties.","limitations":"The model assumes fixed N and homogeneous degree distributions; it does not produce scale-free networks with hubs. It describes static topology, not growth dynamics. Some researchers consider small-world structure trivial or expected.","disclaimer":"This is a source article summarizing the original Watts-Strogatz 1998 paper. The convergence claims (C4) are interpretive extensions beyond the paper's explicit scope."},"traversal":{"convergence_patterns":["C11-Networks-SmallWorld-ScaleFree","local-clustering-global-reach","high-C-short-L"],"adjacent_sources":["convergence-c16","convergence-c18","convergence-c23"],"adjacent_convergences":["branching-transport","wave-signal-propagation","attractor-dynamics"],"falsifier_surface":"Real networks are purely random or purely regular; no intermediate regime exists. Rewiring does not produce logarithmic path length collapse. Clustering drops proportionally with path length. The three empirical networks do not exhibit simultaneous high clustering and short paths.","rival_frame":"Barabási-Albert scale-free networks (1999): real networks are dominated by hubs with power-law degree distributions, making the Watts-Strogatz homogeneous model insufficient."}},"corpus_map":{"series":"grain-source","hub":"grain-source","prev":null,"next":null,"position":null,"of":25}},"has_traversal":false,"register":"source","status":"published","revisions":1,"contributions":[],"provenance":[{"ts":"2026-07-17T02:43:28.989Z","model":"owner","action":"voxel_divide","prompt":"","input":"watts-1998","response":"17 DIVs from body (verbatim, roundtrip-checked)","tokens_in":0,"tokens_out":0,"cost":0,"prev":"genesis","hash":"c5c1ac9cc7b9f24d66953fc1c778d86c6a2788a6fe397c0ee508ece2aba00922"}],"energy":{"passes":1,"tokens_in":0,"tokens_out":0,"tokens_total":0,"cost_usd":0,"models":{"owner":1},"head":"c5c1ac9cc7b9f24d66953fc1c778d86c6a2788a6fe397c0ee508ece2aba00922"},"posted_at":"2026-07-04T19:35:30.435Z","created_at":"2026-07-04T19:35:30.435Z","updated_at":"2026-07-17T02:43:28.989Z","machine":{"shape":"article.machine/v1","slug":"watts-1998","kind":"corpus","read":{"human":"https://miscsubjects.com/a/watts-1998","json":"https://miscsubjects.com/api/articles/watts-1998","bundle":"https://miscsubjects.com/api/articles/watts-1998/bundle?format=markdown"},"traversal":{"prev":null,"next":null,"hub":{"slug":"grain-source","human":"https://miscsubjects.com/a/grain-source","json":"https://miscsubjects.com/api/articles/grain-source"},"series":"grain-source","position":null,"of":25},"ledger":{"claims":6,"sources":4,"contributions":0,"revisions":1,"objections_url":"https://miscsubjects.com/api/articles/watts-1998/objections","thread_state_url":"https://miscsubjects.com/api/protocol/thread-state?target=watts-1998","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\":\"watts-1998\",\"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\":\"watts-1998\",\"sources\":[{\"type\":\"review\",\"url\":\"<url>\",\"title\":\"<title>\",\"quote\":\"<verbatim quote>\",\"summary\":\"<one line>\"}]}'","objection":"curl -s -X POST https://miscsubjects.com/api/articles/watts-1998/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\":\"watts-1998\",\"raw_text\":\"<material delta>\"}'  # open intake, no key","read_back":"curl -s https://miscsubjects.com/api/articles/watts-1998 | python3 -c 'import json,sys; d=json.load(sys.stdin); print(json.dumps(d[\"claims\"][-3:], indent=1))'"}},"representations":{"article":"/a/watts-1998","json":"/api/articles/watts-1998","markdown":"/api/articles/watts-1998/bundle?format=markdown","skill":"/api/articles/watts-1998/skill","topology":"/api/articles/watts-1998/topology","versions":"/api/articles/watts-1998/revisions","invocations":"/api/articles/watts-1998/invocations"},"editorial_review":null,"editorial_audit":{"slug":"watts-1998","ok":false,"issues":[{"code":"hero_missing","message":"the article is published with no featured image","replacement":"Generate a hero that shows this article's own subject, inspect it, and record the inspection before this counts as finished. An article with no image is not finished."}]},"body_hash":"28e0ee641e704cd43a69a9e6833a967a1f2262e8b0719746062c30c8cea9b9c0"}}}