Information Technology Reference
In-Depth Information
make and in putting metrics on the natural variability in our elaborate predictions of the
future behavior of the complex webs in which we are entangled.
In the literature of complex webs there is increasing focus on the search for the
most useful web topology. A conviction shared by a significant portion of the scien-
tists involved in research underlying the search for a network science is that there exists
a close connection between web topology and web function. There is also increasing
interest in web dynamics and how that dynamics is related to web topology and func-
tion. It is convenient to stress that the concept of web dynamics that is leading the
research work of many groups is the time dependence of web topology that is based on
the model put forward by Barabási and Albert (BA) introduced in earlier chapters [
6
].
Web dynamics is interpreted in this chapter to mean the time-dependent change in the
properties of the units making up the nodes of the web. In the next chapter we examine a
model in which the elements of the web dynamically interact with one another in ways
that change their properties.
The condition of scale-free webs refers to the distribution density
θ(
k
)
of the number
of links
k
connecting the nodes, which is typically an inverse power law
1
k
α
.
θ(
k
)
∝
(6.1)
The inverse power-law parameter
α
of the distribution of node links must not be con-
fused with the parameter
discussed in
earlier chapters. However, it is interesting to ask whether a relation exists between the
two distributions and their power-law parameters. This is one of the problems addressed
in this and the next chapter. We limit our discussion in this chapter to pointing out that
according to the model of
preferential attachment
μ
of the waiting-time distribution density
ψ(
t
)
3, as we discuss presently, but
experimental and theoretical arguments indicate that the power-law index is generally in
the interval 1
α
=
3. The highest-degree nodes, those nodes with the greatest number
of connections, are often called “hubs” and are thought to serve specific purposes in
their webs, although what that purpose is depends greatly on the domain of the index.
Figure
6.1
suggests that there are clusters with hubs, the hubs being the local leaders
that may facilitate a cluster reaching a consensus that can then be transmitted to other
clusters by means of long-range interactions. However, there is apparently no reason
why local clusters may not be characterized by all-to-all coupling, which should make
it easier to reach consensus locally. Is a local leader necessary in order to realize local
≤
α
≤
In the literature of complex networks there is the widespread conviction that the real networks on
the right are, unlike the random network on the left, characterized by scale-free distributions.
This property implies the existence of hubs, shown as filled circles on the right, namely nodes
with a number of links much higher than that of the typical nodes.
Figure 6.1.
