Intelligence as Relational Dynamics (Canonical)
Defines intelligence as emergent relational structure, formalizing topology, reciprocity, and adaptation.
Executive KPIs
Formal Framework
Relational system $\mathcal{R}=(E,R,w)$ with entities $E=\{e_i\}$ and symmetric weights $w_{ij}(t)$.
Adjacency $A_{ij}(t)=w_{ij}(t)$, degree $d_i(t)=\sum_j A_{ij}(t)$, Laplacian $L(t)=D(t)-A(t)$.
Consensus/learning update: $$ \mathbf{x}(t+1)=W(t)\,\mathbf{x}(t),\quad W(t)=I-\eta L(t),\quad 0<\eta<\frac{1}{\lambda_{\max}(L)} $$
Kuramoto order parameter: $$ R(t)=\frac{1}{N}\left|\sum_{k=1}^{N} e^{i\theta_k(t)}\right| $$
Mutual information of partitions $X,Y$: $$ I(X;Y)=\sum_{x,y} p(x,y)\log\tfrac{p(x,y)}{p(x)p(y)} $$
Topological Metrics
Connectivity & Clustering
Local clustering: $$ C_i=\frac{2T_i}{k_i(k_i-1)} $$; path length $L$ via average shortest paths.
Communities & Assortativity
Modularity: $$ Q=\frac{1}{2m}\sum_{ij}\Big(A_{ij}-\frac{k_i k_j}{2m}\Big)\,\delta(c_i,c_j) $$; assortativity $r$ (Newman).
Method at a Glance
- Instantiate entities and initial relations; choose update rules for $w_{ij}$.
- Measure $C, L, Q, r$ and order $R(t)$ during learning.
- Probe robustness under edge/agent removals; track recovery time.
- Summarize stability via persistence diagrams and Betti curves.