the theorem guaranteeing that an irreducible non-negative matrix has a unique largest real eigenvalue with a strictly positive eigenvector
mathematical foundation of the collective focus theorem: ensures a unique focus distribution $\pi^*$ exists for the cybergraph
statement: let $A$ be an irreducible non-negative square matrix. then $A$ has a real eigenvalue $\lambda_1 > 0$ such that $\lambda_1 \geq |\lambda|$ for every eigenvalue $\lambda$ of $A$. the eigenvector corresponding to $\lambda_1$ has all strictly positive entries
in cyber, the transition matrix of the cybergraph satisfies irreducibility when the graph is strongly connected. the tru enforces this by adding a teleportation term (damping factor), guaranteeing convergence of diffusion to a unique stationary distribution
this is the same theorem that underlies PageRank. cyber generalizes the result through the tri-kernel: diffusion uses the Perron-Frobenius eigenvector directly, while springs and heat operate on the graph Laplacian spectrum
convergence rate depends on the spectral gap $\lambda_1 - |\lambda_2|$: larger gap means faster consensus on relevance
see collective focus theorem, focus, diffusion, cybergraph, graph theory, tri-kernel