SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 4: Test accuracy on node classification: top three models indicated as  $1^{\mathrm{st}}$ ,  $2^{\mathrm{nd}}$ ,  $3^{\mathrm{rd}}$ .
(a) Undirected graphs.
Film	Squirrel	Chameleon	Citeseer	Pubmed	Cora
GGCN	37.54 ± 1.56	55.17 ± 1.58	71.14 ± 1.84	77.14 ± 1.45	89.15 ± 0.37	87.95 ± 1.05
GPRGNN	34.63 ± 1.22	31.61 ± 1.24	46.58 ± 1.71	77.13 ± 1.67	87.54 ± 0.38	87.95 ± 1.18
FAGCN	35.70 ± 1.00	36.48 ± 1.86	60.11 ± 2.15	77.11 ± 1.57	89.49 ± 0.38	87.87 ± 1.20
GCNII	37.44 ± 1.30	38.47 ± 1.58	63.86 ± 3.04	77.33 ± 1.48	90.15 ± 0.43	88.37 ± 1.25
Geom-GCN	31.59 ± 1.15	38.15 ± 0.92	60.00 ± 2.81	78.02 ± 1.15	89.95 ± 0.47	85.35 ± 1.57
PairNorm	27.40 ± 1.24	50.44 ± 2.04	62.74 ± 2.82	73.59 ± 1.47	87.53 ± 0.44	85.79 ± 1.01
GraphSAGE	34.23 ± 0.99	41.61 ± 0.74	58.73 ± 1.68	76.04 ± 1.30	88.45 ± 0.50	86.90 ± 1.04
GCN	27.32 ± 1.10	53.43 ± 2.01	64.82 ± 2.24	76.50 ± 1.36	88.42 ± 0.50	86.98 ± 1.27
GAT	27.44 ± 0.89	40.72 ± 1.55	60.26 ± 2.50	76.55 ± 1.23	87.30 ± 1.10	86.33 ± 0.48
MLP	36.53 ± 0.70	28.77 ± 1.56	46.21 ± 2.99	74.02 ± 1.90	75.69 ± 2.00	87.16 ± 0.37
CGNN	35.95 ± 0.86	29.24 ± 1.09	46.89 ± 1.66	76.91 ± 1.81	87.70 ± 0.49	87.10 ± 1.35
GRAND	35.62 ± 1.01	40.05 ± 1.50	54.67 ± 2.54	76.46 ± 1.77	89.02 ± 0.51	87.36 ± 0.96
Sheaf (max)	37.81 ± 1.15	56.34 ± 1.32	68.04 ± 1.58	76.70 ± 1.57	89.49 ± 0.40	86.90 ± 1.13
GRAFFNL	35.96 ± 0.95	59.01 ± 1.31	71.38 ± 1.47	76.81 ± 1.12	89.81 ± 0.50	87.81 ± 1.13
GREAD	37.90 ± 1.17	59.22 ± 1.44	71.38 ± 1.30	77.60 ± 1.81	90.23 ± 0.55	88.57 ± 0.66
GraphCON	35.58 ± 1.24	35.51 ± 1.40	49.63 ± 1.89	76.36 ± 2.67	88.01 ± 0.47	87.22 ± 1.48
ACMP	34.93 ± 1.26	40.05 ± 1.53	57.59 ± 2.09	76.71 ± 1.77	87.79 ± 0.47	87.71 ± 0.95
GCN+DropEdge	29.93 ± 0.80	41.30 ± 1.77	59.06 ± 2.04	76.57 ± 2.68	86.97 ± 0.42	83.54 ± 1.06
GAT+DropEdge	28.95 ± 0.76	41.27 ± 1.76	58.95 ± 2.13	76.13 ± 2.20	86.91 ± 0.45	83.54 ± 1.06
FLODE	37.16 ± 1.42	64.23 ± 1.84	73.60 ± 1.55	78.07 ± 1.62	89.02 ± 0.38	86.44 ± 1.17
(b) Directed graphs.
Film	Squirrel	Chameleon
ACM	36.89 ± 1.18	54.4 ± 1.88	67.08 ± 2.04
HLP	34.59 ± 1.32	74.17 ± 1.83	77.48 ± 1.50
FSGNN	35.67 ± 0.69	73.48 ± 2.13	78.14 ± 1.25
GRAFF	37.11 ± 1.08	58.72 ± 0.84	71.08 ± 1.75
FLODE	37.41 ± 1.06	74.03 ± 1.58	77.98 ± 1.05
(c) Heterophily-specific graphs. For Minesweeper, Tolokers and Questions the evaluation metric is the AUROC.
