  We introduce and define three types of small worlds: small worlds based on
the diameter of the network (SWD), those based on the average geodesic distance
between nodes (SWA), and those based on the median geodesic distance (SWMd).
These types of networks are defined as limiting properties of sequences of
sets. We show the exact relation between these three types, namely that each
SWD network is also an SWA network and that each SWA network is also an SWMd
network. Yet, having the small-world property is rather evident, in the sense
that most networks are small-world networks in one of the three ways. We
introduce sequences of distance frequencies, so-called alpha-sequences, and
prove a relation between the majorization property between alpha-sequences and
small-world properties.
