Edge-level inference results across all settings. The TPR and FDR are calculated separately for each of the two subnetworks (|$G_1$||$|V_1|=25$| and |$G_2$||$|V_2|=50$|). The means (standard deviations) of TPR and FDR are summarized based on 100 repeated simulations. TPR is determined by the proportion of edges in |$G_c$| that can be recovered by |$\hat{G}_c$|, and FDR is the proportion of edges in |$\hat{G}_c$| are not in |$G_c$|. TPR = 1 and FDR = 0 suggest a perfect recovery of |$G_c$| by |$\hat{G}_c$|. SICERS outperforms the comparable methods because the objective function can maximize the signal while suppressing noise, and thereby better recovers the underlying true |$G_c$|.
| . | . | . | |$S = 240$| . | |$S = 120$| . | ||||
|---|---|---|---|---|---|---|---|---|
| . | . | Cohen’s |$d$| . | 1.2 . | 0.8 . | 0.5 . | 1.2 . | 0.8 . | 0.5 . |
| SICERS | TPR | |$G_1$| | 1(0) | 0.87(0.2) | 0.91(0.19) | 1(0) | 0.9(0.2) | 0.88(0.2) |
| |$G_2$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | ||
| FDR | |$G_1$| | 0(0) | 0(0) | 0(0.01) | 0(0) | 0.02(0.04) | 0.02(0.04) | |
| |$G_2$| | 0(0) | 0.03(0.04) | 0.09(0.21) | 0(0) | 0.04(0.05) | 0.09(0.19) | ||
| Louvain | TPR | |$G_1$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) |
| |$G_2$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | ||
| FDR | |$G_1$| | 0.21(0.1) | 0.58(0.12) | 0.44(0.11) | 0.25(0.11) | 0.58(0.12) | 0.41(0.16) | |
| |$G_2$| | 0.16(0.06) | 0.03(0.03) | 0.04(0.04) | 0.16(0.05) | 0.02(0.03) | 0.03(0.03) | ||
| Dense | TPR | |$G_1$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) |
| |$G_2$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | ||
| FDR | |$G_1$| | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | |
| |$G_2$| | 0(0) | 0(0) | 0.35(0.27) | 0(0) | 0(0) | 0.52(0.13) | ||
| NBS | TPR | |$G_1$| | 1(0) | NA | NA | 1(0) | NA | NA |
| |$G_2$| | 1(0) | NA | NA | 1(0) | NA | NA | ||
| FDR | |$G_0$| | 0.28(0.05) | NA | NA | 0.21(0.1) | NA | NA | |
| |$G_2$| | 0.59(0.16) | NA | NA | 0.54(0.21) | NA | NA | ||
| BH-FDR | TPR | 1(0) | 0.95(0) | 0.94(0) | 1(0) | 0.94(0.01) | 0.75(0.01) | |
| FDR | 0.18(0.01) | 0.5(0) | 0.54(0) | 0.18(0.01) | 0.5(0) | 0.54(0.01) | ||
| . | . | . | |$S = 240$| . | |$S = 120$| . | ||||
|---|---|---|---|---|---|---|---|---|
| . | . | Cohen’s |$d$| . | 1.2 . | 0.8 . | 0.5 . | 1.2 . | 0.8 . | 0.5 . |
| SICERS | TPR | |$G_1$| | 1(0) | 0.87(0.2) | 0.91(0.19) | 1(0) | 0.9(0.2) | 0.88(0.2) |
| |$G_2$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | ||
| FDR | |$G_1$| | 0(0) | 0(0) | 0(0.01) | 0(0) | 0.02(0.04) | 0.02(0.04) | |
| |$G_2$| | 0(0) | 0.03(0.04) | 0.09(0.21) | 0(0) | 0.04(0.05) | 0.09(0.19) | ||
| Louvain | TPR | |$G_1$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) |
| |$G_2$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | ||
| FDR | |$G_1$| | 0.21(0.1) | 0.58(0.12) | 0.44(0.11) | 0.25(0.11) | 0.58(0.12) | 0.41(0.16) | |
| |$G_2$| | 0.16(0.06) | 0.03(0.03) | 0.04(0.04) | 0.16(0.05) | 0.02(0.03) | 0.03(0.03) | ||
| Dense | TPR | |$G_1$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) |
| |$G_2$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | ||
| FDR | |$G_1$| | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | |
| |$G_2$| | 0(0) | 0(0) | 0.35(0.27) | 0(0) | 0(0) | 0.52(0.13) | ||
| NBS | TPR | |$G_1$| | 1(0) | NA | NA | 1(0) | NA | NA |
| |$G_2$| | 1(0) | NA | NA | 1(0) | NA | NA | ||
| FDR | |$G_0$| | 0.28(0.05) | NA | NA | 0.21(0.1) | NA | NA | |
| |$G_2$| | 0.59(0.16) | NA | NA | 0.54(0.21) | NA | NA | ||
| BH-FDR | TPR | 1(0) | 0.95(0) | 0.94(0) | 1(0) | 0.94(0.01) | 0.75(0.01) | |
| FDR | 0.18(0.01) | 0.5(0) | 0.54(0) | 0.18(0.01) | 0.5(0) | 0.54(0.01) | ||
Edge-level inference results across all settings. The TPR and FDR are calculated separately for each of the two subnetworks (|$G_1$||$|V_1|=25$| and |$G_2$||$|V_2|=50$|). The means (standard deviations) of TPR and FDR are summarized based on 100 repeated simulations. TPR is determined by the proportion of edges in |$G_c$| that can be recovered by |$\hat{G}_c$|, and FDR is the proportion of edges in |$\hat{G}_c$| are not in |$G_c$|. TPR = 1 and FDR = 0 suggest a perfect recovery of |$G_c$| by |$\hat{G}_c$|. SICERS outperforms the comparable methods because the objective function can maximize the signal while suppressing noise, and thereby better recovers the underlying true |$G_c$|.
