# The Evolution of Wikipedia's Norm Network
## 2 Methods
### Influence and Overlap  
An important feature of the norm network is the sphere of influence: the pages that rely on any particular page for context.  
Consider, for example, the norm page \"Neutral Point of View\" (NPOV), a page urging editors to describe article subjects without taking sides. A page that links to NPOV relates its own subject to NPOV in some fashion. For example, among many pages that link to NPOV is \"Propaganda\", an essay urging editors to be wary of using propaganda outlets of authoritarian governments. The Propaganda page links to the NPOV page in order to define the notion of \"undue weight\"; NPOV's content can thus be said to influence the interpretation of what is found on Propaganda.  
Influence is distinct from centrality; centrality measures the extent to which pages link to the page in question. Conversely, influence measures the extent to which the content of that page influences other pages. In our formalism, a node \(p\) can be understood to _influence_ a node \(q\) when \(q\) links to \(p\). Influence need not be direct, however: \(p\) can influence \(q\) if \(q\) links to \(r\) and \(r\) links to \(p\). To measure the non-local influence, we consider random walks on the direction-reversed network.  
More formally, placing a random-walker at node \(p\), we allow her to take \(n\) steps from this starting point along the direction-reversed network; we write the resulting probability distribution over the final position as \(p_{i}\), the probability of the walker ending up at node \(i\). The distribution \(p_{i}\) defines the influence that \(p\) has on \(i\).  
To quantify the distance between two nodes, we then consider the influence _overlap_ between two arbitrary nodes \(p\) and \(q\). Overlap quantifies the extent to which two random walkers, beginning at these nodes, will tend to visit the same pages. If \(p_{i}\) and \(q_{i}\) are the probability distributions associated with the influence of node \(p\) and \(q\), then overlap is defined as:  
\[O(p,q)=\frac{\sum_{i=1}^{N}p_{i}q_{i}}{\left(\left[\sum_{i=1}^{N}p_{i}^{2} \right]\left[\sum_{i=1}^{N}q_{i}^{2}\right]\right)^{1/2}} \tag{1}\]  
For multiple pages, we can compute the average pairwise overlap simply by averaging the overlap between all possible pairs within the set.  
High overlap between \(p\) and \(q\) indicates that two pages influence a large number of common nodes. When \(n\) goes to infinity, the random walkers converge to the stationary distribution, and the overlap is one; conversely, when \(n\) is small, random walkers have less time to encounter each other. We take \(n\) equal to five, larger than the average shortest path (roughly three, in our network), so that nodes are potentially reachable, but much less than the convergence time to the stationary distribution.  
Overlap can be thought of as a measure of the separation of spheres of influence. It invokes only local mechanisms: users traveling from one page to another by the links that connect them. This is in contrast to a measure, such as shortest paths, which is computationally expensive and requires detailed, global knowledge of the network link-structure. In general, for example, the number of nodes an algorithm needs to visit in order to determine the shortest path between two nodes will usually be much larger than the length of the final path.  
Both influence and overlap require us to specify particular nodes of interest; we focus in this work on pairs of high-EC pages, or core norms.