Dependencies

Mutants have fitness \(r{\gt}0\) and residents fitness \(1\). A step chooses a parent \(u\) with probability proportional to its fitness and an offspring position \(w\) uniformly among the neighbours of \(u\) (or \(w=u\) if \(u\) is isolated); the individual at \(w\) is replaced by a copy of the parent.

In one pull (death–Birth) step, a uniformly random vertex \(u\) adopts the type of a uniformly random neighbour \(w\) (an isolated vertex keeps its type), so the pair \((u,w)\) has weight \(\frac1n\cdot \frac1{\deg u}\) for \(w\sim u\). The neutral push (Birth–death) process is the Moran kernel with \(r=1\).

For a configuration with mutant set \(S\), \(\Phi _{\mathrm{push}}(S)=\sum _{v\in S}1/\deg v\) and \(\Phi _{\mathrm{pull}}(S)=\sum _{v\in S}\deg v\).

For a configuration with \(i\) mutant leaves, \(\Phi =q^i\) if the centre is a resident and \(\Phi =\kappa q^i\) if it is a mutant.

On a \(d\)-regular graph, the probability that a step adds a mutant equals \(r\) times the probability that it removes one: both are proportional to the number of edges between the two types, counted from either side.

Let an observable be invariant in expectation, equal to \(1\) on the all-mutant and to \(0\) on the all-resident configuration. If the all-mutant configuration is absorbing and the probability that neither type has taken over tends to zero, the observable is the fixation probability.

The weights of (parent, offspring position) pairs are nonnegative and sum to one.

On a connected graph, the probability that neither type has taken over tends to zero.

On a regular graph, \((1/r)^{\# \text{mutants}}\) is preserved in expectation by one step; when \(r=1\), so is the number of mutants.

On a connected regular graph with \(r\neq 1\), the fixation probability from \(k\) mutants is \((1-r^{-k})/(1-r^{-n})\).

On the complete graph, the fixation probability from \(k\) mutants is \((1-r^{-k})/(1-r^{-n})\).

On a connected regular graph with \(r=1\), the fixation probability is \(k/n\).

On a connected graph, the probability that the pull process has not reached a constant configuration after \(t\) steps tends to \(0\).

On a connected graph with at least two vertices, the fixation probability of a mutant set \(S\) is \(\sum _{v\in S}\frac1{\deg v}\big/\sum _{v\in V}\frac1{\deg v}\) under neutral push and \(\sum _{v\in S}\deg v/(2|E|)\) under pull.

On every finite graph, \(\Phi _{\mathrm{push}}\) is invariant in expectation under neutral push and \(\Phi _{\mathrm{pull}}\) is invariant in expectation under pull.

For \(r\ne 1\), the fixation probability from \(s\) is \((1-\Phi (s))/(1-\kappa q^n)\). A single mutant fixes with probability \((1-q)/(1-\kappa q^n)\) from a leaf, \((1-\kappa )/(1-\kappa q^n)\) from the centre, and \((n(1-q)+(1-\kappa ))/((n+1)(1-\kappa q^n))\) from a uniformly random vertex.

For every \(n\ge 2\), the fixation probability from a uniformly random vertex is larger than Moran’s \((1-1/r)/(1-r^{-(n+1)})\) for \(r{\gt}1\) and smaller for \(0{\lt}r{\lt}1\).

For every \(r{\gt}0\), \(\Phi \) is invariant in expectation under one Birth–death step on the star.

For \(r{\gt}1\), the fixation probability from a uniformly random vertex of the star with \(n\) leaves tends to \(1-1/r^2\) as \(n\to \infty \).

For every \(r{\gt}0\), a single mutant at a uniformly random vertex fixes with probability