### Structural Modelling -

Relations Determining a

World of Connectivity

### 1. Relational Modelling

Structural Modelling or equivalently Relational Modelling is the most basic form of a mathematical model, as it basically avoids the introduction of time or more general sequential changes in the state of the model, representing a certain system. Structural modelling is very basic, but therefore powerful (if interpreted correctly) at the same time. To understand this, consider the problem to understand any given system. We do naturally distinguish between system components and their relations. Relations are often expressed in natural language in statements such as ’Component A influences component B’ or ’Component A is linked to component B’. In mathematical terms, both statements are so-called binary relations, because the sentence references to two (arbitrary) components A and B that are part of the system. The first statement is a so-called directed binary relation, because it can be true that A influences B, but B does not influence A. It is also called a non.symmetric relation for obvious reasons. The second statement is symmetric, because if A is linked to B, then it is automatically also true that B is linked to A.

The two components A and B are part of a set E, which is the set of all system
components or equivalently system elements of the system under observation. We can
therefore see, that set theory is the basis of relational modelling. The set E is called the basic
set or elementary set. Note that we implicitly assume that any element e ∈ E is not further
divisible, here we logically follow the tradition of the early Greek atomists. However, this
indivisibility is just a modelling assumption, not a fact of reality. It is a necessary assumption
without which the model hierachy could not be closed. Sometimes it is necessary to construct
a subset E^{′}, the subset of identified basic system components, where some system
components which have been elements of E have not been identified in the mathematical
abstraction process, for example, because their relevance was considered a-priori
irrelevant.

Relational modelling therefore always starts with an ontological statement: the identification of the basic set E. The second modelling step is then usually the creation of types as identified system classes inside a classification scheme: the system components are bundled into types. Mathematically we introduce equivalent relations on the set E, creating a new set T, the set of types. We will give examples in a moment. Mathematically, typification is again a set theoretic operation, the use of an equivalence relation. Every scientific theory uses types. The classical example is an atomic ensemble. Every identified or unidentified atom is in the set E, and every atom can be uniquely identified as a chemical species, i.e. a type.

Mathematically, if one splits a system into system elements or system types and their relations, one enters the area of graph theory. Depending on the situation, it is sometimes necessary not to pre-impose cardinality in the definition of sets. Graphs are the most basic type of a relational or structural mathematical model. Here we give a definition without specifying the cardinality of any set V , either representing E or T, depending on the modelling choice:

Definition 1.1. [Graph]

A set theoretic structure G = G(V,A,s,t), consisting of two sets V and A, called the
set of vertices and the set of arrows, respectively, and two functions s : A → V , and
t : A → V , called the source and the target function, is called a graph.

Given an arrow a ∈ A we refer to s(a) as the source vertex of a and to t(a) as the target
vertex of a. To draw a graph, we first represent every vertex v ∈ V as a dot, then for every
element a ∈ A we draw an arrow connecting dot s(a) to dot t(a). The connection to binary
relations is as follows: consider again the set V . A binary relation on V , denoted by R_{2}(V ),
is simply a subset R_{2}(V ) ⊆ V ×V . We have two projections π_{1},π_{2} : V ×V → V , and there is
one inclusion i : R_{2}(V ) → V ×V . Therefore, there exist two mappings π_{1} ∘i : R_{2}(V ) → S and
π_{2} ∘ i : R_{2}(V ) → S.

Given the binary relation R_{2}(V ), we can now define a graph G_{R2(V )} = G(V,R_{2}(V ),π_{1} ∘i,π_{2} ∘i).
Given some graph G = G(V,A,s,t), we can extract a binary relation G_{R2(V )}. Indeed, given the
functions s and t, we can define the product function ⟨s,t⟩ : A → V × V . This is not yet a
binary relation, because the product function might not be injective. However, we can define
R_{2}(V ) ⊆ V × V by taking the image of ⟨s,t⟩. Note that in general then the graph
G = G(V,A,s,t) is different from G_{R2(V )} = G(V,R_{2}(V ),π_{1} ∘ i,π_{2} ∘ i), because any multiple
arrows of G(V,A,s,t) between two given vertices are matched into a single arrow in
G(V,R_{2}(V ),π_{1} ∘i,π_{2} ∘i). Typically in applications, such as network theory, it is the directed
graph G_{R2(V )} which is considered. If all binary relations defining R_{2}(V ) := E are symmetric,
then one calls the graph G = G(V,E) unordered, and E is called the set of edges. If V is a
finite set, then also E = R_{2}(V ) is finite, constituting a finite graph. Starting from sets V and
E ⊆ V ×V , creating a graph G = G(V,E), it is clear that we can conversely define the graph
G = G(V,E^{′},s,t), where the new arrow set E^{′} is identical to E in case E is ordered, or
contains two arrows for every e ∈ E in case E is unordered. We have s(v_{1},v_{2}) = v_{1}
and t(v_{1},v_{2}) = v_{2} for any (v_{1},v_{2}) ∈ E^{′} in case E is unordered, and respects the
order of any 2-tupel in case E is an ordered binary relation. We now call G(V,E) a
binary graph (or shortly a graph, if the context is clear), whenever E ⊆ V × V is a
binary relation. In case we have E is ordered, we often replace E by A, and write
G = G(V,A), calling A again the set of arrows. In network theory, one usually starts with
a finite binary graph G(V,E), and then uses some construction process, such as
preferential attachment, to look at the infinite limit of such graphs, then called a
network.