Roman-empire	Minesweeper	Tolokers	Questions
ResNet	65.88 ± 0.38	50.89 ± 1.39	72.95 ± 1.06	70.34 ± 0.76
ResNet+SGC	73.90 ± 0.51	70.88 ± 0.90	80.70 ± 0.97	75.81 ± 0.96
ResNet+adj	52.25 ± 0.40	50.42 ± 0.83	78.78 ± 1.11	75.77 ± 1.24
GCN	73.69 ± 0.74	89.75 ± 0.52	83.64 ± 0.67	76.09 ± 1.27
GraphSAGE	85.74 ± 0.67	93.51 ± 0.57	82.43 ± 0.44	76.44 ± 0.62
GAT	80.87 ± 0.30	92.01 ± 0.68	83.70 ± 0.47	77.43 ± 1.20
GAT-sep	88.75 ± 0.41	93.91 ± 0.35	83.78 ± 0.43	76.79 ± 0.71
GT	86.51 ± 0.73	91.85 ± 0.76	83.23 ± 0.64	77.95 ± 0.68
GT-sep	87.32 ± 0.39	92.29 ± 0.47	82.52 ± 0.92	78.05 ± 0.93
FAGCN	60.11 ± 0.52	89.71 ± 0.31	73.35 ± 1.01	63.59 ± 1.46
CPGNN	63.96 ± 0.62	52.03 ± 5.46	73.36 ± 1.01	65.96 ± 1.95
H2GCN	64.85 ± 0.27	86.24 ± 0.61	72.94 ± 0.97	55.48 ± 0.91
FSGNN	79.92 ± 0.56	90.08 ± 0.70	82.76 ± 0.61	78.86 ± 0.92
GloGNN	59.63 ± 0.69	51.08 ± 1.23	73.39 ± 1.17	65.74 ± 1.19
FAGCN	65.22 ± 0.56	88.17 ± 0.73	77.75 ± 1.05	77.24 ± 1.26
GBK-GNN	74.57 ± 0.47	90.85 ± 0.58	81.01 ± 0.67	74.47 ± 0.86
JacobiConv	71.14 ± 0.42	89.66 ± 0.40	68.66 ± 0.65	73.88 ± 1.16
FLODE	74.97 ± 0.53	92.43 ± 0.51	84.17 ± 0.58	78.39 ± 1.22
Table 5: Selected hyperparameters, learned exponent, step size, and Dirichlet energy in the last layer for real-world datasets.
(a) Undirected.
Dataset
Film	Squirrel	Chameleon	CiteSeer	Pubmed	Cora
learning rate	10-3	2.5 · 10-3	5 · 10-3	10-2	10-2	10-2
weight decay	5 · 10-4	5 · 10-4	10-3	5 · 10-3	10-3	5 · 10-3
hidden channels	256	64	64	64	64	64
num. layers	1	6	4	2	3	2
encoder layers	3	1	1	1	3	1
decoder layers	2	2	2	1	1	2
input dropout	0	1.5 · 10-1	0	0	5 · 10-2	0
decoder dropout	10-1	10-1	0	0	10-1	0
exponent	1.001 ± 0.003	0.17 ± 0.03	0.35 ± 0.15	0.92 ± 0.03	0.82 ± 0.07	0.90 ± 0.02
step size	0.991 ± 0.002	1.08 ± 0.01	1.22 ± 0.03	1.04 ± 0.02	1.12 ± 0.02	1.06 ± 0.01
Dirichlet energy	0.246 ± 0.006	0.40 ± 0.02	0.13 ± 0.03	0.021 ± 0.001	0.015 ± 0.001	0.0227 ± 0.0006
(b) Directed.
Dataset
Film	Squirrel	Chameleon
learning rate	10-3	2.5 · 10-3	10-2
weight decay	5 · 10-4	5 · 10-4	10-3
hidden channels	256	64	64
num. layers	1	6	5
encoder layers	3	1	1
decoder layers	2	2	2
input dropout	0	10-1	0
decoder dropout	0.1	10-1	0
exponent	1.001 ± 0.005	0.28 ± 0.06	0.30 ± 0.11
step size	0.990 ± 0.002	1.22 ± 0.02	1.22 ± 0.05
Dirichlet energy	0.316 ± 0.005	0.38 ± 0.02	0.27 ± 0.04
(c) Heterophily-specific graphs.
Dataset
Roman-empire	Minesweeper	Tolokers	Questions
learning rate	10-3	10-3	10-3	10-2
weight decay	0	0	0	5 · 10-4
hidden channels	512	512	512	128
num. layers	4	4	4	5
encoder layers	2	2	1	2
decoder layers	2	2	2	2
input dropout	0	0	0	0
decoder dropout	0	0	0	0
exponent	0.689 ± 0.038	0.749 ± 0.017	1.053 ± 0.041	1.090 ± 0.046
step size	0.933 ± 0.015	0.984 ± 0.004	0.993 ± 0.009	0.789 ± 0.062
Dirichlet energy	0.059 ± 0.003	0.173 ± 0.019	0.155 ± 0.013	0.092 ± 0.039
Table 6: Node classification accuracy of ordered DSBM graphs: top three models as  $1^{\mathrm{st}}$ ,  $2^{\mathrm{nd}}$  and  $3^{\mathrm{rd}}$ .
