SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathbb {P} \left(\left| \frac {1}{n} \sum_ {i = 1} ^ {n} \epsilon_ {i} - \frac {1}{2} \right| \geq \delta / 2\right) \leq 2 \exp (- n \delta^ {2} / 2).
$$
Notice that  $\sum_{i=1}^{n} \epsilon_i = |C_1|$  and  $|C_0| + |C_1| = n$ , we can conclude that for any  $\delta > 0$ , the probability that the number of nodes in each class concentrates will satisfy
$$
\mathbb {P} \left(\frac {| C _ {0} |}{n}, \frac {| C _ {1} |}{n} \in \left[ \frac {1}{2} - \delta , \frac {1}{2} + \delta \right]\right) \geq 1 - C \exp (- c n \delta^ {2}),
$$
for some constant  $C, c > 0$ .
We now turn to the events  $\{\mathbf{I}_{2,i}(\delta')\}_{i\in [n]}$ . Notice that the node degrees are sums of independent Poisson random variables. It is known that sums of independent Poisson random variables will be another Poisson random variable. Hence, conditioning on  $\theta = (\theta_{i})_{i\in [n]}$ , for each node  $i\in [n]$ , we have
$$
D _ {i i} \sim 1 + \operatorname {P o i} (\frac {n - 1}{2} (p + q) \theta_ {i}),
$$
where  $D_{ii}$  is the degree of node  $i$ , and
$$
\mathbb {E} [ D _ {i i} ] = 1 + \frac {n - 1}{2} (p + q) \theta_ {i}.
$$
To prove that  $\mathbf{I}_{2,i}(\delta')$  will occur with high probability for each  $i \in [n]$ , we introduce the following result [38, Corollary 2.3.7] whose proof can be found in the referred literature:
Lemma C.5 (Corollary 2.3.7 [38]). If  $X \sim \text{Poi}(\lambda)$ , then for  $t \in (0, \lambda]$ , we have
$$
\mathbb {P} (| X - \lambda | \geq t) \leq 2 \exp \left(- \frac {c t ^ {2}}{\lambda}\right).
$$
Here, we can let  $t = \delta' \lambda$  where  $\delta' \in (0,1]$  and get a tail bound as follows:
$$
\mathbb {P} (| D _ {i i} - \mathbb {E} [ D _ {i i} ] | \geq \delta^ {\prime} \mathbb {E} [ D _ {i i} ]) \leq 2 \exp (- \frac {c (\delta^ {\prime} \mathbb {E} [ D _ {i i} ]) ^ {2}}{\mathbb {E} [ D _ {i i} ]}) = 2 \exp (- c \mathbb {E} [ D _ {i i} ] \delta^ {\prime 2}).
$$
It follows that for each  $i\in [n]$  and any  $\delta^{\prime}\in (0,1]$ , we have
$$
\mathbb {P} \left(\frac {D _ {i i}}{n} \in \left[ \frac {1}{2} (p + q) (1 - \delta^ {\prime}) \theta_ {i}, \frac {1}{2} (p + q) (1 + \delta^ {\prime}) \theta_ {i} \right] ^ {\mathrm {c}}\right) \leq C \exp (- c n (p + q) \theta_ {i} \delta^ {\prime 2}),
$$
for some  $C, c > 0$ .
We next consider the events  $\{\mathbf{I}_{3,i}(\delta')\}_{i\in [n]}$ . Observe that for each node  $i$ , we can decompose node degree as  $D_{ii} = D_{ii}^{\mathrm{intra}} + D_{ii}^{\mathrm{inter}}$ , where
$$
D _ {i i} ^ {\text {i n t r a}} = \sum_ {j \in \mathcal {N} (i)} \mathbb {1} \left\{\epsilon_ {j} = \epsilon_ {i} \right\},
$$
$$
D _ {i i} ^ {\text {i n t e r}} = \sum_ {j \in \mathcal {N} (i)} \mathbb {1} \left\{\epsilon_ {j} \neq \epsilon_ {i} \right\}.
