In the paper "Fast unfolding of communities in large networks", the authors show the first definition of modularity: $$ Q = \frac1{2m} \sum_{i,j} \left[A_{ij} - \frac{k_i k_j}{2m}\right]\delta(c_i, c_j). $$ However, a few pages later, they show the $\Delta Q$ definition (the change of modularity when moving a vertex $i$ to a community $c$): $$ \Delta Q = \left[\frac{\Sigma_{in} + k_{i,in}}{2m} - \left(\frac{\Sigma_{tot} + k_i}{2m}\right)^2\right] - \left[\frac{\Sigma_{in}}{2m} - \left(\frac{\Sigma_{tot}}{2m}\right)^2 - \left(\frac{k_i}{2m}\right)^2\right] $$
How did the authors reached to this definition of $\Delta Q$? It cannot be developed from the original modularity definition shown above.
Thank you in advance.
The formula of the change in the modularity is derived from this formula of modularity: $$\sum_{c_i\in C}[{\frac{\sum_{in}^{c_i}}{2m}-\frac{(\sum_{tot}^{c_i})^2}{4m^2}}]$$
In order to show that $$\frac{1}{2m}\sum_{i,j}[A_{i,j}-\frac{k_ik_j}{2m}]\delta(c_i,c_j) = \sum_{c_i\in C}[{\frac{\sum_{in}^{c_i}}{2m}-\frac{(\sum_{tot}^{c_i})^2}{4m^2}}]$$ you should note that
$\sum_{i,j}{k_ik_j}\delta(c_i,c_j)=\sum_{c_i \in C}(\sum_{j \in c_i}{k_j})^2$.
A detailed explanation of the change in modularity formula is explained here: https://www.quora.com/How-is-the-formula-for-Louvain-modularity-change-derived.