(a) Varying edge density.
α*
0.1	0.08	0.05
ChebNet	19.9 ± 0.6	20.0 ± 0.7	20.0 ± 0.7
GCN-D	68.9 ± 2.1	67.6 ± 2.7	58.5 ± 2.0
APPNP-D	97.7 ± 1.7	95.9 ± 2.2	90.3 ± 2.4
GraphSAGE-D	20.1 ± 1.1	19.9 ± 0.8	19.9 ± 1.0
GIN-D	57.3 ± 5.8	55.4 ± 5.5	50.9 ± 7.7
GAT-D	42.1 ± 5.3	39.0 ± 7.0	37.2 ± 5.5
DGCN	84.9 ± 7.2	81.2 ± 8.2	64.4 ± 12.4
DiGraph	82.1 ± 1.7	77.7 ± 1.6	66.1 ± 2.4
DiGraphIB	99.2 ± 0.5	97.7 ± 0.7	89.3 ± 1.7
MagNet	99.6 ± 0.2	98.3 ± 0.8	94.1 ± 1.2
FLODE	99.3 ± 0.1	98.8 ± 0.1	97.5 ± 0.1
(b) Varying net flow.
β*
0.05	0.10	0.15	0.20	0.25	0.30	0.35	0.40
ChebNet	19.9 ± 0.7	20.1 ± 0.6	20.0 ± 0.6	20.1 ± 0.8	19.9 ± 0.9	20.0 ± 0.5	19.7 ± 0.9	20.0 ± 0.5
GCN-D	68.6 ± 2.2	74.1 ± 1.8	75.5 ± 1.3	74.9 ± 1.3	72.0 ± 1.4	65.4 ± 1.6	58.1 ± 2.4	45.6 ± 4.7
APPNP-D	97.4 ± 1.8	94.3 ± 2.4	89.4 ± 3.6	79.8 ± 9.0	69.4 ± 3.9	59.6 ± 4.9	51.8 ± 4.5	39.4 ± 5.3
GraphSAGE-D	20.2 ± 1.2	20.0 ± 1.0	20.0 ± 0.8	20.0 ± 0.7	19.6 ± 0.9	19.8 ± 0.7	19.9 ± 0.9	19.9 ± 0.8
GIN-D	57.9 ± 6.3	48.0 ± 11.4	32.7 ± 12.9	26.5 ± 10.0	23.8 ± 6.0	20.6 ± 3.0	20.5 ± 2.8	19.8 ± 0.5
GAT-D	42.0 ± 4.8	32.7 ± 5.1	25.6 ± 3.8	19.9 ± 1.4	20.0 ± 1.0	19.8 ± 0.8	19.6 ± 0.2	19.5 ± 0.2
DGCN	81.4 ± 1.1	84.7 ± 0.7	85.5 ± 1.0	86.2 ± 0.8	84.2 ± 1.1	78.4 ± 1.3	69.6 ± 1.5	54.3 ± 1.5
DiGraph	82.5 ± 1.4	82.9 ± 1.9	81.9 ± 1.1	79.7 ± 1.3	73.5 ± 1.9	67.4 ± 2.8	57.8 ± 1.6	43.0 ± 7.1
DiGraphIB	99.2 ± 0.4	97.9 ± 0.6	94.1 ± 1.7	88.7 ± 2.0	82.3 ± 2.7	70.0 ± 2.2	57.8 ± 6.4	41.0 ± 9.0
MagNet	99.6 ± 0.2	99.0 ± 1.0	97.5 ± 0.8	94.2 ± 1.6	88.7 ± 1.9	79.4 ± 2.9	68.8 ± 2.4	51.8 ± 3.1
FLODE	99.3 ± 0.1	98.5 ± 0.1	96.7 ± 0.2	92.8 ± 0.1	87.2 ± 0.3	77.1 ± 0.5	63.8 ± 0.3	50.1 ± 0.5
Following Zhang et al. (2021), we train our model in both experiments for 3000 epochs and use early-stopping if the validation accuracy does not increase for 500 epochs. We select the best model based on the validation accuracy after sweeping over a few hyperparameters. We give exact numerical values for the experiments with the standard error in Table 6a and refer to Appendix A.2 for the chosen hyperparameters.
DSBM with varying edge density. In the first experiment, the model is evaluated based on its performance on the DSBM with varying  $\alpha_{i,j} = \alpha^{*}$ ,  $\alpha^{*} \in \{0.1, 0.08, 0.05\}$  for  $i \neq j$ , which essentially changes the density of edges between different clusters. The other probabilities are fixed at  $\alpha_{i,i} = 0.5$ ,  $\beta_{i,i} = 0.5$  and  $\beta_{i,j} = 0.05$  for  $i > j$ . The results are shown in Figure 6 with exact numerical values in Table 6a.
