Page 336 - Becoming Metric Wise

328 Becoming Metric-Wise

10.2.5 Reciprocity
In a directed unweighted network, reciprocity is defined as the observed
probability that if node i points to node j, also j points also to node i.A
friendship network is a typical case for which it makes sense to calculate a
reciprocity value.
In weighted networks such as journal citation networks reciprocity is
defined as follows (Squartini et al., 2013):
Let w ij be the weight of the link between node i and node j, in that
order, where w ii 5 0 for all i. Let rw ij 5 rw ji 5 min(w ij ,w ji ) denote the
reciprocated weight between nodes i and j. Then the reciprocity value W
for the network is:

P P
i j rw ij
(10.12)
W 5 P P
i j w ij
10.2.6 Complex Networks and the Small-World Phenomenon

The term complex networks is not precisely defined, but usually refers to
large networks which are neither regular nor completely random. One
may say that the citation network is a complex network. Two well-known
and much studied classes of complex networks are scale-free networks and
small-world networks. The term scale-free network is used for networks
that have a degree distribution following a power law. Size and rank fre-
quency of hyperlinks between websites follow—approximately a power law
as was observed by Rousseau (1997b) for a small sample and by Faloutsos
et al. (1999) for a much larger set. Also the network of protein interactions
has a power law distribution (Jeong et al., 2001). A small-world network is
characterized as a network exhibiting a high degree of clustering and hav-
ing at the same time a small average distance between nodes. These charac-
teristics distinguish small-world networks from random networks which
also have a small average distance between nodes, but a much smaller
degree of clustering. The small-world phenomenon is related to the notion
of six degrees of separation (Kochen, 1989; Milgram, 1967; Newman &
Watts, 1999). The concept that all persons on earth are linked through
friendship or acquaintanceship in at most six steps was first proposed in
1929 by the Hungarian writer Frigyes Karinthy in a short story called
“Chains” (Braun, 2004). This idea is a practical expression of the concept
that such networks tend to grow exponentially. A related concept is the
Erd˝ os number. The Erd˝ os number, named after the famous Hungarian