$$
Obviously,  $D_{ii}^{\mathrm{intra}} = |C_{\epsilon_i}\cap \mathcal{N}_i|$  and  $D_{ii}^{\mathrm{inter}} = |C_{1 - \epsilon_i}\cap \mathcal{N}_i|$  will concentrate around  $\frac{np\theta_i}{2}$  and  $\frac{np\theta_i}{2}$ , correspondingly. And given the tail bound for  $\{\mathbf{I}_{2,i}(\delta ')\}_{i\in [n]}$ , by a similar argument, we have for each  $i\in [n]$  and any  $\delta^{\prime}\in (0,1]$ ,
$$
\mathbb {P} \left(\mathbf {I} _ {3, i} \left(\delta^ {\prime}\right)\right) \geq 1 - C \exp \left(- c n (p + q) \theta_ {i} \delta^ {\prime 2}\right),
$$
Define the union event  $U(\delta, \delta') = \mathbf{I}_1(\delta) \bigcap_{i \in \mathcal{V}_{\alpha}} \mathbf{I}_{2,i}(\delta') \bigcap_{i \in \mathcal{V}_{\alpha}} \mathbf{I}_{3,i}(\delta')$ . Recall that  $\forall i \in \mathcal{V}_{\alpha}$ , we have  $\theta_i \geq \alpha$ . Thus, we can then choose  $\delta = n^{-1/2 + \epsilon}$  and  $\delta' = (\alpha \log n)^{-1/2 + \epsilon}$ . Since  $p, q = \omega\left(\frac{\log^2 n}{n}\right)$  from Assumption 4.3, by a simple union bound, we have for  $\epsilon > 0$  small enough, for any  $c > 0$  there is  $C > 0$  such that
$$
\mathbb {P} \left(U \left(n ^ {- 1 / 2 + \epsilon}, (\alpha \log n) ^ {- 1 / 2 + \epsilon}\right)\right) \geq 1 - \frac {C}{n ^ {c}}. \tag {3}
$$
Finally, we establish the lower bound for  $\alpha$  indicated in the statement, which is  $\frac{1}{\log n}$ . The reason why we need this lower bound is that if  $\alpha$  is too small, then the subgroups:  $C_0(\alpha), C_1(\alpha)$  will be too sparse that their member nodes' degree is too small to assure the concentration inequalities.
By the definition of the event:  $U(\delta, \delta')$  and union bound, we have
$$
\mathbb {P} \left(\mathbf {I} _ {1} (\delta)\right) \leq C \exp (- c n \delta^ {2})
$$
$$
\leq C \exp (- c n ^ {2 \epsilon})
$$
$$
(p l u g \quad i n \delta = n ^ {- 1 / 2 + \epsilon})
$$
$$
\leq C / n ^ {c}
$$
$$
(\text {i f w e c h o o s e} \epsilon \geq \frac {\log \log n}{2 \log n} > 0),
$$
(by Assumption 4.3)
$$
\begin{array}{l} \mathbb {P} \left(\bigcap_ {i \in \mathcal {V} _ {\alpha}} \mathbf {I} _ {2, i} \left(\delta^ {\prime}\right) \bigcap_ {i \in \mathcal {V} _ {\alpha}} \mathbf {I} _ {3, i} \left(\delta^ {\prime}\right)\right) \leq n \cdot C \exp \left(- c n (p + q) \alpha \delta^ {\prime 2}\right) \\ \leq n \cdot C \exp \left(- c \log^ {2} n \cdot \alpha \delta^ {\prime 2}\right) \\ = C \exp (\log n - c \log^ {2} n \cdot \alpha (\alpha \log n) ^ {- 1 + 2 \epsilon}) \\ (p l u g i n \delta^ {\prime} = (\alpha \log n) ^ {- 1 / 2 + \epsilon}) \\ = C \exp (\log n - c \log n \cdot (\alpha \log n) ^ {2 \epsilon}) \\ = C \exp (\log n \cdot (1 - c \cdot (\alpha \log n) ^ {2 \epsilon})) \\ = C n ^ {1 - c \cdot (\alpha \log n) ^ {2 \epsilon}}. \\ \end{array}
$$
We want to ensure the last term stays in  $O(1 / n^{\beta})$  for some  $\beta > 0$ . Notice that if  $\alpha < \log^{-1}n$ , we cannot find suitable  $\epsilon > 0$  for some small  $c$  to satisfy  $1 - c \cdot (\alpha \log n)^{2\epsilon} < 0$ . Hence, we can conclude that a natural lower bound for  $\alpha$  should be  $\frac{1}{\log n}$ , i.e.,  $\alpha > \frac{1}{\log n}$ .
