---
alias: random walk, markov, exploration, diffusion, freedom
tags: cyber
crystal-type: process
crystal-domain: cyber
---
first operator of the tri-kernel
transition matrix `P = AD⁻¹` governs probability flow across the cybergraph
`π^(t+1) = α P^T φ*^(t) + (1-α)u`
- α = teleport parameter
- u = prior (stake-weighted)
answers: "where does probability flow?"
the exploration component of the cyberank. the full cyberank is the fixed point of all three tri-kernel operators blended together
row-stochastic, preserves probability mass
powers remain local. converges to unique stationary distribution under ergodicity
locality: geometric decay via teleport parameter α
the exploration force — a gas wandering, sampling connections
universal pattern
- physics: gas wandering, sampling
- biology: synaptic chatter, neural noise
- ecology: species dispersal, seed rain
- economics: trade, migration, meme flow
together with springs and heat kernel forms the tri-kernel
see tri-kernel for completeness proof
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