Flow, process and networks

Transition-probability matrix

Lists probabilities of moving from each current state to each next state.

Table
Transition
Relationship

What it answers

How likely is each next state from the current state?

Shape: State-by-state probability grid
What it shows: Persistence and switching probability.

Example

The question

How likely is a customer to move from one status to another next month?

Monthly chance of moving between customer statuses

Rows are this month’s status, columns next month’s; each row adds to 100%. Active customers stay active 85% of the time; at-risk customers recover 30% and are lost 30%.

Illustrative data. It belongs to no company.

When to use it

Use it when

Discrete-state transition models.

Avoid it when

Undefined time steps or mixing counts with probabilities.

How to read it

Choose a current-state row and read probabilities across it.

Note

Rows normally sum to one under a row-stochastic convention.

The data you need

Layout: Transition edge list
Fields: FromState: text; ToState: text; Probability: number in [0,1]
One row is: One state-to-state transition probability. What is the grain?
Rules: For a row-stochastic model, sum probabilities to 1 for each FromState. Include absorbing-state self-transitions.
Optional: Alternative transition counts with a documented normalisation.
Preparation: Aggregate transition counts and normalise each source-state row to one, unless a different stochastic convention is explicitly chosen.
Units: Conditional probabilities per time step. Each source-state row sums 1 under a row-stochastic convention.

An edge list: one row per link, from one node to another, often with a node table beside it.

Worked example

FromA: stayA0.8, moveB0.2. The row sums 1, so a case in A has 20% probability of B next step under the model.

FromState	ToState	Probability
A	A	0.8
A	B	0.2
B	A	0.3
B	B	0.7

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