5. Insights from probability and statistics
One can also draw a graph of the frequency with which outcomes occur. These distributions have characteristic shapes depending on the processes involved in the way data are generated. For discrete situations the sum of the "probabilities" of the outcomes adds to one. For situations where there are a non-discretely infinite collection of outcomes, one requires a curve the area under which (i.e. between the positive valued curve and the axis) is 1. For a fair die, one would think that each outcome would occur about equally often and that the "distribution" of the outcomes is uniform. For other processes the distribution of the outcomes follows the so-called Gauss curve, normal distribution, or bell-shaped curve (scaled so that the area under the curve is 1):
Mathematicians have discovered many distributions other than the uniform distribution and the normal distribution. Among these are the Poisson distribution, the exponential distribution, and the Erlang distribution. Each of these distributions arises from a particular class of applied problems. For example, the Erlang distribution finds many applications in the field of telephony.
One empirical approach to complex networks is to examine the connected underlying triples and compute the statistical distribution of the different types of triples; that is, the percentage of each type of (connected) triple for the different triples that appear in the graph. One now can try to see what networks which have similar distributions of triples have in common.
(Note that in the bottom right graph in Figure 2 there are two edges that intersect at a point that is not a vertex of the graph.)
Using the data one collects on the basis of looking at the distributions of subgraphs of different kinds, one can try to cluster the networks that have different distributions and see what features might explain their similar characteristics.
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