# A Simple Model of Knowledge Scaffolding Applied to Wikipedia Growth
## 4 The Model  
Since a graphical representation of knowledge acquisition is given by an oriented or growing network [31], we could represent knowledge scaffolding as shown in Figure 1. This feed-forward structure can be seen as a layered network where an item at any level depends on other items at lower levels. We chose instead a simpler, linear progression, as shown in Figure 2. This is of course a simplification of the process, because, for example, with a linear 1D model, we cannot put into evidence the structures that group together all simultaneous and independent derivations. The analysis in this case is easier, and it should not impair the conceptual results. As we can consider all items with the same highest prerequisite as simultaneous, we privileged the search for simplicity in order to put into evidence the plausible essentialcomponents of an observed phenomenon and to avoid too many variables that could be tweaked to fit any data set available. The knowledge space is represented by an array \(X_{i}\) of \(N\) items, which can be known (\(X_{i}=1\)) or unknown (\(X_{i}=0\)). We denote by  
\[c(t)=\sum_{i}X_{i} \tag{1}\]  
the size of the knowledge corpus at a certain time. The linear scaffolding process can be modeled by a random proposal of contributions, each of which depends on a certain number \(K\) of prerequisites, and provide a higher contribution. We assume that there is a limit to jumps in knowledge, and therefore, the newly proposed item cannot be at a distance greater than \(L\) from the highest prerequisite (Figure 2). In the structure of our one-dimensional model, the distance between any two nodes can be roughly seen as the difference in prerequisites, or already known items of knowledge, needed to obtain the considered item, and \(L\) can be seen as the higher limit of how many consecutive knowledge items could be obtained using only the known items up to a certain point. Let us denote by \(m\) the index of the highest known item in the corpus, i.e., \(X_{m}=1\) and \(X_{j}=0\)\(\forall j>m\). The corpus can contain holes, i.e., \(X_{j}=0\) with \(j<m\). A contribution cannot refer to prerequisites higher than \(m\), but could be rejected because it is based on holes in the present corpus. Redundant contributions (i.e., articles providing already known items) are not considered, even if they are valid. These rejected pages are the equivalents of those marked as duplicates (and subsequently merged) in Wikipedia. The corpus filling proceeds by randomly choosing \(K\) items, \(j_{1},\ldots,j_{k},\ldots,j_{K}\), with \(j_{k}\leq m\), as the prerequisite for a new contribution. It may happen that two or more of the \(j_{k}\) correspond to the same item, since they are chosen at random in the interval \(0,\ldots,m\). We denote by \(j_{M}\) the largest values of the selected \(j_{k}\), and we extract a random integer \(i\) such that  
\[i\in(j_{M}+1,\ldots,j_{M}+L) \tag{2}\]  
as a candidate for the new piece of knowledge. Figure 2: The linear scaffolding model. In this version, there are no well-defined levels. Known items are denoted by ones in a linear array of elements (\(X\)), and unknown ones by zeros. The “highest” known item is at position \(m\). New items (\(i\)) become known (\(X_{i}=1\)) if all random prerequisites \(j_{k}\) are known. The maximum knowledge jump \(L\) limits the choice of \(i\) so that \(1\leq i-j_{M}\leq L\), where \(j_{M}\) is the highest of prerequisites \(j_{k}\). In the figure, the proposed new item \(i\) is based on prerequisites \(j_{1}\), \(j_{2}=J_{M}\), and \(j_{3}\) (\(K=3\)). If all the prerequisites are known (i.e., \(X_{j_{k}}=1\)\(\forall k\)) and the piece of knowledge \(X_{i}\) is not already known (i.e., \(X_{i}=0\)), then this contribution is added to the corpus (\(X_{i}=1\) and \(c\) is incremented by one), and if \(i>m\), then \(m=i\). In any case, the time \(t\) is incremented by one.