Abstract | This work extends what is known so far for a basic model of
evolutionary antagonis
m in undirected ne
tworks (graphs).
More specif-
ically, this work studies the generalized Moran process, as introduced
by Lieberman, Hauert, and Nowak [Nature, 433:312-316, 2005], where
the individuals of a population reside on the vertices of an undirected
connected graph. The initial population has a single
mutant
of a
fitness
value
r
(typically
r>
1), residing at some vertex
v
of the graph, while
every other vertex is initially occupied by an individual of fitness 1. At
every step of this process, an individual (i.e. vertex) is randomly chosen
for reproduction with probability proportional to its fitness, and then it
places a copy of itself on a random neighbor, thus replacing the individ-
ual that was residing there. The main quantity of interest is the
fixation
probability
, i.e. the probability that eventually the whole graph is occu-
pied by descendants of the mutant. In this work we concentrate on the
fixation probability when the mutant is initially on a specific vertex
v
,
thus refining the older notion of Lieberman et al. which studied the fix-
ation probability when the initial mutant is placed at a random vertex.
We then aim at finding graphs that have many “strong starts” (or many
“weak starts”) for the mutant. Thus we introduce a parameterized no-
tion of
selective amplifiers
(resp.
selective suppressors
)ofevolution.We
prove the existence of
strong
selective amplifiers (i.e. for
h
(
n
)=
Θ
(
n
)
vertices
v
the fixation probability of
v
is at least 1
−
c
(
r
)
n
for a func-
tion
c
(
r
) that depends only on
r
), and the existence of quite strong
selective suppressors. Regarding the traditional notion of fixation prob-
ability from a random start, we provi
de strong upper and lower bounds:
first we demonstrate the non-existence of “strong universal” amplifiers,
and second we prove the
Thermal Theorem
which states that for any
undirected graph, when the mutant starts at vertex
v
, the fixation prob-
ability at least (
r
−
1)
/
(
r
+
deg
v
deg
min
). This theorem (which extends the
“Isothermal Theorem” of Lieberman et al. for regular graphs) implies
an almost tight lower bound for the usual notion of fixation probability.
Our proof techniques are original and are based on new domination ar-
guments which may be of general interest in Markov Processes that are
of the general birth-death type. |