Mathematics
Markov Chains and Their Steady-State Distributions
Quick fact
Google's original PageRank algorithm treats web surfing as a Markov chain, where the steady-state probability of being on a page determines that page's importance ranking.
Why this is interesting
Have you ever wondered how a random process like the weather or a stock price can settle into a predictable pattern over time? Markov chains explain how systems with no memory still reach a stable long-run behavior.
Read the full explanation
Understanding Markov Chains and Their Steady-State Distributions
Imagine a simple board game where you move based on a die roll, but your next position depends only on where you are now, not on how you got there. That's the core idea of a Markov chain: the future depends only on the present, not the past. In formal terms, this is called the Markov property. A Markov chain consists of a set of states and probabilities of moving between them each step. These probabilities are captured in a transition matrix, where each row sums to 1. If you start in some initial state and repeatedly apply the transition matrix, the probabilities of being in each state evolve. The remarkable thing is that for many chains, these probabilities converge to a fixed distribution that doesn't change anymore—this is the steady-state distribution. It tells you the long-run proportion of time the system spends in each state, regardless of where it started.
A deeper explanation
The steady-state distribution emerges because the Markov chain acts like a linear transformation on probability vectors. The transition matrix P has a special property: its largest eigenvalue is 1, and the corresponding eigenvector (normalized) gives the steady-state distribution. As you repeatedly multiply a probability vector by P, the component along this eigenvector persists while other components shrink—if the chain is regular (irreducible and aperiodic). This is why the chain converges: the memory of the initial state fades away. This convergence is the foundation of Markov Chain Monte Carlo (MCMC), where we design a chain whose steady state is the distribution we want to sample from. The steady-state concept also underlies PageRank and many models in physics, biology, and economics, making it a powerful lens for understanding long-term behavior in stochastic systems.