modularity
Table of Contents
- Notation
- \(A_{i,j}\) is entry in adjacency matrix of node i and node j
- \(\delta(c(i),c(j)\) returns 1 if \(i\) and \(j\) is of the same class, and 0 if \(i\) and \(j\) is of different classes
- \(m\) - number of links, (\(2m\) is also sum of degrees)
- definition \[ Q = \frac{1}{2m}\sum_{i,j}(A_{i,j} - \frac{\text{deg} i * \text{deg} j }{2m} )\delta(c(i),c(j)) \]
Backlinks
homophily
tendency of people associate with others whom they perceive as being similar to themselves. To measure homophily,
assortativity coefficient
with assumption that modularity can go as high as \[ Q_{max }= 1 - \frac{1}{2m}\sum_{i,j} \frac{\text{deg} i * \text{deg} j }{2m} \delta(c(i),c(j)) \] when All \(A_{i,j}\) values are \(1\)(there’s 2m links in the sum in total.), the assortativity coefficient is defined as \[ \frac{Q}{Q_{max}} \] interpretation - normalized modularity. The higher this value, the more connected everyone to everyone