EvoLudoLab: Moran process on the Frucht graph

Color code:	Residents	Mutants
	New resident	New mutant

Payoff code:	Residents	Mutants

Fixation and absorption times on the Frucht graph

On the Frucht graph the expected times until a single mutant has fixed 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. In fact, due to the lack of further structural symmetries (in addition to regularity) the fixation times are different for each vertex. 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. Because the Frucht graph is a circulation 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

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

--fitness <r,m>

fitness of residents r and of mutants m.

--init <r,m>

initial frequencies of residents r and mutants m. Frequencies that do not add up to 100% are scaled accordingly.

--inittype <type>

type of initial configuration:

frequency

random distribution with given frequency

uniform

uniform random distribution

monomorphic

monomorphic initialization

mutant

single mutant in homogeneous population of another type. Mutant and resident types are determined by the types with the lowest and highest frequency, respectively (see option --init).

stripes

stripes of traits

kaleidoscopes

(optional) configurations that produce evolutionary kaleidoscopes for deterministic updates (players and population). Not available for all types of games.