  The degree distribution, referred to as the delta-sequence of a network is
studied. Using the non-normalized Lorenz curve, we apply a generalized form of
the classical majorization partial order. Next, we introduce a new class of
small worlds, namely those based on degree centralities of networks. Similar to
a previous study, small worlds are defined as sequences of networks with
certain limiting properties. We distinguish between three types of small
worlds: those based on the highest degree, those based on the average degree,
and those based on the median degree. We show that these new classes of small
worlds are different from those introduced previously based on the diameter of
the network or the average and median distance between nodes. However, there
exist sequences of networks that qualify as small worlds in both senses of the
word, with stars being an example. Our approach enables the comparison of two
networks with an equal number of nodes in terms of their small-worldliness.
Finally, we introduced neighboring arrays based on the degrees of the zeroth
and first-order neighbors and proved that for trees, equal neighboring arrays
lead to equal delta-arrays.
