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Networks

mathematician Paul (Pa ´l) Erd˝ os, is a way of describing the collaboration
distance in scientific articles between an author and Erd˝ os. Erd˝ os himself
has an Erd˝ os number zero. Those who have collaborated with Erd˝ os have
an Erd˝ os number equal to one (more than 500 persons), those that have
collaborated with a person who has Erd˝ os number one have Erd˝ os number
two, and so on. Of course, scientists active in fields far away from mathe-
matics have no Erd˝ os number (or one may say that their Erd˝ os number is
infinity). Gla ¨nzel and Rousseau showed that many informetricians have a
finite Erd˝ os number (Gla ¨nzel & Rousseau, 2005).
A review on complex networks is provided in Newman (2003). In
this work the author pays attention to three main aspects in the study of
complex networks: statistical properties such as path lengths and degree
distributions, creating models for networks, and the effects of structure
on system behavior. He points out that much progress has been made on
the first two aspects, but that less in known about the third. Another
review of network science can be found in (Watts, 2004).

Motifs
Baraba ´si and Oltvai (2004) pointed out that the occurrence of motifs seems
to be a general property of all real networks. Motifs are small, local patterns
of interactions that occur in complex networks and that are characterized
by the fact that they occur significantly more than expected (Milo et al.,
2002). To be precise, a motif M is a pattern (such as a triangle) that re-
occurs in a network. Mathematically, a motif can be said to be the name of
the equivalence class of a set of isomorphic subgraphs. Motifs have been
shown to occur in transcription networks, signaling and neuronal net-
works, economics, information and social networks and other types of net-
works (Krumov et al., 2011; Sporns & Ko ¨tter, 2004; Zhang et al., 2014).
In all these networks motifs serve as basic building blocks. Although in
Milo et al. (2002) a set of three criteria is proposed for a subgraph to be a
motif, these criteria can only be described as “ad hoc.” At the moment
there does not seem to exist a precise mathematical definition of a motif.

10.3 H-INDICES IN NETWORKS
10.3.1 The Lobby Index and the h-index of a Network
Schubert, Korn and Telcs (Korn et al., 2009; Schubert et al., 2009) intro-
duced the lobby index as a centrality parameter for nodes, while Schubert
and Soos (Schubert & Soos, 2010) defined the h-index of a network as