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rewiring events occur at a rate w[SI] and every rewiring event gives rise to exactly
one SS-link, the total rate at which rewiring creates SS-links is simply w[SI].
Summing all the terms, the dynamics of the rst moments can be described by
the balance equations
d
dt
[SS] = (r + w)[SI]p[SSI] ;
(18.2)
d
dt
[II] = p([SI] + [ISI])2r[II] :
(18.3)
Again, these equations do not yet constitute a closed model, but depend on the un-
known third moments [SSI] and [ISI]. However, the rst order-moment expansion
captures the eect of rewiring. While we will return to the equation above later,
a feasible way of closing the system is to approximate the second moments by a
mean-eld-like approximation: the moment-closure approximation.
Let us start by approximating [ISI]. One half of the ISI-triplet is actually an SI-
link, which we know occurs at the density [SI]. In order to approximate the number
ISI-triplets running through a given link we have to calculate the expectation value
of the number of additional infected nodes that are connected to the susceptible
node. For this purpose let us assume that the susceptible node of the given SI-link
has an expected number ofhqilinks in addition to the one that is already occupied
in the SI-link. Every one of these links is an SI-link with probability [SI]=(hkiS).
(Here, we have neglected the fact that we have already used up one of the total
number of SI-links. This assumption is good if the number of SI-links is reasonably
large.) Taking the density of SI-links and the probability that they connect to
additional SI-link into account we obtain
[ISI] =
[SI]
2
S
(18.4)
where =hqi=hkiremains to be determined. The quantityhqithat appears in
is the so-called mean excess degree. Precisely speaking it denotes the expected
number of additional links that are found by following a random link.
The mean excess degree of a network is governed by two opposing eects (New-
man, 2003): One the one hand we are only counting the additional links, so that
for a node of given degree k the excess degree is q = k1. On the other hand, we
have reached the node by following a link and therefore have a higher probability
to arrive at a node of high degree. Depending on the network topology thehqican
therefore be larger or smaller thanhki. It is a special property of Erdos-Renyi ran-
dom graphs that both eects cancel, so thathqi=hkiand = 1. Smaller values of
are found in homogeneous networks such as regular lattices while networks with a
wider degree distribution generally correspond to larger (and in scale-free networks
even diverging) values of .
In the present example the network topology changes dynamically in time. For
the sake of simplicity I will assume that 1. Note, that this could be called
