{"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)","register":"oip_protocol","tags":["philosophy","oip","convergence-catalogue","node","systems-theory"],"category":null,"style":{},"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_material":"Specifies the generative mechanism and resulting degree distribution for scale-free topology."},{"id":"c4","text":"Weak ties bridge otherwise disconnected clusters in networks.","section":"C11 — Networks / Small-World / Scale-Free","tier":"anecdotal","source_ids":[],"source_status":"unsourced","why_material":"Identifies a key bridging role of weak connections in maintaining small-world properties."}],"sources":[],"prov":{"model":"Fable 5 (Claude Code)","action":"write"}}