Information Technology Reference
In-Depth Information
2
Webs, trees and branches
In the previous chapter we examined examples of hyperbolic and inverse power-law
distributions dating back to the beginning of the last century, addressing biodiversity,
urban growth and scientific productivity. There was no discussion of physical structure
in these examples because they concerned the counting of various quantities such as
species, people, cities, words, and scientific publications. It is also interesting to exam-
ine the structure of complex physical phenomena, such as the familiar irregularity in the
lightning flashes shown in Figure
1.13
. The branches of the lightning bolt persist for
fractions of a second and then blink out of existence. The impression left is verified in
photographs, where the zigzag pattern of the electrical discharge is captured. The time
scale for the formation of the individual zigs and zags is on the order of milliseconds and
the space scales can be hundreds of meters. So let us turn our attention to webs having
time scales of milliseconds, years or even centuries and spatial scales from millimeters
to kilometers.
All things happen in space and time, and phenomena localized in space and time
are called events. Publish a paper. Run a red light. It rains. The first two identify an
occurrence at a specific point in time with an implicit location in space; the third implies
a phenomenon extended in time over a confined location in space. But events are mental
constructs that we use to delineate ongoing processes so that not everything happens at
once. Publishing a paper is the end result of a fairly long process involving getting
an idea about a possible research topic, doing the research, knowing when to gather
results together into a paper, writing the paper, sending the manuscript to the appropriate
journal, reading and responding to the referees' criticism of the paper, and eventually
having the paper accepted for publication. All of these activities are part of the scientific
publishing process, so it would appear that any modeling of the process should involve
an understanding of each of the individual steps. Yet in the previous chapter we saw that
Lotka was able to determine the distribution of papers published without such detailed
understanding. This is both the strength and the weakness of statistical/probabilistic
models. Such models enable the scientist to find patterns in data from complex webs,
often without connecting that structure to the mechanisms known to be operating within
the web.
In this chapter we explore ways of relating the complexity of a web to the form of the
empirical distribution function. In fact we are interested in the inverse problem: given
an empirical distribution, what is entailed about the underlying complex web producing
the distribution? One tool that we find particularly useful is the geometrical notion of
