## 3 Materials and Methods
### Instruments and Settings
#### 3.2.1 Formalisation of the Evaluation Space  
We differentiated between students and authors. A given student can be represented as two different authors if they participate in two different courses. This happens whenever a student does not pass the course and must repeat it the next academic year. First, we performed a formal characterisation to identify the different elements involved in the interaction process. We considered an examination space as an \(n\)-tuple **S** = {**A, e, d, c, q, s, iv, gl**}, where:  
**A** is the set of authors: **A** = {a\({}_{i}\), i = 1... n\({}_{A}\)};  
**e** is the set of program exercises (assignments) to be answered: **e** = {e\({}_{k}\), k = 1... n\({}_{e}\)};  
**d** is the set of answers to the different exercises of **e**: **d** = {d\({}_{ki}\), k = 1... n\({}_{p}\); I = 1... n\({}_{A}\)};  
they are referred to as _documents_;  
**c** is a _classification category_ for each pair of matched documents: **c** = {C\({}_{m}\), m = 1... n\({}_{C}\)};  
when two documents are compared, two _percentages of similarity_ are returned (comparing the proportion of one document included in the other and vice versa); those percentages were discretised and summarised into these classes with the following interpretation: C\({}_{1}\) = no relation; C\({}_{2}\) = weak relation; C\({}_{3}\) = strong relation; C\({}_{4}\) = obvious copy;  
**q** represents a set of _behaviour categories_: **q** = {Q\({}_{individuals}\), Q\({}_{soft}\), Q\({}_{hard}\), Q\({}_{cheat}\)} of network nodes;\(\mathbf{s}\) is the final score obtained by each author: \(\mathbf{s}=\{\text{s}_{\text{i}},\text{ i}=1\dots\text{ n}_{\text{A}}\}\); it is a number in the interval [0,10], which is the scale used in the Spanish academic system;  
\(\mathbf{iv}\) represents the set of discrete values of the intensity of a mutual relationship;  
\(\mathbf{g}\) represents a set of clusters of authors: \(\mathbf{g}=\{1,\dots,\text{ n}_{\text{g}}\}\). Given \(\mathbf{S}\), we can now establish the relationships between its elements, \(\mathbb{N}\) being the set of natural numbers. Let \(\mathbf{F}\) be the set of functions, \(\mathbf{F}=\{\text{Auth},\text{Sol},\text{Similar},\text{Similar}\_\text{ cod},\text{Similar}\_\text{linearity},\text{Author}\_\text{behaviour},\text{Path}\_\text{matrix},\text{Group}\}\), being:  
\(\text{Sol}\): \(\mathbf{A}\times\mathbf{e}\rightarrow\mathbf{d}\). This function assigns the answer of the author to a given exercise. This is called a _document_ throughout this article. \(\mathbf{Auth}\): \(\mathbf{d}\rightarrow\mathbf{A}\). This auxiliary function assigns its author to each document. \(\mathbf{Similar}\): \(\mathbf{d}\times\mathbf{d}\rightarrow\) [0,1] \(\times\) [0,1] \(\times\) [_null_, _left_, _right_]. This function assigns to each pair of _documents_ the two _percentages of similarity_ and the orientation of the potential copy. This is discussed in Section 3.3.1. \(\text{Similar\_intensity}\): [0,1] \(\times\) [0,1] \(\times\) [_null_, _left_, _right_] \(\rightarrow\)\(\mathbf{iv}\). This function assigns an _intensity value_ for each pair of _document similarity percentages_. It is a measure of the strength of the relationship between both. \(\text{Similar\_cod}\): \(\mathbf{iv}\rightarrow\mathbf{c}\). This function transforms the _intensity value_ into a _classification category_. Compounding _Similar_intensity_ and _Similar_cod_, a _classification category_ can be obtained for each pair of _documents_. This is detailed in Section 3.3.1. \(\text{Author\_behaviour}\): \(\mathbf{A}\rightarrow\mathbf{q}\). _Author_behaviour_ assigns a _behaviour category_ to each author, taking into account the average of all the _intensity values_ of all documents authored by [PERSON] and any other author. This is detailed in Section 3.3.2. \(\text{Path\_matrix}\): \(\mathbf{A}\times\mathbf{A}\times\mathbf{q}\rightarrow\{0,1\}\).