| . | . | . | |$S = 240$| . | |$S = 120$| . | ||||
|---|---|---|---|---|---|---|---|---|
| . | . | Cohen’s |$d$| . | 1.2 . | 0.8 . | 0.5 . | 1.2 . | 0.8 . | 0.5 . |
| SICERS | TPR | |$G_1$| | 1(0) | 0.87(0.2) | 0.91(0.19) | 1(0) | 0.9(0.2) | 0.88(0.2) |
| |$G_2$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | ||
| FDR | |$G_1$| | 0(0) | 0(0) | 0(0.01) | 0(0) | 0.02(0.04) | 0.02(0.04) | |
| |$G_2$| | 0(0) | 0.03(0.04) | 0.09(0.21) | 0(0) | 0.04(0.05) | 0.09(0.19) | ||
| Louvain | TPR | |$G_1$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) |
| |$G_2$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | ||
| FDR | |$G_1$| | 0.21(0.1) | 0.58(0.12) | 0.44(0.11) | 0.25(0.11) | 0.58(0.12) | 0.41(0.16) | |
| |$G_2$| | 0.16(0.06) | 0.03(0.03) | 0.04(0.04) | 0.16(0.05) | 0.02(0.03) | 0.03(0.03) | ||
| Dense | TPR | |$G_1$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) |
| |$G_2$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | ||
| FDR | |$G_1$| | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | |
| |$G_2$| | 0(0) | 0(0) | 0.35(0.27) | 0(0) | 0(0) | 0.52(0.13) | ||
| NBS | TPR | |$G_1$| | 1(0) | NA | NA | 1(0) | NA | NA |
| |$G_2$| | 1(0) | NA | NA | 1(0) | NA | NA | ||
| FDR | |$G_0$| | 0.28(0.05) | NA | NA | 0.21(0.1) | NA | NA | |
| |$G_2$| | 0.59(0.16) | NA | NA | 0.54(0.21) | NA | NA | ||
| BH-FDR | TPR | 1(0) | 0.95(0) | 0.94(0) | 1(0) | 0.94(0.01) | 0.75(0.01) | |
| FDR | 0.18(0.01) | 0.5(0) | 0.54(0) | 0.18(0.01) | 0.5(0) | 0.54(0.01) | ||
| . | . | . | |$S = 240$| . | |$S = 120$| . | ||||
|---|---|---|---|---|---|---|---|---|
| . | . | Cohen’s |$d$| . | 1.2 . | 0.8 . | 0.5 . | 1.2 . | 0.8 . | 0.5 . |
| SICERS | TPR | |$G_1$| | 1(0) | 0.87(0.2) | 0.91(0.19) | 1(0) | 0.9(0.2) | 0.88(0.2) |
| |$G_2$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | ||
| FDR | |$G_1$| | 0(0) | 0(0) | 0(0.01) | 0(0) | 0.02(0.04) | 0.02(0.04) | |
| |$G_2$| | 0(0) | 0.03(0.04) | 0.09(0.21) | 0(0) | 0.04(0.05) | 0.09(0.19) | ||
| Louvain | TPR | |$G_1$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) |
| |$G_2$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | ||
| FDR | |$G_1$| | 0.21(0.1) | 0.58(0.12) | 0.44(0.11) | 0.25(0.11) | 0.58(0.12) | 0.41(0.16) | |
| |$G_2$| | 0.16(0.06) | 0.03(0.03) | 0.04(0.04) | 0.16(0.05) | 0.02(0.03) | 0.03(0.03) | ||
| Dense | TPR | |$G_1$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) |
| |$G_2$| | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | 1(0) | ||
| FDR | |$G_1$| | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | 0(0) | |
| |$G_2$| | 0(0) | 0(0) | 0.35(0.27) | 0(0) | 0(0) | 0.52(0.13) | ||
| NBS | TPR | |$G_1$| | 1(0) | NA | NA | 1(0) | NA | NA |
| |$G_2$| | 1(0) | NA | NA | 1(0) | NA | NA | ||
| FDR | |$G_0$| | 0.28(0.05) | NA | NA | 0.21(0.1) | NA | NA | |
| |$G_2$| | 0.59(0.16) | NA | NA | 0.54(0.21) | NA | NA | ||
| BH-FDR | TPR | 1(0) | 0.95(0) | 0.94(0) | 1(0) | 0.94(0.01) | 0.75(0.01) | |
| FDR | 0.18(0.01) | 0.5(0) | 0.54(0) | 0.18(0.01) | 0.5(0) | 0.54(0.01) | ||
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