# A Simple Model of Knowledge Scaffolding Applied to Wikipedia Growth
## 8 Discussion and Conclusions  
We presented a simple model of knowledge dynamics in which topics are arranged as an acyclic network.  
The process of knowledge scaffolding (accumulation) is a random process in which the size of the corpus \(c\) and the maximum knowledge \(m\) grow with time following a power law with exponent \(\alpha=\frac{1}{2}\), once we have fixed the connectivity \(K\) and the maximum distance  
Figure 12: Plot of the average value of the maximum height of corpus (approximating the corpus site) \(m(t)\) versus time \(t\) (continuous line), and the logistic growth of the user base \(M(t)\) (dotted line).  
Figure 13: (a) Log–log plot of the maximum height of corpus (approximating the corpus site) \(m(t)\) versus time \(t\) for early times. (b) The log–log plot of \(m(t)\) versus \(t\). The dotted line marks the exponent \(1/2\).  
from the previous knowledge \(L\). Asymptotically, we always obtain an approximately constant number of holes. Both those trends are independent of the values of \(K\) and \(L\).  
We have found an analytical approximation of the process, based on a death-birth Markov process, exhibiting an asymptotic power law for \(L=1\). By ignoring the presence of holes, an assumption justified by the fact that their number remains constant and is therefore negligible compared to the asymptotic trends of the corpus, the same approximation can be used for \(L>1\) as well.  
We have then confronted our model with the English Wikipedia's growth, which shows initial exponential or large-exponent power-law growth, followed by a slowdown.  
We assume that the initial growth was mainly due to the increasing number of users. The subsequent phase was dominated by the filling of gaps and splitting of long pages, a phase still affected by the previous growth trend. The asymptotic phase, which is more similar to our model, should consist in a stable user base and growth dominated by new findings and discoveries, still coupled to the process of filling the gaps.  
We have observed some similarities between our model and the latest Wikipedia growth, even if Wikipedia pages are not acyclic (but Wikipedia categories are), and its degree distribution is compatible with a kind of preferential attachment, and therefore, it does not correspond to a fixed connectivity \(K\).  
Despite this, we found a power law in both trends, and the exponent measured for Wikipedia decreased with time. We think that the present exponent of Wikipedia's trend is due to the fact that it has not yet reached its final stage.  
We have shown that it is possible to include a growing number of contributors in our model, and assuming that their number follows exponential or logistic (which shows en early exponential-like behavior) growth, with constant activity per user, it is also possible to approximate the first phases of Wikipedia's growth.  
We think that we have provided a constructive agent-based model that reproduces some of the main aspects of Wikipedia's growth. Clearly, a more detailed model of user behavior should be developed and checked against experimental data.  
Conceptualization, F.B.; methodology, F.B.; software, F.B. and G.d.B.C.; validation, G.d.B.C.; formal analysis, F.B.; investigation, F.B. and G.d.B.C.; writing--original draft preparation, F.B.; writing--review and editing, F.B. and G.d.B.C.; visualization, G.d.B.C.; supervision, F.B. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
All Wikipedia data used here can be freely downloaded from the Wikimedia Foundation.  
The authors declare no conflict of interest. The authors had no role in the collection of data from Wikipedia.  
All code is available upon request.