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Fig. 18.2. Structure of adaptive networks. Plotted is the mean nearest-neighbor degree
h
k
nn
i
(top) and the degree distribution
k
for susceptible nodes (bottom, circles) and infected nodes
(bottom, dots) depending on the degree k. (Left) Indiscriminate rewiring: the network is a
random graph with Poissonian degree distributions and vanishing degree correlation. (Center) No
local dynamics (p = r = 0): the infected and the susceptible nodes separate into two unconnected
random subgraphs. (Right) Adaptive network with rewiring and local dynamics (w = 0:3, r =
0:002, p = 0:008): the degree distributions are broadened considerably and a strong assortative
degree correlation appears. Figure reprinted from Gross et al. (2006).
In the rst case (left column) rewiring is done independently of the epidemic
state, in this case there is no adaptive topological change and the network becomes
a random graph. Although nodes of high degree have a slightly higher probabil-
ity of being infected, the eect of this asymmetry is hardly visible in the degree
distribution; both the infected and the susceptible nodes independently follow al-
most exactly the Poisson distribution, that would be expected in a random graph.
The neighbor-degree ishk
nn
i= 21, independently of the degree of the node under
consideration, which implies that the degree correlation vanishes.
In the second case (center column) all nodes remain in their initial state, however
all SI-links are converted into SS-links by rewiring. To investigate this in more detail
let us denote the prevalence of the disease, i.e. the density of infected nodes by I =
1S, where S is the density of of susceptible nodes. Likewise, the average density
of the links per node is denoted by [SS], [SI], and [II], respectively. So, eectively
all quantities are normalized to the total number of nodes N. In the initial random
conguration the link densities are approximately [SS] =hkiS
2
=2, [II] =hkiI
2
=2,
and [SI] =hki=2[SS][II] =hkiSI. Since all links that are initially SI-links end
up as SS-links we obtain the nal link densities [SI] = 0, [SS] =hki(1I
2
)=2, and
[II] =hki(I
2
)=2. Note that, although the susceptible nodes are isolated from the
infected, the links within each of the two subpopulations are still placed randomly,
so that the infected nodes as well as the susceptible nodes both follow a Poissonian
degree distribution, albeit with dierent mean. Consequently the degree correlation
is still zero within each subpopulation. However, since susceptible individuals have
