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N
C
=15
N
C
=20
N
C
=10
Fig. 16.1. Frequency dependent steady state dynamics. Results of active linking dynamics for
a population size of N = 30 individuals. Cooperators are located in the \inner-rim", and are
represented by blue circles, whereas defectors are located in the \outer-rim", and are represented
by red circles. In this way, CC-links (solid cyan lines) live only within the \inner-rim", whereas
CD-links (solid red lines) occupy the space between the rims while DD-links (solid grey lines)
cross the entire region of the gure. Each panel depicts a snapshot in the steady state of the
active-linking dynamics, associated with a dierent (and xed) frequency of C and D players. The
parameters determining the active linking dynamics are:
C
=
D
= 0:5,
CC
= 0:5,
CD
= 0:25
and
DD
= 0:5.
interesting case we discuss below is
2
, where the system has xed points with
intermediate ranges of X, Y and Z. Rescaling and in an appropriate way (note
that the equation contains squares of and linear terms of ) does not change the
xed points of the system, but aects the overall timescale of active linking. When
this process is coupled with strategy dynamics, such changes can be crucial.
While the above is probably the simplest possibility to model linking dynamics,
more sophisticated choices are possible, taking for example the number of existing
links of a node into account. However, to address some general properties of the
coevolution between links and strategies, we concentrate on the simplest choice rst.
In the steady state, the number of links of the three dierent types is given by
2
C
X
= X
m
2
C
+
CC
= X
m
CC
;
C
D
C
D
+
CD
= Y
m
CD
;
Y
= Y
m
(16.2)
2
D
2
D
+
DD
= Z
m
DD
:
Z
= Z
m
Here,
CC
,
CD
, and
DD
are the fractions of active CC, CD and DD links in
the steady states. Examples of population structures attained under steady-state
dynamics for three dierent combinations of (N
C
;N
D
) are shown in Fig. 16.1.
16.2.2. Strategy dynamics
Next, we address the dynamics of the strategies at the nodes. We consider the
stochastic dynamics of a nite population, i.e. we restrict ourselves to nite net-
