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Object Invocation Protocol · protocol specification

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

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§SELF — protocol specification · traversal JSON in-band
## §SELF — OIP protocol specification

**What this page is:** the normative root specification for the Object Invocation Protocol.

**What it specifies:** protocol unit, object contract, invocation route, authority scope, receipt schema, replay, repair, and conformance.

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**Proof rule:** an action is not proven by intent, description, or a 200. It is proven by the ledger and the OIP receipt for the invocation.

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

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"] }

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