DSBM with varying net flow. In the other scenario, the model is tested on how it performs when the net flow from one cluster to another varies. This is achieved by keeping  $\alpha_{i,j} = 0.1$  constant for all  $i$  and  $j$ , and allowing  $\beta_{i,j}$  to vary from 0.05 to 0.4. The other probabilities are fixed at  $\alpha_{i,i} = 0.5$  and  $\beta_{i,i} = 0.5$ . The results are shown in Figure 6 with exact numerical values in Table 6b.
A.3 Ablation Study
We perform an ablation study on Chameleon and Squirrel (directed, heterophilic), and Citeseer (undirected, homophilic). For this, we sweep over different model options using the same hyperparameters
Table 7: Selected hyperparameters for DSBM dataset.
(a) Varying edge density.
α*
0.1	0.08	0.05
learning rate	5·10-3	5·10-3	5·10-3
decay	1·10-3	1·10-3	5·10-4
input dropout	1·10-1	2·10-1	1·10-1
decoder dropout	1·10-1	5·10-2	1·10-1
hidden channels	256	256	256
(b) Varying net flow.
β*	0.4
0.05	0.1	0.15	0.2	0.25	0.3	0.35
learning rate	5·10-3	5·10-3	1·10-3	1·10-3	1·10-3	1·10-3	1·10-3	1·10-3
decay	1·10-3	1·10-3	5·10-4	1·10-3	1·10-3	1·10-3	5·10-4	1·10-3
input dropout	1·10-1	1·10-1	2·10-1	1·10-1	1·10-1	2·10-1	5·10-2	2·10-1
decoder dropout	1·10-1	1·10-1	5·10-2	5·10-2	5·10-2	1·10-1	2·10-1	1·10-1
hidden channels	256	256	256	256	256	256	256	256
via grid search. The test accuracy corresponding to the hyperparameters that yielded maximum validation accuracy is reported in Table 8.
The ablation study on Chameleon demonstrates that all the components of the model (learnable exponent, ODE framework with the Schrödinger equation, and directionality via the SNA) contribute to the performance of FLODE. The fact that performance drops when any of these components are not used suggests that they all play crucial roles in the model's ability to capture the structure and evolution of heterophilic graphs. It is important to note that the performance appears to be more dependent on the adjustable fraction in the FGL than on the use of the ODE framework, illustrating that the fractional Laplacian alone can effectively capture long-range dependencies. However, when the ODE framework is additionally employed, a noticeable decrease in variance is observed.
From Theory to Practice. We conduct an ablation study to investigate the role of depth on Chameleon, Citeseer, Cora, and Squirrel datasets. The results, depicted in Figure 8, demonstrate that the neural ODE framework enables GNNs to scale to large depths (256 layers). Moreover, we see that the fractional Laplacian improves over the standard Laplacian in the heterophilic graphs which is supported by our claims in Section 5.2. We highlight that using only the fractional Laplacian without the neural ODE framework oftentimes outperforms the standard Laplacian with the neural ODE framework. This indicates the importance of the long-range connections built by the fractional Laplacian.
We further demonstrate the close alignment of our theoretical and experimental results, which enables us to precisely anticipate when the models will exhibit HFD or LFD behaviors. In this context, we calculate parameters (according to Theorem D.5) and illustrate at each depth the expected and observed behaviors. For Squirrel and Chameleon, which are heterophilic graphs, we observe that both their theoretical and empirical behaviors are HFD. Additionally, the learned exponent is small. In contrast, for Cora and Citeseer, we see the opposite.
Finally, we employ the best hyperparameters in Table 5a to solve both fractional heat and Schrödinger graph ODEs, further substantiating the intimate link between our theoretical advancements and practical applications.
Table 8: Ablation study on node classification task: top two models are indicated as  ${1}^{\text{st }}$  and  ${2}^{\text{nd }}$
(a) Chameleon (directed, heterophilic).
Update Rule	Test Accuracy	Dirichlet Energy
D	xt+1 = xt - ihLαxtW	77.79 ± 1.42	0.213 (t=5)
xt+1 = xt - ihL xtW	75.72 ± 1.13	0.169 (t=6)
xt+1 = -i LαxtW	77.35 ± 2.22	0.177 (t=4)
xt+1 = -i L xtW	69.61 ± 1.59	0.178 (t=4)
U	xt+1 = xt - ihLαxtW	73.60 ± 1.68	0.131 (t=4)
xt+1 = xt - ihL xtW	70.15 ± 0.86	0.035 (t=4)
xt+1 = -i LαxtW	71.25 ± 3.04	0.118 (t=4)
xt+1 = -i L xtW	67.19 ± 2.49	0.040 (t=4)
D	xt+1 = xt - hLαxtW	77.33 ± 1.47	0.378 (t=6)
xt+1 = xt - hL xtW	73.55 ± 0.94	0.165 (t=6)
xt+1 = - LαxtW	74.12 ± 3.60	0.182 (t=4)
xt+1 = - L xtW	68.47 ± 2.77	0.208 (t=4)
(b) Squirrel (directed, heterophilic).
