# EvoLudoLab: Moran process on the Tietze graph

Color code: Residents Mutants New resident New mutant
 Payoff code: Residents Mutants

## Fixation and absorption times on Tietze's graph

On Tietze's graph the expected times until a single mutant fixes in the population, or until either one of the absorbing states is reached (homogenous population of residents or mutants), depends on which vertex the initial mutant arose. However, because Tietze's graph exhibits some structural symmetries, the fixation times are identical for groups of vertices. More specifically there are three groups of vertices, two of size three (vertices \(\{2, 5, 8\}\) and \(\{9, 10, 11\}\)) and one of size six (vertices \(\{0, 1, 3, 4, 6, 7\}\), with identical fixation times.

The histogram shows the statistics for the conditional fixation times of a single mutant, i.e. the expected time it takes for a mutant to take over the population, the expected time for the mutant to go extinct (fixation of resident), as well as the expected time to reach an absorbing states for each vertex where the initial mutant was placed. The initial mutant is sequentially placed on every vertex. For reference, the corresponding fixation times for the original Moran process, or, equivalently, for the spatial Moran process on circulations is indicated by a dark red line (may not be visible if its outside the range of displayed values).

Also check out the statistics for the fixation probabilities of mutants and residents. Tietze's graph is also a circulation and hence the probabilities are the same for all vertices and are also identical to the original Moran process (indicated by the dark red line). However, also note that a considerable number of sample runs are required to obtain a good estimate of the the fixation probabilities. Unfortunately, an even significantly larger number of samples is required for accurate estimates of fixation times because of large variation - fixation times appear to be following an exponential distribution. To make matters worse, differences in fixation times between different nodes are rather small and hence it becomes very challenging to prove the differences through simulations. Fortunately, however, for sufficiently small graphs the fixation times can be calculated numerically and simulations easily confirm the trends observed in the exact numerical analysis.

For the simulations, the fitness of residents is set to \(1\) and that of mutants to \(2\). Thus, a single mutant has approximately a \(50\%\) chance to take over the population.

### Data views

 Snapshot of the spatial arrangement of strategies. 3D view of snapshot of the spatial arrangement of strategies. Time evolution of the strategy frequencies. Snapshot of the spatial distribution of payoffs. 3D view of snapshot of the spatial distribution of payoffs. Time evolution of average population payoff bounded by the minimum and maximum individual payoff. Snapshot of payoff distribution in population. Degree distribution in structured populations. Statistics of fixation probability for each vertex where the initial mutant arose. Statistics of conditional fixation times of residents and mutants as well as absorption time for each vertex where the initial mutant arose. Message log from engine.

## Game parameters

The list below describes only the few parameters related to the evolutionary dynamics of residents and mutants with fixed fitness (constant selection). Numerous other parameters are available to set population structures or update rules on the player as well as population level.

--resident <r>
fitness of resident r.
--mutant <m>
fitness of mutant m.
--initfreqs <m:r>
initial frequencies of residents r and mutants m. Frequencies that do not add up to 100% are scaled accordingly. If either frequency is zero, the population is initialized to a homogenous state with just a single, randomly placed individual of the opposite type.