Table 1.

Metrics used to quantify algorithm performance. Here Θ¹ and Θ² denote the true inverse covariance matrices, and Θ̂¹ and Θ̂² denote the two estimated inverse covariance matrices. Here 1{A} is an indicator variable that equals one if the event A holds, and equals zero otherwise. (1) Metrics based on recovery of the support of Θ¹ and Θ². Here t₀ = 10⁻⁶. (2) Metrics based on identification of perturbed nodes and co-hub nodes. The metrics PPC and TPPC quantify node perturbation, and are applied to PNJGL, FGL, and GL. The metrics PCC and TPCC relate to co-hub detection, and are applied to CNJGL, GGL, and GL. We let t_s = μ + 5.5σ, where μ is the mean and σ is the standard deviation of ${{‖ {\hat{V}}_{- i, i} ‖}_{2}}_{i = 1}^{p}$ (PPC or TPPC for PNJGL), ${{‖ {({\hat{Θ}}^{1} - {\hat{Θ}}^{2})}_{- i, i} ‖}_{2}}_{i = 1}^{p}$ (PPC or TPPC for FGL/GL), ${{‖ {\hat{V}}_{- i, i}^{1} ‖}_{2}}_{i = 1}^{p}$ and ${{‖ {\hat{V}}_{- i, i}^{2} ‖}_{2}}_{i = 1}^{p}$ (PPC or TPPC for CNJGL), or ${{‖ {\hat{Θ}}_{- i, i}^{1} ‖}_{2}}_{i = 1}^{p}$ and ${{‖ {\hat{Θ}}_{- i, i}^{2} ‖}_{2}}_{i = 1}^{p}$ (PPC or TPPC for GGL/GL). However, results are very insensitive to the value of t_s, as is shown in Appendix G. (3) Frobenius error of estimation of Θ¹ and Θ².

(1)

Positive edges:

\sum_{i < j} (1 {∣ {\hat{Θ}}_{i j}^{1} ∣ > t_{0}} + 1 {∣ {\hat{Θ}}_{i j}^{2} ∣ > t_{0}})

True positive edges:

\sum_{i < j} (1 {∣ Θ_{i j}^{1} ∣ > t_{0} and ∣ {\hat{Θ}}_{i j}^{1} ∣ > t_{0}} + 1 {∣ Θ_{i j}^{2} ∣ > t_{0} and ∣ {\hat{Θ}}_{i j}^{2} ∣ > t_{0}})

(2)

Positive perturbed columns (PPC):

\begin{array}{l} PNJGL : \sum_{i = 1}^{p} 1 {{‖ {\hat{V}}_{- i, i} ‖}_{2} > t_{s}}; \\ FGL / GL : \sum_{i = 1}^{p} 1 {{‖ {({\hat{Θ}}^{1} - {\hat{Θ}}^{2})}_{- i, i} ‖}_{2} > t_{s}} \end{array}

True positive perturbed columns (TPPC):
PNJGL: Σ_{i∈I_p} 1{||V̂₋_i,i||₂ > t_s};
FGL/GL: Σ_{i∈I_p} 1{||(Θ̂¹ − Θ̂²)₋_i,i||₂ > t_s},
where I_P is the set of perturbed column indices.
Positive co-hub columns (PCC):

\begin{array}{l} CNJGL : \sum_{i = 1}^{p} 1 {{‖ {\hat{V}}_{- i, i}^{1} ‖}_{2} > t_{s} and {‖ {\hat{V}}_{- i, i}^{2} ‖}_{2} > t_{s}}; \\ GGL / GL : \sum_{i = 1}^{p} 1 {{‖ {\hat{Θ}}_{- i, i}^{1} ‖}_{2} > t_{s} and {‖ {\hat{Θ}}_{- i, i}^{2} ‖}_{2} > t_{s}} \end{array}

True positive co-hub columns (TPCC):

\begin{array}{l} CNJGL : \sum_{i \in I_{c}} 1 {{‖ {\hat{V}}_{- i, i}^{1} ‖}_{2} > t_{s} and {‖ {\hat{V}}_{- i, i}^{2} ‖}_{2} > t_{s}}; \\ CGL / GL : \sum_{i \in I_{c}} 1 {{‖ {\hat{Θ}}_{- i, i}^{1} ‖}_{2} > t_{s} and {‖ {\hat{Θ}}_{- i, i}^{2} ‖}_{2} > t_{s}} \end{array}

where I_C is the set of co-hub column indices.

(3)

Error:

\sqrt{\sum_{i < j} {(Θ_{i j}^{1} - {\hat{Θ}}_{i j}^{1})}^{2}} + \sqrt{\sum_{i < j} {(Θ_{i j}^{2} - {\hat{Θ}}_{i j}^{2})}^{2}}