Update Rule	Test Accuracy	Dirichlet Energy
D	xt+1 = xt - ihLαxtW	74.03 ± 1.58	0.38 ± 0.02
xt+1 = xt - ihL xtW	64.04 ± 2.25	0.35 ± 0.02
xt+1 = -i LαxtW	64.25 ± 1.85	0.46 ± 0.01
xt+1 = -i L xtW	42.04 ± 1.58	0.29 ± 0.05
U	xt+1 = xt - ihLαxtW	64.23 ± 1.84	0.40 ± 0.02
xt+1 = xt - ihL xtW	55.19 ± 1.52	0.26 ± 0.03
xt+1 = -i LαxtW	61.40 ± 2.15	0.43 ± 0.01
xt+1 = -i L xtW	41.19 ± 1.95	0.20 ± 0.02
D	xt+1 = xt - hLαxtW	71.86 ± 1.65	0.50 ± 0.01
xt+1 = xt - hL xtW	59.34 ± 1.78	0.43 ± 0.03
xt+1 = - LαxtW	42.91 ± 7.86	0.32 ± 0.08
xt+1 = - L xtW	35.37 ± 1.69	0.25 ± 0.05
U	xt+1 = xt - hLαxtW	62.95 ± 2.02	0.61 ± 0.08
xt+1 = xt - hL xtW	52.19 ± 1.17	0.51 ± 0.07
xt+1 = - LαxtW	59.04 ± 0.02	0.44 ± 0.02
xt+1 = - L xtW	39.69 ± 1.54	0.20 ± 0.02
(c) Citeseer (undirected, homphilic).
Update Rule	Test Accuracy	Dirichlet Energy
xt+1 = xt - ihLαxtW	78.07 ± 1.62	0.021 (t=5)
xt+1 = xt - ihL xxtW	77.97 ± 2.29	0.019 (t=4)
xt+1 = -i LαxtW	77.27 ± 2.10	0.011 (t=6)
xt+1 = -i L xxtW	77.97 ±2.23	0.019 (t=4)
Table 9: Learned  $\alpha$  and spectrum of W. According to Theorem 5.3, we denote FD :=  $\lambda_{K}(\mathbf{W})\mathrm{f}_{\alpha}(\lambda_{1}(\mathbf{L})) - \lambda_{1}(\mathbf{W})$  and FD :=  $\Im (\lambda_K(\mathbf{W}))\mathrm{f}_\alpha (\lambda_1(\mathbf{L})) - \Im (\lambda_1(\mathbf{W}))$  for the fractional heat (H) and Schrödinger (S) graph ODEs, respectively. The heterophilic graphs Squirrel and Chameleon exhibit HFD since FD < 0, while the homophilic Cora, Citeseer, Pubmed exhibit LFD since FD > 0.
λ1(L)	Film -0.9486	Squirrel -0.8896	Chameleon -0.9337	CiteSeer -0.5022	Pubmed -0.6537	Cora -0.4826
H	α	1.008 ± 0.007	0.19 ± 0.05	0.37 ± 0.14	0.89 ± 0.06	1.15 ± 0.08	0.89 ± 0.01
λ1(W)	-2.774 ± 0.004	-1.62 ± 0.03	-1.81 ± 0.02	-1.76 ± 0.01	-1.66 ± 0.06	-1.81 ± 0.01
λK(W)	2.858 ± 0.009	2.21 ± 0.03	2.29 ± 0.05	2.28 ± 0.06	1.1 ± 0.3	2.32 ± 0.01
FD	0.367 ± 0.001	-0.54 ± 0.02	-0.42 ± 0.04	0.52 ± 0.02	0.97 ± 0.09	0.60 ± 0.01
S	α	1.000 ± 0.002	0.17 ± 0.03	0.34 ± 0.11	0.90 ± 0.07	0.76 ± 0.07	0.90 ± 0.02
∑(λ1(W))	-2.795 ± 0.001	-1.68 ± 0.01	-1.79 ± 0.01	-1.70 ± 0.04	-1.74 ± 0.01	-1.78 ± 0.01
∑(λK(W))	2.880 ± 0.002	2.21 ± 0.03	2.46 ± 0.02	2.29 ± 0.07	0.98 ± 0.09	2.30 ± 0.02
FD	0.4945 ± 0.0001	-0.48 ± 0.03	-0.62 ± 0.03	0.46 ± 0.06	1.03 ± 0.05	0.59 ± 0.01
Figure 8: Ablation study on the effect of different update rules and different number of layers on undirected datasets. The x-axis shows the number of layers  $2^{L}$  for  $L \in \{0, \dots, 8\}$ . FD is calculated according to Theorem 5.3.
B Appendix for Section 3
Proposition 3.3. Let  $\mathcal{G}$  be a directed graph with SNA  $\mathbf{L}$ . For every  $\lambda \in \lambda (\mathbf{L})$ , it holds  $|\lambda |\leq 1$  and  $\lambda (\mathbf{I} - \mathbf{L}) = 1 - \lambda (\mathbf{L})$ .