Thus, combining Equation 3 with this fact, we complete the proof.
Next, in Lemma C.6, we claim that given the adjacency matrix  $\mathbf{A}$ , class memberships  $(\epsilon_i)_{i\in [n]}$ , degree-corrected factors  $(\theta_{i})_{i\in [n]}$  and a pre-defined threshold  $\alpha >0$ , then with high probability, the convoluted node features  $\tilde{\mathbf{x}}_i\approx \frac{p\boldsymbol{\mu} + q\boldsymbol{\nu}}{p + q}$  for  $i\in C_0(\alpha)$  and  $\tilde{\mathbf{x}}_i\approx \frac{q\boldsymbol{\mu} + p\boldsymbol{\nu}}{p + q}$  for  $i\in C_1(\alpha)$ .
Lemma C.6 (Convoluted Feature Concentration). Given  $\alpha \in (\frac{1}{\log n}, n]$ ,  $C_0(\alpha), C_1(\alpha)$  defined by Definition 4.1, and  $\mathcal{V}_{\alpha} = C_0(\alpha) \cup C_1(\alpha)$ . Conditionally on  $\mathbf{A}$ ,  $(\epsilon_i)_{i \in [n]}$  and  $(\theta_i)_{i \in [n]}$ , we have that for any  $c > 0$  and some  $C > 0$ , with probability at least  $1 - \frac{C}{n^c}$ , for every node  $i \in \mathcal{V}_{\alpha}$  and any unit vector  $\mathbf{w}$ ,
$$
\left| \left\langle \tilde {\mathbf {x}} _ {i} - \frac {p \boldsymbol {\mu} + q \boldsymbol {\nu}}{p + q}, \mathbf {w} \right\rangle (1 + o (1)) \right| = O \left(\sqrt {\frac {\log n}{d n (p + q) \alpha}}\right) f o r i \in C _ {0} (\alpha),
$$
$$
\left| \left\langle \tilde {\mathbf {x}} _ {i} - \frac {q \boldsymbol {\mu} + p \boldsymbol {\nu}}{p + q}, \mathbf {w} \right\rangle (1 + o (1)) \right| = O \left(\sqrt {\frac {\log n}{d n (p + q) \alpha}}\right) f o r i \in C _ {1} (\alpha).
$$
Proof. Since  $(\mathbf{X}, \mathbf{A})$  is sampled from DC-CSBM  $(\pmb{\mu}, \pmb{\nu}, p, q, \theta)$ , when conditioning on  $(\epsilon_{i})_{i \in [n]}$ , we have node  $i$ 's node feature  $\mathbf{x}_{i} \sim \mathcal{N}(m_{i}, \frac{1}{d} I)$  where  $\pmb{m}_{i} = \pmb{\mu}$  if  $i \in C_{0}$  and  $\pmb{m}_{i} = \pmb{\nu}$  if  $i \in C_{1}$ . We can also write
$$
\mathbf {x} _ {i} = (1 - \epsilon_ {i}) \boldsymbol {\mu} + \epsilon_ {i} \boldsymbol {\nu} + \frac {g _ {i}}{\sqrt {d}},
$$
where  $g_{i}\sim \mathcal{N}(\mathbf{0},\mathbf{I})$  is standard normal vector.
On the other hand, conditioning on the adjacency matrix  $\mathbf{A}$  and class memberships  $\epsilon = (\epsilon_{i})_{i\in [n]}$ , the mean of the convoluted feature of node  $i$  can be written as
$$
m (i) = \mathbb {E} [ \tilde {\mathbf {x}} _ {i} | \mathbf {A}, \epsilon ] = \frac {1}{D _ {i i}} \sum_ {j \in [ n ]} \tilde {\mathbf {A}} _ {i j} \boldsymbol {m} _ {j},
$$
by the definition of the graph convolution operation  $(\tilde{\mathbf{x}}_i = [\mathbf{D}^{-1}\tilde{\mathbf{A}}\mathbf{X}]_i)$ .
