For signed weighted networks (i.e. networks with positive and negative edges), the
calculation of the modularity Q is problematic. Gomez, Jensen, and Arenas (2009) explain that,
when calculating modularity Q for unweighted (Newman & Girvan, 2004) or weighted networks
(Fan, Li, Zhang, Wu, & Di, 2007), the term \(\frac{k_{u}}{2m}\) indicates the probability of
node \(u\) making connections with other nodes in the network, if connections between nodes
were random. Gomez, Jensen, and Arenas (2009) discuss how, when networks are signed, the
positive and negative edges cancel each other out and the term \(\frac{k_{u}}{2m}\) loses its
probabilistic meaning. To deal with this limitation, Gomez, Jensen, and Arenas (2009) proposed modularity Q for signed
weighted networks, generalised to fuzzy modularity Q for signed weighted networks:
$$Q=(\frac{2w^{+}}{2w^{+}+2w^{-}})(\frac{1}{2m^{+}}) \sum_{c\epsilon_C} \sum_{u,v\epsilon_V} \alpha_{cu}^{+} \alpha_{cv}^{+}
(A_{uv}^{+}-\frac{k_{u}^{+}k_{v}^{+}}{2m})-
(\frac{2w^{-}}{2w^{+}+2w^{-}})(\frac{1}{2m^{-}}) \sum_{c\epsilon_C} \sum_{u,v\epsilon_V} \alpha_{cu}^{-} \alpha_{cv}^{-}
(A_{uv}^{-}-\frac{k_{u}^{-}k_{v}^{-}}{2m})$$
where the sign + indicates positive edge weights and the sign - indicates negative edge weights, respectively.