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For strategy dynamics, we adopt the pairwise comparison rule [Szabo and Toke
(1998); Blume (1993)], which has been recently shown to provide a convenient frame-
work of game dynamics at all intensities of selection [Traulsen et al. (2007, 2006)].
According to this rule, two individuals from the population, A and B are randomly
selected for update (only the selection of mixed pairs can change the composition of
the population). The strategy of A will replace that of B with a probability given
by the Fermi function (from statistical physics)
1
1 + e
(
A
B
)
:
p =
(16.6)
The reverse will happen with probability 1 p. The quantity , which in physics
corresponds to an inverse temperature, controls the intensity of selection. For
1, we can expand the Fermi function in a Taylor series and recover weak selection,
which can be viewed as a high temperature expansion of the dynamics [Nowak
et al. (2004); Traulsen et al. (2007)]. For 1, the intensity of selection is high.
In the limit ! 1, the Fermi function reduces to a step function: In this case
the individual with the lower payo will adopt the strategy of the other individual
regardless of the payo dierence.
The quantity of interest in nite population dynamics is the xation probability
, which is the probability that a single mutant individual of one type takes over a
resident population with N 1 individuals of another type.
16.2.3. Separation of timescales
The system of coevolving strategies and links is characterized by two timescales:
One describing the linking dynamics (
a
), the second one describing strategy dy-
namics (
e
). We can obtain analytical results in two limits, where both timescales
are separated. Dening the ratio W =
e
=
a
, separation of time scales will occur
for W 1 and W 1.
16.2.3.1. Fast strategy dynamics
If strategies change fast compared to changes of the network structure, active linking
does not aect strategy dynamics. Thus, the dynamics is identical to the evolution-
ary game dynamics on a xed network. Such systems have been tackled by many
authors for a long time [Nowak and May (1992); Herz (1994); Lindgren and Nordahl
(1994); Szabo and Toke (1998); Hauert (2002); Vainstein and Arenzon (2001); Szabo
and Vukov (2004); Abramson and Kuperman (2001); Ebel and Bornholdt (2002a);
Holme et al. (2003); Santos and Pacheco (2005); Ohtsuki et al. (2006); Santos et al.
(2006b, 2008)]. The diculty of an analytical solution for such systems is deter-
mined by the topology of the network, which corresponds to an initial condition in
our case. Analytical solutions are feasible only for few topologies. One important
limiting case leading to analytical solutions are complete networks corresponding