Thus, for any unit vector  $\mathbf{w}$ , we have
$$
\tilde {\mathbf {x}} _ {i} \cdot \mathbf {w} = \frac {1}{D _ {i i}} \sum_ {j \in [ n ]} \tilde {\mathbf {A}} _ {i j} \langle \mathbf {x} _ {j}, \mathbf {w} \rangle = \langle m (i), \mathbf {w} \rangle + \frac {1}{D _ {i i} \sqrt {d}} \sum_ {j \in [ n ]} \tilde {\mathbf {A}} _ {i j} \cdot \langle g _ {j}, \mathbf {w} \rangle . \tag {4}
$$
Let us define  $F_{i} = \frac{1}{D_{ii}\sqrt{d}}\sum_{j\in [n]}\tilde{\mathbf{A}}_{ij}\cdot \langle g_j,\mathbf{w}\rangle$  and observe that  $\langle g_j,\mathbf{w}\rangle$  is a standard Gaussian random variable for all  $j\in [n]$ . Thus, we have that  $F_{i}\sim \mathcal{N}(0,\frac{1}{dD_{ii}})$ , conditioning on the adjacency matrix  $\mathbf{A}$ . Now we introduce Borell's inequality [2] to give a high-probability bound of  $|F_{i}|$  for all  $i\in \mathcal{V}_{\alpha}$ .
Lemma C.7 (Borell's Inequality, Theorem 2.1.1 in Adler et al. [2]). Let  $F_{i}\sim \mathcal{N}(0,\sigma_{F_{i}}^{2})$  for each  $i\in \mathcal{V}_{\alpha}$ . Then for any  $K > 0$ , we have
$$
\mathbb {P} \big (\max _ {i \in \mathcal {V} _ {\alpha}} F _ {i} - \mathbb {E} [ \max _ {i \in \mathcal {V} _ {\alpha}} F _ {i} ] > K \big) \leq \exp (- \frac {K ^ {2}}{2 \max _ {i \in \mathcal {V} _ {\alpha}} \sigma_ {F _ {i}} ^ {2}}).
$$
We can further define the event  $Q_{\alpha} = Q_{\alpha}(t) = \{\max_{i\in \mathcal{V}_{\alpha}}|F_i|\leq t\}$ . Observe that
$$
\begin{array}{l} \mathbb {P} \left(Q _ {\alpha} ^ {\mathrm {c}} | \mathbf {A}\right) = \mathbb {P} \left(\max  _ {i \in \mathcal {V} _ {\alpha}} \left| F _ {i} \right| > t | \mathbf {A}\right) \\ \leq 2 \mathbb {P} (\max  _ {i \in \mathcal {V} _ {\alpha}} F _ {i} > t | \mathbf {A}) \\ = 2 \mathbb {P} \left(\max  _ {i \in \mathcal {V} _ {\alpha}} F _ {i} - \mathbb {E} \left[ \max  _ {i \in \mathcal {V} _ {\alpha}} F _ {i} \right] > t - \mathbb {E} \left[ \max  _ {i \in \mathcal {V} _ {\alpha}} F _ {i} \right] | \mathbf {A}\right). \\ \end{array}
$$
If we let the union event  $U \coloneqq U(n^{-1/2 + \epsilon}, (\alpha \log n)^{-1/2 + \epsilon})$  defined the same as in Lemma C.4, then by Lemma C.7 and the definition of  $\mathcal{V}_{\alpha}$ ,
$$
\begin{array}{l} \mathbb {P} \left(Q _ {\alpha} ^ {\mathbf {c}}\right) \leq \mathbb {P} \left(U \cap Q _ {\alpha} ^ {\mathbf {c}}\right) + \mathbb {P} \left(U ^ {\mathbf {c}}\right) \\ \leq 2 \exp (- c ^ {\prime} (t - \mathbb {E} [ \max  _ {i \in \mathcal {V} _ {\alpha}} F _ {i} ]) ^ {2} \cdot d n (p + q) \alpha) + \frac {1}{n ^ {c}}, \\ \end{array}
$$
for any  $c > 0$  and some  $c' > 0$ .
