Page 338 - Becoming Metric Wise

330 Becoming Metric-Wise

an indicator for complete networks. The lobby index of node n in an
unweighted network is defined as the largest integer k such that n has at
least k neighbors with a degree, d(n), at least k. The degree h-index of a
network is defined as, “A network’s degree h-index is h if not more than
h of its nodes have a degree not less than h” (Schubert et al., 2009).

10.3.2 The h-degree (Zhao et al., 2011)
In many real networks, the strength or weight of a link is an important
parameter, leading to the notion of weighted (or valued) networks. We
define the node strength of a node in a weighted network as the sum of
the strengths (or weights) of all its links (Barrat et al., 2004). In weighted
networks, we use the term node strength, denoted as d s (n), not to be con-
fused with the term node degree which we only use in unweighted net-
works (or for the underlying unweighted network where each weight is
equal to one).
The lobby index mentioned in Section 10.3.1 can be adapted to a
weighted network as follows, the w-lobby index (weighted network
lobby index) of node n is defined as the largest integer k such that node n
has at least k neighbors with node strength at least k. Next we define the
h-degree of a node in a weighted network.
Definition: The h-degree (Zhao et al., 2011)
The h-degree (d h ) of node n in a weighted network is equal to d h (n),
if d h (n) is the largest natural number such that n has d h (n) links each with
strength at least equal to d h (n).

Properties
• In directed weighted networks, one can define an in-h-degree and an
out-h-degree.
• In a weighted network with N nodes, the highest possible h-degree is
N 2 1. This happens in a star network where the center is linked to
the other nodes with a weight equal to (at least) N 2 1.
• In a ring where each weight is at least 2, each node has h-degree 2.
• In unweighted networks, the h-degree of a node is either 1 (a noniso-
lated node) or 0 (an isolated node). Also in weighted networks isolated
nodes have d h 5 0.
Schubert introduced the coauthor partnership ability (2012), which is
a special case of the h-degree (Rousseau, 2012). Indeed, when consider-
ing a coauthorship network, then Schubert’s coauthor partnership ability
index is exactly the h-degree in this network, where weights are the