every non-negative primitive matrix has a unique positive eigenvector. this is the theorem that guarantees focus converges
named after Oskar Perron (1907, positive matrices) and Georg Frobenius (1912, extension to non-negative matrices)
the theorem
three forms, each extending the previous.
positive matrices (Perron, 1907)
let $A$ be a real $n \times n$ matrix with all entries strictly positive ($A_{ij} > 0$). then:
- $A$ has a unique real eigenvalue $\rho(A)$ equal to its spectral radius — the Perron root
- $\rho(A) > |\lambda|$ for every other eigenvalue $\lambda \neq \rho(A)$
- the corresponding eigenvector $\mathbf{v}$ has all positive entries — the Perron vector
- $\mathbf{v}$ is unique up to scaling
non-negative primitive matrices (Frobenius, 1912)
a non-negative matrix $A \geq 0$ is primitive if it is irreducible and aperiodic — equivalently, $A^k > 0$ for some $k$. the same conclusions hold: unique Perron root dominating all others, unique positive Perron vector.
stochastic chains (the case that matters)
a row-stochastic matrix $P$ (all entries $\geq 0$, each row sums to 1) that is irreducible and aperiodic satisfies:
$$\exists!\; \pi^* \in \Delta^{n-1} : \pi^* P = \pi^*, \quad \pi^*_i > 0 \;\forall\, i$$
$\pi^*$ is the unique stationary distribution. from any starting distribution $\mu^{(0)}$:
$$\left\|\mu^{(0)} P^t - \pi^*\right\|_1 \leq C \cdot (1 - \lambda)^t \to 0$$
where $\lambda$ is the spectral gap $1 - |\lambda_2|$ and $\lambda_2$ is the second-largest eigenvalue by modulus.
why primitive = irreducible + aperiodic
irreducibility: from any state $i$, every state $j$ is reachable with positive probability in finitely many steps. on a directed graph, this is strong connectivity.
aperiodicity: the chain is not periodic — there is no $d \geq 2$ such that the chain always returns to a state in multiples of $d$ steps. a single self-loop is sufficient for aperiodicity. the teleport term $\alpha$ in the diffusion operator guarantees this.
without irreducibility: $\pi^*$ may not be unique (absorbing states create multiple stationary distributions). without aperiodicity: $P^t$ oscillates rather than converges.
application to the cybergraph
the tri-kernel composite operator $\mathcal{R}$ acts on probability distributions over particles. it converges to $\pi^*$ because its induced transition matrix satisfies both conditions.
irreducibility: the graph is strongly connected (any particle reachable from any other through the link structure). the teleport term in the diffusion operator $\mathcal{D}$ provides a global path between all particle pairs with probability $\alpha > 0$.
aperiodicity: the teleport also breaks periodicity — any state has a positive self-transition probability $\alpha/|P|$.
therefore by Perron-Frobenius:
| property | formal guarantee |
|---|---|
| focus $\pi^*$ exists | unique stationary distribution |
| all $\pi^*_p > 0$ | every particle has positive attention |
| $\pi^*$ is independent of initial distribution | any $\phi^{(0)} \to \pi^*$ |
| convergence rate is geometric | $\|\phi^{(t)} - \pi^*\|_1 \leq C(1-\lambda_2)^t$ |
the rate is controlled by the spectral gap $\lambda_2 = 1 - |\lambda_2(P)|$. larger gap = faster convergence. the gap is bounded below by the Cheeger constant (graph connectivity) and above by $1$ (trivial convergence in one step). see spectral gap for the graph-theoretic analysis.
in PageRank and cyberank
Larry Page and Sergey Brin implicitly invoked Perron-Frobenius in PageRank (1998): the recursive definition "the importance of a page is the sum of importances of pages linking to it" is an eigenvalue equation $\pi = \pi P$. Perron-Frobenius guarantees this has a unique positive solution when $P$ is irreducible and aperiodic — which the damping factor (teleport parameter) ensures.
cyberank is PageRank generalized: instead of a flat importance score, the stationary distribution $\pi^*$ over particles weighted by neuron stakes and ICBS market prices. the convergence guarantee is the same theorem, applied to a richer transition matrix.
the Banach fixed-point connection
Perron-Frobenius guarantees existence and uniqueness. the collective focus theorem (CFT) proves convergence via the Banach fixed-point theorem: the composite operator $\mathcal{R}$ is a contraction with coefficient $\kappa < 1$, so $\mathcal{R}^t \phi \to \pi^*$ for any $\phi$. the two approaches are complementary — Perron-Frobenius via spectral theory, CFT via metric contraction. the spectral gap $\lambda_2$ is the Banach contraction coefficient $\kappa$ viewed from the eigenvalue side.
see collective focus theorem for the contraction proof. see spectral gap for the rate analysis. see Oskar Perron for biography. see cyber/focus for the engineering implementation.