Table 2.

Penalty terms used in the penalty methods.

Methods	Mathematical notation	Characteristics
Li & Li, 2008 [27]	${\displaystyle \sum }_{u~v}{\left(\frac{{\beta }_u}{\sqrt{d_u}}-\frac{{\beta }_v}{\sqrt{d_v}}\right)}^{2}w\left(u,v\right)$   Here, du is the degree of freedom for gene u, recording the sum of weights for all genes connected to gene u. w(u, v) is the weight for the edge between genes u and v.	Aims at smoothing the β coefficients over the network, ignoring that neighboring genes might have β's in opposite directions.
[55]	${\displaystyle \sum }_{u~v}{\left(\frac{{\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\widetilde{{\beta }_u})\beta }_u}{\sqrt{d_u}}-\frac{{\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\widetilde{{\beta }_v})\beta }_v}{\sqrt{d_v}}\right)}^{2}w\left(u,v\right)$   Here, $\widetilde{{\beta }_u}$ is the estimated value of β coefficient for gene u, and sign (x) represents the sign of x, if x>0 sign(x)=1; x<0 sign(x)=-1; otherwise sign(x)=0.	Accounts for that two connected genes might have β's with different signs, but may not work well since it is difficult to estimate the signs for β's.
[30]	${\displaystyle \sum }_{u~v}{\left[{\left(\frac{{\beta }_u}{\sqrt{d_u}}\right)}^{\gamma }+{\left(\frac{{\beta }_v}{\sqrt{d_v}}\right)}^{\gamma }\right]}^{1/\gamma }w\left(u,v\right),\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\gamma >1$	Shrinks the weighted β's of two neighboring genes towards each other, but the estimates may be severely biased.
[26, 56]	for γ = ∞, it becomes ${\displaystyle \sum }_{u~v}\max \left(\frac{{|\beta }_u|}{\sqrt{d_u}},\frac{{|\beta }_v|}{\sqrt{d_v}}\right)w\left(u,v\right)$	A 2-step procedure is used to reduce biases; it is proved that this performs better than that with smaller γ
[57]	${\displaystyle \sum }_{u~v}\left|I\left(\frac{\left|{\beta }_u\right|}{d_u}\ne 0\right)-I\left(\frac{\left|{\beta }_v\right|}{d_v}\ne 0\right)\right|w\left(u,v\right)$   Here, I (x) is an indicator. If the condition x is true I(x)=1, otherwise its value is 0.	Encourages simultaneous selection of neighboring genes in the network. But the Indictor function I is not continuous and thus needs special care.
The generalize elastic net: [29]	${\lambda }_1{\displaystyle \sum }_u{D}_u\left|{\beta }_u\right|+\frac{{\lambda }_2}{2}{\beta }^{T}P\beta$   Here D and P are additional penalty weights for individual genes (gene-level penalty) and gene pairs (pathway-level penalty).	Includes the network-constrained penalty term by [27] as a special case, capable of accommodating any positive semi-definite measure of dissimilarity between pairs of genes.