Simple Markov Chain Steady State Demo
Explore the long-run behavior of small Markov chains. Enter a transition matrix, choose an initial distribution, and watch power iteration converge to the steady state distribution.
Explore Markov Chain Steady States
Enter a transition matrix to discover the long-run behavior of your Markov chain. Watch how the distribution converges to the steady state through power iteration.
Quick Start:
- Choose the number of states (2-6)
- Enter transition probabilities in the matrix
- Select an initial distribution
- Click "Compute Steady State" to run power iteration
What is a Markov Chain?
A Markov chain is a mathematical model where the next state depends only on the current state, not on the history. The steady state is the distribution the chain converges to after many steps.
Try a Preset Example
Use the "Load Preset Example" dropdown to explore common Markov chain patterns like weather transitions, absorbing states, or PageRank-style link structures.
Start by setting up your chain above
Understanding Markov Chains and Steady States
What is a Markov Chain?
A Markov chain is a mathematical model that describes a sequence of possible events where the probability of each event depends only on the state attained in the previous event. This property is called the Markov property or "memorylessness."
Common examples include weather patterns (tomorrow's weather depends on today's), customer behavior (next purchase depends on current status), and random walks (next position depends on current position).
The Transition Matrix
A transition matrix P encodes all the probabilities of moving between states. The entry P[i][j] represents the probability of transitioning from state i to state j. Key properties:
- All entries are non-negative (probabilities can't be negative)
- Each row sums to 1 (you must go somewhere, including staying put)
- The matrix is square (same states for "from" and "to")
This tool automatically normalizes rows that don't sum to 1, making it forgiving of input errors.
What is the Steady State Distribution?
The steady state (or stationary) distribution is a probability distribution that remains unchanged after applying the transition matrix. Mathematically, if π is the steady state distribution:
This means that once the chain reaches the steady state, the probability of being in each state stays constant over time, even as the system continues to transition between states.
Power Iteration Method
This tool uses power iteration to find the steady state. Starting from an initial distribution π0, we repeatedly multiply by the transition matrix:
For well-behaved chains, this sequence converges to the steady state distribution. We stop when the change between iterations (measured by L1 distance) falls below a tolerance threshold.
Important Chain Properties
- Irreducible: Every state can eventually be reached from every other state. Chains with isolated groups of states are reducible.
- Aperiodic: The chain doesn't get stuck in deterministic cycles. Having self-loops (P[i][i] > 0) often ensures aperiodicity.
- Absorbing: A state is absorbing if once entered, the chain never leaves (P[i][i] = 1). Absorbing states strongly influence long-run behavior.
An irreducible and aperiodic chain has a unique steady state that doesn't depend on the initial distribution. Reducible or periodic chains may have multiple steady states or no steady state at all.
Interpreting the Initial Distribution
The initial distribution represents where the system starts. Three common choices:
- Uniform: Equal probability across all states (no prior knowledge)
- Single state: The system definitely starts in one specific state
- Custom: You have prior knowledge about starting probabilities
For irreducible aperiodic chains, the initial distribution doesn't affect the final steady state—only how long it takes to get there.
Common Applications
- PageRank: Google's algorithm models web surfing as a Markov chain
- Customer churn: Modeling customer lifecycle states (new, active, churned)
- Weather forecasting: Simple weather state transitions
- Queueing theory: Modeling system states in service systems
- Reliability: Component failure and repair states
- Games: Board game positions and dice outcomes
Limitations of This Demo
This is a simplified educational tool with important limitations:
- Size limit: Only 2-6 states (real applications may have thousands)
- Power iteration only: May not converge for all chains (eigenvalue methods are more robust)
- Heuristic checks: Irreducibility and aperiodicity tests are approximate
- Numerical precision: May have issues with very small probabilities
- No continuous-time: Only discrete-time Markov chains are modeled
Mathematical Foundation
For those interested in the theory, the steady state distribution is the left eigenvector of P corresponding to eigenvalue 1:
The Perron-Frobenius theorem guarantees that for irreducible aperiodic chains, there exists a unique steady state distribution with all positive entries.
Frequently Asked Questions
Each row represents all possible transitions from one state. Since you must go somewhere (including staying in the same state), the probabilities must add up to 1 (100%).
This tool automatically normalizes rows that don't sum to 1, but large deviations are flagged as they may indicate input errors.
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