Definition
x_v = (1/λ) · sum of x_u over neighbors u. Vector form: A x = λ x. Take largest eigenvalue's eigenvector.
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Power iteration
Initialize x = 1s. Iterate x ← A x / ||A x||. Converges to principal eigenvector.
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Convergence
Perron-Frobenius theorem: connected graph → unique positive eigenvector. Convergence rate depends on ratio λ₁ / λ₂.
Related
Katz centrality: eigenvector + small self-boost. PageRank: eigenvector of scaled adjacency + damping.