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Fig. 18.4. Two parameter bifurcation diagram showing the dependence on the rewiring rate w
and the infection probability p. In the white and light gray regions there is only a single attractor,
which is a healthy state in the white region and an endemic state in the light gray region. In the
medium gray region both of these states are stable. Another smaller region of bistability is shown
in dark gray. Here, a stable healthy state coexists with a stable epidemic cycle. The transition lines
between these regions correspond to transcritical (dash-dotted), fold (dashed), Hopf (continuous),
and cycle fold (dotted) bifurcations. Note that the fold and transcritical bifurcation lines emerge
from a cusp bifurcation at p = 0:0001, w = 0. The rewiring rate is r = 0:0002. Figure reprinted
from Gross et al. (2006).
18.4.3. Accuracy and extensions of the moment closure
approximation
The derivation of the Eqs. (18.5){(18.7) has involved approximations at various
stages. Nevertheless the predicted bifurcation diagrams are in very good agree-
ment with the numerical results. In the literature much attention has been paid
to inaccuracies in the factor . However, the results presented here show that the
approximation = 1 yields good results even for networks with a relatively wide
degree distribution.
One reason for the good agreement between theory and simulation is that adap-
tive networks are particularly well suited to be treated by the moment-closure ap-
proximation; The ongoing topological evolution of an adaptive network means that
there is a constant mixing of the topology. One could say that, over time, the
adaptive network is an ensemble of itself. It is therefore more accessible to mean-
eld-like approximations. In addition the mixing reduces spatial correlations. This
may explain why moment closure seems to yield better results on adaptive networks
than on static ones.
In my experience ODEs found by moment closure are in good agreement with
the explicit numerical simulation of adaptive networks as long as the network under
consideration is suciently large (N10
5
). In small networks deviations can
appear because of stochastic uctuations, which for instance in the SIS model can
lead to premature extinction of the disease. However, as real world networks are
mostly large this deviation can be seen as a shortcoming of the simulation rather
