Hyperon AI Algorithms+ECAN (Economic Attention Networks)+ECAN Deep Dive

Core Mechanisms

Two Importance Currencies. Every Atom carries two dynamically updated scalar values:

• Short-Term Importance (STI): Captures immediate, context-dependent relevance. High-STI atoms are in the Attentional Focus — the working memory subset actively available for reasoning and learning.
• Long-Term Importance (LTI): Reflects longer-horizon expected utility — how consistently an Atom has contributed to successful inference, learning, or goal-directed behavior over time.

Economic Dynamics. ECAN implements an attention economy with several interacting mechanisms:

• Importance diffusion: STI spreads through Hebbian-weighted associative links. When two atoms are co-activated, their Hebbian link strengthens and future STI diffuses more readily between them. Three diffusion modes exist: Attentional Focus-only (AF, faster, narrower), Whole-AtomSpace (WA, broader, slower), and Fringe spreading (K-hop, see below).
• Rent collection: Atoms in the Attentional Focus pay "rent" — their STI decays over time unless refreshed by ongoing relevance. This prevents working memory from growing unboundedly.
• Forgetting: Atoms whose STI falls below threshold and whose LTI is low may be evicted from active processing, reclaiming resources.
• Hebbian link creation/updating: Co-occurrence patterns between atoms generate and strengthen associative links, which in turn guide future attention flow.

Fringe Spreading (Third Diffusion Mode). The AF-only and WA diffusion modes have a known gap: atoms 2-3 hops from the Attentional Focus receive negligible STI even when they are highly relevant (e.g., question-answer ImplicationLinks connected to AF words only via intermediate nodes). Fringe spreading addresses this by diffusing STI K hops out from each AF atom (typically K=2 or K=3), analogous to a "fringe of consciousness." This is faster than WA diffusion but broader than AF-only. A design tension remains between fringe spreading and intra-AF attractor dynamics: variable-K spreading (Matt Iklé's proposal to replace AF-only entirely) risks slowing attractor formation by orders of magnitude. (mailing-list-backed: Fringe-of-consciousness-importance-spreading, 2016)

Directional Diffusion and Hebbian/Non-Hebbian Split. STI diffusion follows a tunable split: x% along HebbianLinks, (1-x)% along non-Hebbian links (e.g., InheritanceLinks, EvaluationLinks). Early implementation revealed that STI could get "stuck" on intermediate structures (e.g., ListLinks inside EvaluationLinks) because diffusion originally only flowed along ordered link targets, not back to source atoms. The fix required diffusion along both incoming and outgoing sets. For calibrating non-Hebbian weights, a three-step empirical methodology was proposed: (1) run Hebbian-only diffusion, (2) measure co-activation frequency for each (link-type, source-node-type, target-node-type) combination, (3) derive "virtual Hebbian weights" from the resulting statistics. (mailing-list-backed: Importance diffusion, 2014)

Mathematical / Formal Foundations

Information Geometry. The 2011 paper by Iklé and Goertzel formalizes ECAN using information geometry, casting attention allocation as movement on a statistical manifold where the distance metric reflects information-theoretic cost of attention reallocation.

Fluid-Dynamic Extension (Proposed). The 2025 whitepaper (§5.3–5.4) proposes extending ECAN with an incompressible fluid dynamics model where attention mass (STI values) becomes a conserved fluid density \(\rho(t,x)\) advected by a divergence-free velocity field \(u(t,x)\). Key proposed properties:

• The velocity field is derived from an optimal control policy solving a Hamilton-Jacobi-Bellman (HJB) boundary-value problem
• The incompressibility constraint \(\nabla \cdot u = 0\) enforces strict attention budget conservation
• Pressure acts as a shadow price for attention congestion
• Between ECAN's discrete economic updates, attention would evolve via the continuity equation with the HJB-derived velocity field, providing goal-aware routing that complements ECAN's diffusion

The continuity equation governing attention transport:

\[\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho\, u) = 0\]

• Variables: \(\rho(t,x)\) = attention density (STI mass) at position \(x\) and time \(t\); \(u(t,x)\) = velocity field derived from HJB optimal control
• Assumptions: Incompressibility (\(\nabla \cdot u = 0\)) enforces strict conservation of total attention budget
• Meaning: Attention mass is neither created nor destroyed — it flows through the knowledge graph along goal-optimal paths between ECAN's discrete economic steps
• Source: Goertzel (2025), Hyperon Whitepaper §5.3–5.4

This fluid-dynamic layer would not replace ECAN's economic mechanisms but provide a continuous-time transport substrate between ECAN's discrete steps. The whitepaper notes this is particularly useful for credit assignment in multi-hop causal chains where traditional diffusion may lag.