Proof. We show that the numerical range  $\mathcal{W}(\mathbf{L}) = \{\mathbf{x}^{\mathsf{H}}\mathbf{L}\mathbf{x} : \mathbf{x}^{\mathsf{H}}\mathbf{x} = 1\}$  satisfies  $\mathcal{W}(\mathbf{L}) \subset [-1,1]$ . As  $\mathcal{W}(\mathbf{L})$  contains all eigenvalues of  $\mathbf{L}$  the thesis follows.
Let  $\mathbf{A}$  be the adjacency matrix of  $\mathcal{G}$  and  $\mathbf{x} \in \mathbb{C}^N$  with  $\mathbf{x}^{\mathrm{H}}\mathbf{x} = 1$ . Applying the Cauchy-Schwartz inequality in (2) and (3), we get
$$
\begin{array}{l} \left| \mathbf {x} ^ {\mathsf {H}} \mathbf {L} \mathbf {x} \right| \stackrel {(1)} {\leq} \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {N} a _ {i, j} \frac {\left| x _ {i} \right| \left| x _ {j} \right|}{\sqrt {d _ {i} ^ {i n} d _ {j} ^ {o u t}}} \\ = \sum_ {i = 1} ^ {N} \frac {\left| x _ {i} \right|}{\sqrt {d _ {i} ^ {i n}}} \sum_ {j = 1} ^ {N} a _ {i, j} \frac {\left| x _ {j} \right|}{\sqrt {d _ {j} ^ {o u t}}} \\ \stackrel {(2)} {\leq} \sum_ {i = 1} ^ {N} \frac {| x _ {i} |}{\sqrt {d _ {i} ^ {i n}}} \sqrt {\sum_ {j = 1} ^ {N} a _ {i , j} \frac {| x _ {j} | ^ {2}}{d _ {j} ^ {o u t}} \sum_ {j = 1} ^ {N} a _ {i , j}} \\ = \sum_ {i = 1} ^ {N} | x _ {i} | \sqrt {\sum_ {j = 1} ^ {N} a _ {i , j} \frac {| x _ {j} | ^ {2}}{d _ {j} ^ {o u t}}} \\ \stackrel {(3)} {\leq} \sqrt {\sum_ {i = 1} ^ {N} | x _ {i} | ^ {2} \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {N} a _ {i , j} \frac {| x _ {j} | ^ {2}}{d _ {j} ^ {o u t}}} \\ = \sum_ {i = 1} ^ {N} \left| x _ {i} \right| ^ {2}, \\ \end{array}
$$
where we used  $a_{i,j}^2 = a_{i,j}$ . We have  $\sum_{i=1}^{N}|x_i|^2 = \mathbf{x}^\mathsf{H}\mathbf{x} = 1$  such that  $\mathcal{W}(\mathbf{L}) \subset [-1,1]$  follows. The second claim follows directly by  $(\mathbf{I} - \mathbf{L})\mathbf{v} = \mathbf{v} - \lambda \mathbf{v} = (1 - \lambda)\mathbf{v}$ .
Proposition 3.5. Let  $\mathcal{G}$  be a directed graph with SNA L. Then  $1 \in \lambda(\mathbf{L})$  if and only if the graph is weakly balanced. Suppose the graph is strongly connected; then  $-1 \in \lambda(\mathbf{L})$  if and only if the graph is weakly balanced with an even period.
Proof. Since the numerical range is only a superset of the set of eigenvalues, we cannot simply consider when the inequalities (1) - (3) in the previous proof are actual equalities. Therefore, we have to find another way to prove the statement. Suppose that the graph is weakly balanced, then
$$
\sum_ {j = 1} ^ {N} a _ {i, j} \left(\frac {k _ {j}}{\sqrt {d _ {j} ^ {\mathrm {o u t}}}} - \frac {k _ {i}}{\sqrt {d _ {i} ^ {\mathrm {i n}}}}\right) = 0, \forall j \in \{1, \dots , N \}.
$$
We will prove that  $\mathbf{k} = (k_{i})_{i = 1}^{N}$  is an eigenvector corresponding to the eigenvalue 1,
$$
(\mathbf {L k}) _ {i} = \sum_ {j = 1} ^ {N} \frac {a _ {i , j}}{\sqrt {d _ {i} ^ {\mathrm {i n}} d _ {j} ^ {\mathrm {o u t}}}} k _ {j} = \frac {1}{\sqrt {d _ {i} ^ {\mathrm {i n}}}} \sum_ {j = 1} ^ {N} \frac {a _ {i , j}}{\sqrt {d _ {j} ^ {\mathrm {o u t}}}} k _ {j} = \frac {1}{\sqrt {d _ {i} ^ {\mathrm {i n}}}} \sum_ {j = 1} ^ {N} \frac {a _ {i , j}}{\sqrt {d _ {i} ^ {\mathrm {i n}}}} k _ {i} = \frac {1}{d _ {i} ^ {\mathrm {i n}}} \left(\sum_ {j = 1} ^ {N} a _ {i, j}\right) k _ {i} = k _ {i}.
