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and, so far, only general statements as \the probability to be generous is correlated
with the number of social links of an individual" can be made [Branas-Garza et al.
(2007)].
One important property of social networks that is seldom addressed in theoret-
ical studies is that real world social networks are not static. Instead, we make new
friends and lose touch with old ones, depending on the kind of interaction we have
with them. This makes social networks an example of an adaptive network [Gross
and Blasius (2008)]. The basic idea is that interactions which benet both partners
last longer than interactions where one partner is exploited by the other. Here, we
discuss such an approach, which leads to analytical results in certain limits. These
serve as important starting points for further developments.
16.2.ActiveLinking:RandomLinkFormation
We break down the model into two parts: Evolutionary dynamics of strategies
(or behaviors) of the individuals associated with nodes in a network whose links
describe social interactions. The adaptive nature of the social interactions leads to
a network linking dynamics. We consider two-player games of cooperation in which
individuals can choose to give help to the opponent (to cooperate, C), or to refuse
to do so (to defect, D). The network is of constant size with N nodes. The number
of links, however, is not constant and changes over time. There are N
C
individuals
that cooperate and N
D
= N N
C
individuals that defect.
16.2.1. Linking dynamics
An interaction between two players occurs if there is a link between these players.
Links are formed at certain rates and have specic life-times. We denote by X(t)
the number of CC links at time t. Similarly, Y (t) and Z(t) are the number of
CD and DD links at time t. The maximum possible number of CC, CD and DD
links is given by X
m
= N
C
(N
C
1)=2, Y
m
= N
C
N
D
, and Z
m
= N
D
(N
D
1)=2,
respectively. Suppose cooperators form new links at rate
C
and defectors form
new links at rate
D
. Thus, CC links are formed at a rate
2
C
, CD links are formed
at a rate
C
D
and DD links are formed at a rate
2
D
. The death rates of CC, CD
and DD links are given by
CC
,
CD
and
DD
, respectively. If the number of links
is large, we can model the dynamics of links by dierential equations. We obtain a
system of three ordinary dierential equations for the number of links
X =
2
C
(X
m
X)
CC
X ;
Y =
C
D
(Y
m
Y )
CD
Y ;
Z =
2
D
(Z
m
Z)
DD
Z :
(16.1)
For
2
, the network is almost complete, which recovers the results for well-
mixed populations. For
2
, the network is sparse with few links. The most
