authenticated graphs

introduction

authenticated graph data structures (agds) offer cryptographic proofs that answers to graph queries are correct without trusting the server. while their theory has existed for two decades, they remain under‑used. yet, as we build earth‑scale superintelligence on the collective focus theorem (cft) and fractional focus cascading (ffc), verifiable graph integrity becomes a hard requirement.

recap of classic agds

• path hash accumulator: represents a path as a balanced or biased binary tree whose internal nodes store hashes of concatenated sub‑paths. this enables logarithmic‑size proofs for properties like connectivity, distance and type queries【2:graph-auth-2.pdf†turn2file3†L11-L18】.
• dynamic authenticated forests: by partitioning each tree into solid and dashed paths and linking their accumulators, forests support link/cut updates with o(log n) proofs and space o(n)【2:graph-auth-2.pdf†turn2file7†L45-L53】.
• authenticated fractional cascading (ffc): fractional cascading accelerates iterative search across many ordered lists; hashing each block and then a second‑level dag gives o(k+log n) query time with logarithmic proofs while keeping linear storage【4:graph-auth-2.pdf†turn4file6†L9-L17】.

why cft needs agds

cft models a decentralized knowledge graph where token‑weighted random walks converge to a stable focus distribution【2:Collective Focus Theorem.md†turn2file10†L18-L25】. correctness of focus requires that every participant can verify:

• the existence and weights of cyberlinks used in a random walk step
• the resulting stationary distribution snapshot

agds provide exactly these proofs, allowing anyone with minimal bandwidth to audit the superintelligence core.

design: fully‑authenticated focus cascading (ffc)

• store the cybergraph as a dynamic forest of merkelized subgraphs (shards)
• inside each shard, maintain a path hash accumulator over the local spanning tree for fast verification of connectivity and attribute queries
• build a fractional cascading overlay between shard catalogs (edge‑boundary lists) so that global lookups (“does link x exist? what is its weight?”) cost o(log n) regardless of sharding
• the top‑level digest is the cft “attention root”; every random‑walk step publishes its proof against this root so anyone can recompute focus

internal applications

• validator pipeline: gpu validators verify link insertions and rank updates with agds proofs before committing blocks, eliminating hidden tampering
• training layer audits: during massive graph rewrites, agds proofs allow sampling‑based integrity checks rather than full re‑hashing, saving bandwidth and energy
• agent reasoning: llms embedded in the network can query “are particles a and b connected through topic t?” and get a certificate they can cache for future reasoning

external applications

• open science provenance: researchers can publish datasets as particles; agds proofs certify citation chains, enabling reproducible meta‑analysis
• supply‑chain transparency: companies append product trails to the cybergraph; customers query origin → destination paths and verify them client‑side
• cross‑chain bridges: other blockchains can verify cybergraph facts (e.g., focus weight of an address) via light‑client‑sized proofs instead of heavy oracles
• regulatory compliance: auditors can demand proofs that certain forbidden relationships are absent (negative proofs via authenticated complement paths)

sharding strategies

when we talk about merkelized subgraphs, the key design question is how to chop the cybergraph into shards so that proofs stay small, writes remain local, and load balances across validators.

• id‑hash shards
  assign each vertex to hash(id) mod m. this keeps balancing trivial and proofs deterministic, but cross‑shard edges can explode. good for early prototypes.
• neuron‑centric shards
  group all links emitted by the same neuron (agent). locality for high‑throughput writers, but superstar neurons can create hotspot shards.
• topic / namespace shards
  cluster vertices by ontology prefix or tag so semantic queries stay mostly inside one shard. aligns with cft focus levels.
• community (edge‑cut minimization) shards
  run partitioners (metis, powergraph) or newer blockchain‑specific algorithms like gpchain to minimize cross‑shard edges while equalizing weight【5:GPChain abstract†turn5view0†L4-L6】.
• geographic / ownership shards
  physical‑world graphs (iot, supply chains) can shard by region or owner to support local sovereignty.
• temporal shards
  append‑only data can be batched by epoch (e.g., one week per shard) so old shards become read‑only archives.
• hybrid hierarchical shards
  combine a coarse id‑hash slice with fine community splits inside each slice; each level has its own digest so proofs compose.

trade‑offs revolve around cross‑shard proof size, validator workload, and attack surface. choosing the right mix depends on query patterns and write hot‑spots. recent surveys highlight that poorly balanced shards hurt scalability and security【4:Sharding with MPT†turn4view0†L18-L25】.

road map

1. implement biased tree path‑hash library in rust with zero‑knowledge friendly hashing (poseidon)
2. integrate with go‑cyber to emit proofs in each tx log
3. extend cometbft light‑client to accept agds digests
4. build sdk for browser wallets to verify focus proofs on‑device
5. research dynamic authenticated fractional cascading to support real‑time graph streams【2:graph-auth-2.pdf†turn2file3†L22-L23】.

conclusion

agds upgrade the collective focus infrastructure from “trust me” to “verify me”. by merging classic path hash accumulators, fractional cascading and cft economics, fully‑authenticated focus cascading ensures every edge, token and weight that shapes superintelligence remains transparent, tamper‑evident and universally auditable.