$$
For the other direction, suppose that there exists  $\mathbf{x} \in \mathbb{R}^N$  such that  $\mathbf{x} \neq 0$  and  $\mathbf{x} = \mathbf{L}\mathbf{x}$ . Then for all  $i \in \{1, \dots, N\}$
$$
0 = (\mathbf {L} \mathbf {x}) _ {i} - x _ {i} = \sum_ {j = 1} ^ {N} \frac {a _ {i , j}}{\sqrt {d _ {i} ^ {\mathrm {i n}} d _ {j} ^ {\mathrm {o u t}}}} x _ {j} - x _ {i} = \sum_ {j = 1} ^ {N} \frac {a _ {i , j}}{\sqrt {d _ {i} ^ {\mathrm {i n}} d _ {j} ^ {\mathrm {o u t}}}} x _ {j} - \sum_ {j = 1} ^ {N} \frac {a _ {i , j}}{d _ {i} ^ {\mathrm {i n}}} x _ {i}
$$
$$
= \sum_ {j = 1} ^ {N} \frac {a _ {i , j}}{\sqrt {d _ {i} ^ {\mathrm {i n}}}} \left(\frac {x _ {j}}{\sqrt {d _ {j} ^ {\mathrm {o u t}}}} - \frac {x _ {i}}{\sqrt {d _ {i} ^ {\mathrm {i n}}}}\right),
$$
hence, the graph is weakly balanced.
By Perron-Frobenius theorem for irreducible non-negative matrices, one gets that  $\mathbf{L}$  has exactly  $h$  eigenvalues with maximal modulus corresponding to the  $h$  roots of the unity, where  $h$  is the period of  $\mathbf{L}$ . Hence,  $-1$  is an eigenvalue of  $\mathbf{L}$  if and only if the graph is weakly balanced and  $h$  is even.
Proposition 3.6. For every  $\mathbf{x} \in \mathbb{C}^{N \times K}$ , we have
$$
\mathfrak {R} \left(\operatorname {t r a c e} \left(\mathbf {x} ^ {\mathsf {H}} (\mathbf {I} - \mathbf {L}) \mathbf {x}\right)\right) = \frac {1}{2} \sum_ {i, j = 1} ^ {N} a _ {i, j} \left\| \frac {\mathbf {x} _ {i}}{\sqrt {d _ {i} ^ {i n}}} - \frac {\mathbf {x} _ {j}}{\sqrt {d _ {j} ^ {o u t}}} \right\| _ {2} ^ {2},
$$
Moreover, there exists  $\mathbf{x} \neq 0$  such that  $\mathcal{E}(\mathbf{x}) = 0$  if and only if the graph is weakly balanced.
Proof. By direct computation, it holds
$$
\begin{array}{l} \frac {1}{2} \sum_ {i, j = 1} ^ {N} a _ {i, j} \left\| \frac {x _ {i , :}}{\sqrt {d _ {i} ^ {\mathrm {i n}}}} - \frac {x _ {j , :}}{\sqrt {d _ {j} ^ {\mathrm {o u t}}}} \right\| _ {2} ^ {2} \\ = \frac {1}{2} \sum_ {i, j = 1} ^ {N} a _ {i, j} \sum_ {k = 1} ^ {K} \left| \frac {x _ {i , k}}{\sqrt {d _ {i} ^ {\mathrm {i n}}}} - \frac {x _ {j , k}}{\sqrt {d _ {j} ^ {\mathrm {o u t}}}} \right| ^ {2} \\ = \frac {1}{2} \sum_ {i, j = 1} ^ {N} a _ {i, j} \sum_ {k = 1} ^ {K} \left(\frac {x _ {i , k}}{\sqrt {d _ {i} ^ {\mathrm {i n}}}} - \frac {x _ {j , k}}{\sqrt {d _ {j} ^ {\mathrm {o u t}}}}\right) ^ {*} \left(\frac {x _ {i , k}}{\sqrt {d _ {i} ^ {\mathrm {i n}}}} - \frac {x _ {j , k}}{\sqrt {d _ {j} ^ {\mathrm {o u t}}}}\right) \\ = \frac {1}{2} \sum_ {i, j = 1} ^ {N} \sum_ {k = 1} ^ {K} a _ {i, j} \frac {\left| x _ {i , k} \right| ^ {2}}{d _ {i} ^ {\mathrm {i n}}} + \frac {1}{2} \sum_ {i, j = 1} ^ {N} \sum_ {k = 1} ^ {K} a _ {i, j} \frac {\left| x _ {j , k} \right| ^ {2}}{d _ {j} ^ {\mathrm {o u t}}} \\ - \frac {1}{2} \sum_ {i, j = 1} ^ {N} \sum_ {k = 1} ^ {K} a _ {i, j} \frac {x _ {i , k} ^ {*} x _ {j , k}}{\sqrt {d _ {i} ^ {\text {i n}} d _ {j} ^ {\text {o u t}}}} - \frac {1}{2} \sum_ {i, j = 1} ^ {N} \sum_ {k = 1} ^ {K} a _ {i, j} \frac {x _ {i , k} x _ {j , k} ^ {*}}{\sqrt {d _ {i} ^ {\text {i n}} d _ {j} ^ {\text {o u t}}}} \\ = \frac {1}{2} \sum_ {i = 1} ^ {N} \sum_ {k = 1} ^ {K} | x _ {i, k} | ^ {2} + \frac {1}{2} \sum_ {j = 1} ^ {N} \sum_ {k = 1} ^ {K} | x _ {j, k} | ^ {2} - \frac {1}{2} \sum_ {i, j = 1} ^ {N} \sum_ {k = 1} ^ {K} a _ {i, j} \frac {x _ {i , k} ^ {*} x _ {j , k}}{\sqrt {d _ {i} ^ {\text {i n}} d _ {j} ^ {\text {o u t}}}} - \frac {1}{2} \sum_ {i, j = 1} ^ {N} \sum_ {k = 1} ^ {K} a _ {i, j} \frac {x _ {i , k} x _ {j , k} ^ {*}}{\sqrt {d _ {i} ^ {\text {i n}} d _ {j} ^ {\text {o u t}}}} \\ = \sum_ {i = 1} ^ {N} \sum_ {k = 1} ^ {K} | x _ {i, k} | ^ {2} - \frac {1}{2} \sum_ {i, j = 1} ^ {N} \sum_ {k = 1} ^ {K} a _ {i, j} \frac {(\mathbf {x} ^ {\mathsf {H}}) _ {k , i} x _ {j , k}}{\sqrt {d _ {i} ^ {\text {i n}} d _ {j} ^ {\text {o u t}}}} - \frac {1}{2} \sum_ {i, j = 1} ^ {N} \sum_ {k = 1} ^ {K} a _ {i, j} \frac {x _ {i , k} (\mathbf {x} ^ {\mathsf {H}}) _ {k , j}}{\sqrt {d _ {i} ^ {\text {i n}} d _ {j} ^ {\text {o u t}}}} \\ = \sum_ {i = 1} ^ {N} \sum_ {k = 1} ^ {K} \left| x _ {i, k} \right| ^ {2} - \frac {1}{2} \sum_ {i, j = 1} ^ {N} \sum_ {k = 1} ^ {K} a _ {i, j} \frac {\left(\mathbf {x} ^ {\mathsf {H}}\right) _ {k , i} x _ {j , k}}{\sqrt {d _ {i} ^ {\text {i n}} d _ {j} ^ {\text {o u t}}}} - \frac {1}{2} \left(\sum_ {i, j = 1} ^ {N} \sum_ {k = 1} ^ {K} a _ {i, j} \frac {\left(\mathbf {x} ^ {\mathsf {H}}\right) _ {k , i} x _ {j , k}}{\sqrt {d _ {i} ^ {\text {i n}} d _ {j} ^ {\text {o u t}}}}\right) ^ {*} \\ = \Re \left(\sum_ {i = 1} ^ {N} \sum_ {k = 1} ^ {K} \left| x _ {i, k} \right| ^ {2} - \sum_ {i, j = 1} ^ {N} \sum_ {k = 1} ^ {K} a _ {i, j} \frac {x _ {i , k} ^ {*} x _ {j , k}}{\sqrt {a _ {i} ^ {\ln} d _ {j} ^ {\mathrm {o u t}}}}\right) \\ = \Re \left(\operatorname {t r a c e} \left(\mathbf {x} ^ {\mathsf {H}} (\mathbf {I} - \mathbf {L}) \mathbf {x}\right)\right). \\ \end{array}
$$
The last claim can be proved as follows. For simplicity, suppose  $\mathbf{x} \in \mathbb{R}^N$ . The “ $\Leftarrow$ ” is clear since one can choose  $\mathbf{x}$  to be  $\mathbf{k}$ . To prove the “ $\Rightarrow$ ”, we reason by contradiction. Suppose there exists a  $\mathbf{x} \neq 0$  such that  $\mathcal{E}(\mathbf{x}) = 0$  and the underlying graph is not weakly connected, i.e.,
$$
\forall \tilde {\mathbf {x}} \neq \mathbf {0}, \left| \sum_ {j = 1} ^ {N} a _ {i, j} \left(\frac {\tilde {x} _ {j}}{\sqrt {d _ {j} ^ {\text {o u t}}}} - \frac {\tilde {x} _ {i}}{\sqrt {d _ {i} ^ {\text {i n}}}}\right) \right| > 0, \forall i \in \{1, \dots , N \},
$$
Then, since  $\mathbf{x}\neq 0$