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than the moment-closure approximation. Nevertheless, in large networks the closure
approximation (rather than the moment expansion itself) is the main source of
inaccuracy in the derivation of the ODE system. Let us therefore discuss three
approaches which avoid or amend this approximation. The starting point for this
discussion are the Eqs. (18.2), (18.3) and Eq. (18.1).
One way to avoid the approximation of the second moments is to treat them also
as dynamical variables. In this case their dynamics have to be captured by balance
equations which will in turn depend on the on the third moments. While this only
shifts the problem of closure up by one level. It is reasonable to assume that closure
at the level of third moments will yield a more precise approximation than closure
at the level of second moments. Apart from the gained accuracy, closure at the
third level would allows include processes that act on triplets or triangles rather
than on nodes or links. However, apart from a discussion in Peyrard et al. (2008)
moment closure at higher moments has so far received little attention because of
the diculties involved. In particular the number of equations in the model grows
combinatorially with the order of the expansion and the number of states in the
model.
Instead of extending the moment expansion to higher orders one can also try
to increase the accuracy of the approximation at the second order. In this case the
challenge is to nd a way to accurately compute the expected density of certain
triplets (e.g. [SSI] and [ISI]) from a given density of nodes and links(e.g. I, [SS],
[II]). This may be done for instance by a hybrid analytical/numerical approach
proposed in Gross and Kevrekidis (2008). In this case the bifurcations are computed
numerically on the level of ODEs. But, whenever the bifurcation software needs to
evaluate the system-level equations, it starts a series of a few, very short, individual-
based detailed-level simulations. From appropriately initialized simulations good
approximations of the true triplet densities are then extracted. While this procedure
provides more accurate results than analytical closure it is faster than full simulation
of the system and yields information that would be dicult to extract by simulation
alone.
Yet another approach would be to treat the second moments as unknown func-
tions of the link and node densities. Even in this general case the methods of
dynamical system theory could be applied to extract many dynamical properties of
the system. For instance the approach of generalized models can be used to inves-
tigate the stability of stationary solutions, nd transitions to oscillatory behavior,
and provide evidence for chaotic dynamics without the need to restrict the unknown
functions in the model to specic functional forms (Gross and Feudel, 2006).
18.5. Interpretation and Extensions of the Model
The adaptive SIS model which we have considered so far is clearly conceptual in
nature. In order to facilitate the mathematical analysis we have focused on the
