### 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.

1.1. Graphs and Relations.

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 R2(V ), is simply a subset R2(V ) V ×V . We have two projections π12 : V ×V V , and there is one inclusion i : R2(V ) V ×V . Therefore, there exist two mappings π1 i : R2(V ) S and π2 i : R2(V ) S.

Given the binary relation R2(V ), we can now define a graph GR2(V ) = G(V,R2(V )1 i,π2 i). Given some graph G = G(V,A,s,t), we can extract a binary relation GR2(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 R2(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 GR2(V ) = G(V,R2(V )1 i,π2 i), because any multiple arrows of G(V,A,s,t) between two given vertices are matched into a single arrow in G(V,R2(V )1 i,π2 i). Typically in applications, such as network theory, it is the directed graph GR2(V ) which is considered. If all binary relations defining R2(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 = R2(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(v1,v2) = v1 and t(v1,v2) = v2 for any (v1,v2) 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.