Webs, trees and branches - Complex Webs: Anticipating the Improbable - page 46

a fractal and consequently we devote some space to a discussion of fractal webs. It
has been thirty years since Benoît Mandelbrot [ 44 ] introduced the notion of fractals
into science to capture the spatial irregularity of the geometry of certain objects and
the temporal burstiness of certain time series. This introduction touches on some of the
applications that have been made since the publication of Mandelbrot's seminal topics,
but we leave for subsequent chapters how such fractal webs change in time. Modeling
the dynamics of fractals is of fairly recent origin and much of that discussion breaks
new ground, particularly with regard to the webs they describe.
Anyone who has lived in California for any length of time knows the disorientation
resulting from the floor of a building suddenly lurching like the deck of a ship and
coffee spilling from your cup. For some reason this is particularly disconcerting if the
building is your home. One's first earthquake can be exciting and has convinced more
than one person that the price of California's sunshine is too high. The magnitudes of
these quakes and the time intervals between them have been recorded for over a hundred
years so it is not surprising that quakes have empirical laws associated with both their
size and their frequency. Earthquake magnitudes are measured by the Richter scale, the
logarithm to the base 10 of the maximum amplitude of the detected motion, and the
cumulative distribution of Californian earthquakes is given in Figure 2.1 , which clearly
follows the Gutenberg-Richter law. The linear horizontal scale is actually the exponent
of the factor of ten and therefore this is a log-log plot of the data. It is evident that the
distribution in the number of earthquakes of a given size versus the magnitude of the
quake follows an inverse power law beyond the shaded region. However, the full data set
can be fit using the hyperbolic distribution ( 1.41 ) derived using the maximum-entropy
argument.
Another consideration that invariably arises after experiencing an earthquake is how
long it will be before the “big one” hits. Will the devastating earthquake that will level a
city occur next month, next year or ten years from now? Data have been collected on the
time intervals between earthquakes of a given magnitude and are shown in Figure 2.2 .
One of the oldest geophysical laws is that of Omori, who in 1895 [ 55 ] determined that
the time interval between earthquakes of a given magnitude follows an inverse power
law with a slope of

− 1. With more recent data on California earthquakes it is possible to

The number of earthquakes in California from January 1910 to May 1992, as recorded in the
Berkeley Earthquake Catalog. The data are taken from the National Geophysical Data Center
www.ngdc.noaa.gov. Adapted with permission from [ 53 ].

Figure 2.1.