I'm trying to self-learn network analysis. It is very abstract to me how a negative network can be constructed based on the formula of Modularity. Version 1 or Version 2.
Version 1: $Q=\frac{1}{2m} \sum \limits_{i,j}\left(A_{ij} - \frac{k_i k_j}{2m}\right)\delta(c_i,c_j)$
Version 2: $Q=\sum \limits_{c=1}^{n_c}\left(\frac{l_{c}}{m} - \left(\frac{d_c}{m}\right)^2\right)$
By definition from the original paper: https://arxiv.org/pdf/0803.0476.pdf:
Version 1, $m$ is defined as $\frac{1}{2}\sum \limits_{i,j}A_{ij}$ where $A_{ij}$ represents the weight of the edge between i and j, thus $\frac{_}{2} $is the average fraction of this weight that would be assigned to node j or the marginal probability of j.
$k_i$ and $k_j$ are defined as $\sum \limits_{i or j}A_{iorj}$ as the sum of the weights of the edges attached to vertex i or j.
Version 2, $n_c$ is the number of clusters, $l_c$ is the sum of edges joining vertices in module c, and $d_c$ is the sum of the degrees of the vertices of c.
How should I account for the randomized part? Could anyone provide the matrix of a negative modularity network? .
Thank you!