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

=== Operad O(n) of collaborative actions becomes weighted operad: each n-ary operation carries an internal affinity matrix between participants. ===

Composition rule:
* When substituting subteams, transform affinity matrices by embedding and recomputing inter-subteam affinities (e.g., average pairwise affinities or using linking heuristics).

Team formation algorithm (operadic):
* Given candidate agents {a_i}, compute team score: </code>`<code> TeamScore = λ1 · ∑_{i<j} W_{ij}  + λ2 · DiversityScore({a_i}) + λ3·SkillCoverage <syntaxhighlight>- Homophily-heavy teams maximize pairwise W; heterophily-heavy teams prefer high DiversityScore and complementary skills.

=== Proposition 1 (Homophily increases modularity). In an affinity-weighted graph where link update follows ΔW ∝ Aff, steady-state modularity Q increases with homophily bias. ===

Sketch: If Aff correlates with similarity clusters, edges form mostly within clusters → high modularity. (Standard from network theory; can be formalized using assortativity metrics.)

Proposition 2 (Heterophily increases bridging reachability). Introducing heterophilous affinity for complementary nodes that maximize a bridging objective reduces average path length and increases giant-component connectivity.

Sketch: Edges between dissimilar clusters create shortcuts and reduce diameter; formalizable via percolation thresholds.

Proposition 3 (Affinity-weighted EM-algebra stability). If EM algebra α_W uses affinity-weighted aggregation with convex combination weights derived from W and weights sum to 1, then α_W is contractive under appropriate metric, yielding stability of organization interpretation.

Sketch: Contractiveness follows if weights define a stochastic matrix with spectral radius <1 for deviation dynamics; can use Banach fixed-point.

=== Compute for agent network at time t: ===
* Aff assortativity (Newman assortativity but with continuous attributes): correlation between node attribute similarity and W_{ij}.
* Average affinity per agent: ā_i = ∑_j W_{ij} / N.
* Modularity Q_w for weighted graph using W.
* Average neighbor diversity (entropy of neighbor attribute distribution weighted by W).
* Bridging centrality influenced by heterophilous edges.

=== Add primitives: ===

</syntaxhighlight>
affinity(a,b) : Float    -- returns Aff(a,b)
recruit(target_aff_threshold) : O Agent    -- build team by threshold
rewire(agent, policy) : Action   -- update W_{·,·} by policy
endorse(a,b,weight) : Effect     -- agent a increases trust in b by weight
diversify(team, k) : O Team      -- return k-member team maximizing complementarity
homophily_policy(alpha)         -- param object governing link updates
heterophily_policy(beta, objective) -- param object

<syntaxhighlight>
Semantics:
* recruit uses affinity scores to assemble a team tree using join_W.
* rewire updates adjacency matrix according to ΔW rule above.
* diversify solves small optimization.

=== Map concepts to services: ===
* Affinity Service: stores θ embeddings, computes Aff(a,b), provides APIs GET /affinity?a=&b=; indexes by worker portfolio vectors.
* Matchmaker / Recruiter: compiles teams using homophily/heterophily policy and emits Tree (Obatala) manifests.
* Network Dynamics Engine: runs link update rules (ΔW), rewire scheduling, and stores histories in kogi-state.
* Trust & Reputation Service: derives trust from interactions—used by EM handlers to weight decisions.
* Diversity Dashboard: computes diversity and bridging metrics for governance.
* Policy Engine: defines organizational bias parameters (α, β, λ_i) enforced at EM-algebra time.

Security/ethics:
* monitor for echo chambers (extreme homophily) and enforced diversity thresholds.
* fairness rules: enforce combatting exclusion (e.g., require minimum external edges).

=== - Sweep α (homophily strength) from 0 to 1, measure Q (modularity), average path length, and minority accessibility. Plot phase transitions. ===
* Introduce occasional heterophilous seeding (probability p), measure long-term innovation metric (e.g., novel skill composition success).
* Test team performance: compare homophily-driven vs heterophily-driven team formation on tasks requiring exploration vs exploitation.

=== Affinity computation (vector embeddings e_A ∈ ℝ^d): ===

</syntaxhighlight>
Sim(A,B) = cosine(e_A, e_B) = (e_A · e_B) / (||e_A|| ||e_B||)
Comp(A,B) = 1 - Sim(A,B)   (or use task-specific complementarity)
Aff(A,B) = σ( α·Sim(A,B) + β·Trust(A,B) + γ·HistoryScore(A,B) )
ΔW_{ij} = η ( Aff(A_i,A_j) - W_{ij} ) + ζ Success_{ij} - δ W_{ij}

<syntaxhighlight>
where Success_{ij} is normalized payoff from recent collaborations.

=== Team capability aggregation (weighted coend): ===

</syntaxhighlight>
TeamCapability = ∫^{i ∈ Team} Aff(TeamLead, i) · Capability(i)

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Consensus observation (affinity-weighted coend):

</syntaxhighlight>
SharedObservation = ∫^{j} Aff(Leader, j) · P_j(E)

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Affinity as profunctor composition:

</syntaxhighlight>
Aff^{(2)}(A,C) = ∫^{B} Aff(A,B) ⊗ Aff(B,C)

<syntaxhighlight>
(⊗ a t-norm; coend integrates over intermediate agents.)

=== - Make O enriched over V = [0,1] so that Hom carries affinity weight. ===
* join_W semantics: when μ flattens nested O(O A), re-weight children by internal affinities: </syntaxhighlight> μ_W( Node(op, [m1,...,mk]) ) = Node(op, normalize( map scale_children(mi, Aff_matrix) ) ) </code>``
* EM-algebras α_W include affinity in algebra interpretation.