Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 110 minutes, as an unvalidated planning estimate. This collection contains 3 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
State the row convention, dangling policy, restart distribution and time-step meaning before calculating.
Separate existence, uniqueness and convergence; check stationary candidates by multiplication.
Attempt the changed models before reading public solutions. Exact-arithmetic proofs and rounded numerical observations supply different evidence.
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A graph has A→B, B→A and no outgoing edge from C. Repair C by uniform redistribution over all three vertices. Construct the row-stochastic matrix and find its stationary distribution. Prove uniqueness, then show that a point mass at A does not converge. Add direct restart with probability 1/2 and restart distribution concentrated at C. Find and verify the new stationary distribution and justify convergence from every initial distribution. Repeat the restarted stationary calculation when C is instead repaired with a self-loop. Explain why a restart vector with zero coordinates does not invalidate uniqueness, but does prevent a blanket claim that every stationary coordinate is positive.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let Q have rows (1/6,5/12,5/12), (1/6,1/6,2/3), and (1/3,1/3,1/3). It comes from uniform dangling repair of A→B,C; B→C; C dangling, followed by uniform direct restart with α=1/2. For the exact probability row x=(1/4,3/10,9/20), calculate xQ, the L1 residual and the residual-based bound on ||x−π||₁, where π is the unique stationary row. For probability rows x and π, prove each coordinate error is at most half their L1 distance and deduce that an observed coordinate gap greater than an L1 error bound certifies its strict order in π. Use this to certify both C>B and B>A here. Finally verify π=(8/33,10/33,15/33) independently and compare the bound with the actual L1 error. Explain what extra evidence would be needed if x and the residual were only rounded software output.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A model adds a restart state S to two dangling page states A and B. Each page moves to S with probability 1; S moves to A or B with probability 1/2 each. Write its three-state matrix, solve stationary balance, and prove it is strongly connected but a start at S does not converge. What distribution is seen at the page-visit instants? Compare it with a different two-state model that repairs each page by a self-loop and then performs direct uniform restart with probability 1/2 on every transition. Find that model’s matrix and stationary distribution and prove convergence. Explain why dropping S from a displayed diagram without changing the clock and transition rule is not a valid identification of the two models.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
A graph has A→B, B→A and no outgoing edge from C. Repair C by uniform redistribution over all three vertices. Construct the row-stochastic matrix and find its stationary distribution. Prove uniqueness, then show that a point mass at A does not converge. Add direct restart with probability 1/2 and restart distribution concentrated at C. Find and verify the new stationary distribution and justify convergence from every initial distribution. Repeat the restarted stationary calculation when C is instead repaired with a self-loop. Explain why a restart vector with zero coordinates does not invalidate uniqueness, but does prevent a blanket claim that every stationary coordinate is positive.
Let Q have rows (1/6,5/12,5/12), (1/6,1/6,2/3), and (1/3,1/3,1/3). It comes from uniform dangling repair of A→B,C; B→C; C dangling, followed by uniform direct restart with α=1/2. For the exact probability row x=(1/4,3/10,9/20), calculate xQ, the L1 residual and the residual-based bound on ||x−π||₁, where π is the unique stationary row. For probability rows x and π, prove each coordinate error is at most half their L1 distance and deduce that an observed coordinate gap greater than an L1 error bound certifies its strict order in π. Use this to certify both C>B and B>A here. Finally verify π=(8/33,10/33,15/33) independently and compare the bound with the actual L1 error. Explain what extra evidence would be needed if x and the residual were only rounded software output.
A model adds a restart state S to two dangling page states A and B. Each page moves to S with probability 1; S moves to A or B with probability 1/2 each. Write its three-state matrix, solve stationary balance, and prove it is strongly connected but a start at S does not converge. What distribution is seen at the page-visit instants? Compare it with a different two-state model that repairs each page by a self-loop and then performs direct uniform restart with probability 1/2 on every transition. Find that model’s matrix and stationary distribution and prove convergence. Explain why dropping S from a displayed diagram without changing the clock and transition rule is not a valid identification of the two models.
