Fig 1.
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.
Fig 2.
Invasion vs. fixation on a weighted directed graph.
(a) The studied graph. Node dependent fitness for both types is found in the table just below. The mutant has greater fitness than the resident at a single node, while at another node the mutant has the same fitness as the resident plus some δ. Using the invasion criterion proposed in the main text (i.e. Δλ = 0) we predicted the value of δ required to make the mutant neutral. (b) the fixation probability for different δ values. Fixation probabilities were estimated from 20000 simulations of the process for each point for the first plots and 106 simulations for the second and the third plot. The initial mutant is equally likely to appear in any one node. The horizontal dashed line gives the neutrality level (i.e. fixation probability equal to initial frequency). The vertical solid line identifies the neutral mutant according to our criterion. When invasion predicts fixation, fixation probabilities at the left of the intersection between the vertical (criterion) line and the horizontal (neutrality) line should be smaller than 1/N (i.e. deleterious mutation), while fixation probabilities at the right of this intersection should be greater than 1/N (i.e. beneficial mutation). In the third plot, invasion analysis erroneously classifies all mutants as successful invaders.
Fig 3.
Comparison of approximate fixation probabilities (dotted line) of a beneficial mutant against exact results (solid line) in undirected graphs of different sizes.
Fitness is node independent. Resident fitness is normalized to 1 and mutant fitness is 1 + s. The initial mutant is equally likely to appear in any node. The horizontal dashed line gives neutrality, i.e. 1/N. Along the vertical dashed line, the selective advantage s at NΔλ/σ2 = 1 is reported. Exact results for the complete graph under Bd and dB are given in [25]. Exact results for the star under Bd are given in [2]. Exact results for the cycle under dB are given in [30]. Approximations are computed from Eq (9) where the quantity Δλ is retrieved from the Perron root of .
Fig 4.
Effective size for large undirected graphs under both update processes.
Fig 5.
Effective size in networks under Birth-death update.
The average effective size for three kinds of networks is computed in a random sample (with replacement) of 100 connected networks for each size and network kind. Dashed lines indicate Ne = N and . For random networks, the probability of including an edge in the Erdős-Réyni algorithm was set to 0.75. For small-world networks, a 4-neighbors lattice was generated with a probability of 0.2 of link rewiring. Scale-free networks were generated by linear preferential attachment [39]. Random networks and small-world networks have an average effective size exactly aligned with that of a complete graph.