# Example of matrix population models for graphs.

(a) The population resides on the graph **W**. (b)-(d) We build all relevant matrix population models for both the Bd process (left) and the dB process (right). Each matrix column represents a node. (b)-(c) A column reports the per-time-step probability that the corresponding node contributes the individual in the row node. (b) Matrices correspond to the average matrix model in the neutral case (i.e. only residents). (c) Matrices correspond to the expected matrix model in the presence of a single mutant at node 2. (d) Matrices correspond to invasion matrices. For invasion, the column reports the per-time-step probability that the corresponding (mutant) node contributes the individual in the row node assuming all nodes other than the focal column node are resident. Comparing (c) and (d) helps understanding how invasion matrices are constructed: column 2 for each update process is shared by the matrices. The invasion matrix is built by binding the mutant columns obtained from the construction of a matrix like the one in (c) for each possible mutant position.