Information Technology Reference
In-Depth Information
necessary to postulate a form for the probability
W
(
k
j
)
that the node
j
of degree
k
j
is
deactivated by either failure or attack [
18
]:
k
j
N
W
(
k
j
)
=
,
(6.107)
j
=
1
k
j
where the superscript
can be used to represent external attacks, internal failures or the
superposition of the two.
Using the probability (
6.107
) Gallos
et al
.[
18
] studied the web tolerance under
various attack strategies. They determined that, in an intentional attack, only a little
knowledge about the highly connected nodes, the Don in the above crime scenario, is
necessary in order to dramatically change the threshold relative to the random case.
They point out that the Internet, for example, can be significantly damaged when only
a small fraction of hubs is known to the malicious hacker. The news is not all negative,
however. On the positive side, in the immunization of populations, even if we know
very little about the virus spreaders, that is, the probability of identifying them is low,
the spreading threshold can be reduced significantly with very little knowledge.
β
6.3
Deterministic web models
The search for the power-law index
3 motivated many researchers to refo-
cus their investigations away from random processes with preferential attachment to
deterministic processes. The first model that we illustrate here is that of [
7
].
α<
6.3.1
Hierarchal webs
In Figure
6.21
we show the prescription adopted to generate a hierarchal model web. We
can interpret the sketch as follows. At
n
=
0 we have very poor resolution and perceive
the web as a single black dot. At
n
1 the resolution is a little bit better and we see
that the black dot is actually a set of three black dots, with the central black dot linked
to the two side black dots by a link for each of them. The better resolution of
n
=
=
2
shows that each of the three black dots of
n
1 is, in turn, a set of three black dots and
that the central set is linked to the two side sets by two links per set. In other words,
the single links of
n
=
=
1 are a coarse-grained representation of two links. The better
resolution of
n
3 correspond to eight
long-range links, and so on. All the bottom points of the graph are connected to the top
point, which is called the root.
For instance, with
i
=
3 shows that the four long-range links of
n
=
2, which is in fact the degree of the root at
the first step of Figure
6.21
. If we keep proceeding with the next steps, we see that there
are hubs, namely nodes with a large number of links that are equivalent to the root at
i
=
1, we obtain
k
R
=
=
1. It is easy to assess that after
n
−
i
more steps, namely at the
n
th step, the number
of nodes with
k
R
links is
