. 2021 Jun 18;7:e600. doi: 10.7717/peerj-cs.600

Table 1. Community scoring functions f(S) for weighted and unweighted networks.

	Unweighted f(S)	weighted f(S)
↑ Internal density	$\frac{m_{S}}{n_{S} (n_{S} - 1) ∕ 2}$	$\frac{{\tilde{m}}_{s}}{n_{S} (n_{S} - 1) ∕ 2}$
↑ Edges Inside	m_S	${\tilde{m}}_{S}$
↑ Average Degree	$\frac{2 m_{S}}{n_{S}}$	$\frac{2 {\tilde{m}}_{S}}{n_{S}}$
↓ Expansion	$\frac{c_{s}}{n_{s}}$	$\frac{{\tilde{c}}_{s}}{n_{s}}$
↓ Cut Ratio	$\frac{c_{s}}{n_{s} (n - n_{s})}$	$\frac{{\tilde{c}}_{s}}{n_{s} (n - n_{s})}$
↓ Conductance	$\frac{c_{s}}{2 m_{s} + c_{s}}$	$\frac{{\tilde{c}}_{s}}{2 {\tilde{m}}_{s} + {\tilde{c}}_{s}}$
↓ Normalized Cut	$\frac{c_{s}}{2 m_{s} + c_{s}} + \frac{c_{s}}{2 (m - m_{s}) + c_{s}}$	$\frac{{\tilde{c}}_{s}}{2 {\tilde{m}}_{s} + {\tilde{c}}_{s}} + \frac{{\tilde{c}}_{s}}{2 (\tilde{m} - {\tilde{m}}_{s}) + {\tilde{c}}_{s}}$
↓ Maximum ODF	${max}_{u \in S} \frac{\| \{(u, v) \in E : v ⁄ \in S\} \|}{d (u)}$	${max}_{u \in S} \frac{\sum_{v ⁄ \in S} w_{u v}}{\tilde{d} (u)}$
↓ Average ODF	$\frac{1}{n_{s}} \sum_{u \in S} \frac{\| \{(u, v) \in E : v ⁄ \in S\} \|}{d (u)}$	$\frac{1}{n_{s}} \sum_{u \in S} \frac{\sum_{v ⁄ \in S} w_{u v}}{\tilde{d} (u)}$