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Mathematics · unit version 1

Directed random walks and stationary guarantees

Specify graph transition policies, separate stationary balance from convergence, and prove what a direct restart guarantees.

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Readiness and preparation

Check your starting point

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What does a row of an adjacency matrix record?

Transition probabilities must keep departure and arrival conventions consistent.

Compare reasoning and find a refresher

In the source-row convention, entry (i,j) records the directed edge from i to j. A stochastic transition row additionally assigns nonnegative probabilities summing to one.

Review adjacency and intermediate vertices

Does a stationary distribution guarantee convergence from every starting distribution?

A fixed probability row and a limiting trajectory are different claims.

Compare reasoning and find a refresher

No. A periodic transition can have a unique stationary row while other starting rows cycle forever. State extra hypotheses when claiming convergence.

Review two-state transitions

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Audit the transition model and its conclusion

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Lesson sequence

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  1. Turn a directed graph into a probability model

    Lesson details and objectives

    Specify outgoing choices and dangling behavior, prove mass conservation, and distinguish weighted path probabilities from graph counts.

    • Construct source-row transition probabilities from a directed graph and an explicit dangling policy.
    • Prove preservation of probability and expand multi-step forecasts into weighted paths.
    • Compare alternative policies and distinguish exact model calculations from evidence about real behavior.

    6 practice tasks.

  2. Separate stationary balance from convergence

    Lesson details and objectives

    Solve invariant probability equations and use finite counterexamples to distinguish existence, uniqueness and limiting behavior.

    • Solve and verify stationary balance equations under an explicit row-vector transition convention and dangling-state policy.
    • Prove counterexamples separating existence, uniqueness and convergence, and establish convergence directly in a contracting example.
    • Use a finite-simplex Cesàro argument for existence while identifying exactly which stronger claims it does not prove.

    6 practice tasks.

  3. Certify convergence and ranking under direct restart

    Lesson details and objectives

    Prove a contraction, construct its stationary distribution and convert a residual into a justified ranking certificate.

    • Prove L1 nonexpansion for row-stochastic matrices and contraction for direct restart.
    • Construct and verify the stationary series, including boundary and zero-coordinate cases.
    • Distinguish current-residual, next-iterate and ranking error certificates from rounded numerical evidence.

    6 practice tasks.