Editing Openai/693343d7-a38c-8012-a67c-11cbed4c0fd9 (section)

=== Homophily/heterophily modulate: ===
* Perception filter: P_A(E) biased toward cues from high-aff neighbors (weighting). Replace P with weighted perception: ``<code> P_A(E) = ∑_{j} W_{A,j} · obs_j(E) / ∑_j W_{A,j} <syntaxhighlight>- Memory update: </syntaxhighlight> M_{t+1} = MemoryUpdate(M_t, S_t, A_t, feedback;  W_{·,·}) </code>`<code> with weight averaging over sources proportional to W.
* Trust / credibility: Trust_{A→B} derived from past interaction success and Aff. Trust influences how much agent A accepts model updates from B.
* Learning rates: agents may adopt different learning rates for inputs from homophilous vs heterophilous partners: </code>`<code> α_update = α_0 · (1 + κ·HomPref(A,B)) <syntaxhighlight>
=== ## ===

Turn Agt into a V-enriched category where V = ([0,1], ≤, ×) or another semiring of affinities.
* Objects: agents.
* Hom-objects: Hom_V(A,B) = W_{A,B} (an element of V).
* Composition uses enrichment monoid (e.g., multiplication or t-norm): </syntaxhighlight> (W ∘ V)_{i,k} = max_j T( W_{i,j}, W_{j,k} ) </code>`<code> where T is a t-norm (e.g., min, product).

This captures the idea that two-step trust/affinity composes in a graded way.

==== Model affinity as a profunctor: ====

<syntaxhighlight>Aff : Agtᵒᵖ × Agt → [0,1]

</syntaxhighlight>

which is natural in both arguments: it assigns a weight to every ordered pair. Profunctor composition gives transitive affinity.

Coends aggregate:

<syntaxhighlight>CollectiveAffinity(A) = ∫^{B} Aff(A,B) · Capability(B)

</syntaxhighlight>

(Here · combines affinity weight with capability vector — a weighted coend).

==== If Sim: Agt×Agt → ℝ and Comp similarly, define: ====

<syntaxhighlight>HomophilyTransform : Sim ⇒ Aff
HeterophilyTransform : Comp ⇒ Aff

</syntaxhighlight>

These are natural transformations encoding how similarity/complementarity produce affinities.

==== Refine the perception fibration π : Obs → World into affinity-indexed fibers: ====

For each agent A and affinity threshold θ, the fiber Obs_{w}^{A,θ} = observations of world-state w that A will accept from agents with Aff≥θ.

Sections now select not only perspective but an affinity filter.