Note 2: Minimal requirements to represent data as a "complex network"

Not all well-defined relations lend
themselves to a meaningful representation
as a complex network. This figure is under CC:BY
with a reference to Prof. Dr. Katharina A. Zweig
or to this blogpost.

Have you ever asked yourself what the minimal requirement is to turn something into a "complex network"? Well, mathematically seen, it is the following:

Note 2. Mathematically, a relation R on a given set
of entities or objects is just an arbitrary choice of pairs
of these entities (objects), denoted by R ⊆ O × O. In
principle, any relation can be represented as a graph.
(Zweig2016)

So, the minimal requirements are actually - minimal. While mathematically possible, not all relations gain from being represented as a graph and by being treated as a "complex network". Look, for example at the set of all living humans that own at least one ID card and connect any two of them if their oldest ID-card's ID number shares the last digit. This is surely a relation, but it is also surely a relationship between humans that will not be any better analyzed by turning it into a complex network.

Why is this so? The whole idea of complex network analysis is to understand the interaction structure of entities in a complex system. Complex systems are those with emergent phenomena. Emergence often - well - emerges, when interactions between pairs of entities change the interactions of other pairs of entities because of the interactions between the pairs, i.e., when indirect effects are transferred via the interactions. Brandes et al. express it like this:

By postulating a friendship network in (say) a school class-
room of 25 students, we have taken a theoretical step that
is non-trivial. We have supposed that separate individ-
uals are not an adequate representation, moreover that
even separate dyads are insufficient; rather, that there
is a unity within the classroom that makes it proper to
talk of “a” network, not 25 children or 300 dyads. To con-
ceptualize the classroom in network terms is an implicit
(and strong) claim that connectedness across individual
elements is fundamentally important, so that the class-
room can be thought of as one “system”. (Brandes2013)

I believe that this feature that turns a set of pairs (or dyads) into "one system" is a network process that induces indirect effects via the relationship that binds the pairs together in one network. Thus, the answer to the question is: while mathematically any relation defined on a finite set of entities is good enough for a representation as a graph, semantically, not all relations make sense to be represented as a complex network. And since a relation can once represent a meaningful relationship and the very same relationship can also represent a meaningless relationship (like the one above), it is not the relation itself that decides about its "networkability". It is the relationship the relation represents.

Reference:

(Brandes2013) Brandes, U.; Robins, G.; McCranie, A. & Wasserman, S.: "What is Network Science?", Network Science, 2013, 1, Editorial
(Zweig2016) Katharina A. Zweig: Network Analysis Literacy, ISBN 978-3-7091-0740-9, Springer Vienna, publication expected Dec 2016