By the definition of event  $U(\delta, \delta')$ , we can derive the upper bound of  $\max_{i \in \mathcal{V}_{\alpha}} \sigma_{F_i}$  as follows:
$$
\max _ {i \in \mathcal {V} _ {\alpha}} \sigma_ {F _ {i}} = \max _ {i \in \mathcal {V} _ {\alpha}} \sqrt {\frac {1}{d D _ {i i}}} \leq \max _ {i \in \mathcal {V} _ {\alpha}} \sqrt {\frac {1}{d \frac {n}{2} (p + q) (1 - \delta^ {\prime}) \theta_ {i}}} \leq \sqrt {\frac {2}{d n (p + q) (1 - \delta^ {\prime}) \alpha}}.
$$
Since  $\delta^\prime$  is determined, we have for some constant  $k^{\prime},k > 0$
$$
\mathbb {E} \left[ \max  _ {i \in \mathcal {V} _ {\alpha}} F _ {i} \right] \leq k ^ {\prime} \sqrt {\log n} \max  _ {i \in \mathcal {V} _ {\alpha}} \sigma_ {F _ {i}} \leq k \sqrt {\frac {\log n}{d n (p + q) \alpha}}.
$$
By choosing  $t = C' \sqrt{\frac{\log n}{dn(p + q)\alpha}}$  for some large constant  $C' > k > 0$ , we can obtain
$$
t - \mathbb {E} [ \max  _ {i \in \mathcal {V} _ {\alpha}} F _ {i} ] \geq C ^ {\prime} \sqrt {\frac {\log n}{d n (p + q) \alpha}} - k \sqrt {\frac {\log n}{d n (p + q) \alpha}} > 0.
$$
$$
\begin{array}{l} \mathbb {P} (U \cap Q _ {\alpha}) \geq 1 - \mathbb {P} (U ^ {\mathbf {c}}) - \mathbb {P} (Q _ {\alpha} ^ {\mathbf {c}}) \\ \geq 1 - \frac {2}{n ^ {c}} - 2 \exp (- c ^ {\prime} (t - \mathbb {E} [ \max _ {i \in \mathcal {V} _ {\alpha}} F _ {i} ]) ^ {2} d n (p + q) \alpha) \\ \geq 1 - \frac {2}{n ^ {c}} - \frac {2}{n ^ {c ^ {\prime} (C ^ {\prime} - k) ^ {2}}}. \\ \end{array}
$$
Recall that when conditioning on the event  $U$ , we have
$$
m (i) = \frac {p \boldsymbol {\mu} + q \boldsymbol {\nu}}{p + q} (1 + o (1)) \quad \text {f o r} i \in C _ {0} (\alpha), \tag {5}
$$
$$
m (i) = \frac {q \boldsymbol {\mu} + p \boldsymbol {\nu}}{p + q} (1 + o (1)) \quad \text {f o r} i \in C _ {1} (\alpha). \tag {6}
$$
Thus, on the event  $U \cap Q_{\alpha}$ , we have for each node  $i \in \mathcal{V}_{\alpha}$
$$
| \langle \tilde {\mathbf {x}} _ {i} - m (i), \mathbf {w} \rangle | = O \left(\sqrt {\frac {\log n}{d n (p + q) \alpha}}\right),
$$
Now we are ready to prove Theorem 4.4.
Proof. Recall the definition of linear separability, we need to find some unit vector  $\mathbf{v} \in \mathbb{R}^d$  and  $b \in \mathbb{R}$  such that
$$
\langle \tilde {\mathbf {x}} _ {i}, \mathbf {v} \rangle + b <   0 \quad \text {f o r} i \in C _ {0} (\alpha),
$$
$$
\langle \tilde {\mathbf {x}} _ {i}, \mathbf {v} \rangle + b > 0 \quad \text {f o r} i \in C _ {1} (\alpha).
$$
We fix  $\tilde{\mathbf{v}} = \frac{1}{2\gamma} (\pmb {\nu} - \pmb {\mu})$  and  $\tilde{b} = -\frac{\langle\pmb{\mu} + \pmb{\nu},\tilde{\mathbf{v}}\rangle}{2}$ , where  $\gamma = \frac{1}{2}\| \pmb {\mu} - \pmb {\nu}\| _2$ . By Assumption 4.3, Lemma C.4 and C.6, with probability at least  $1 - O(n^{-c})$  for any  $c > 0$ , for all  $i\in C_0(\alpha)$ , we have
$$
\begin{array}{l} \langle \tilde {\mathbf {x}} _ {i}, \tilde {\mathbf {v}} \rangle + \tilde {b} \\ = \frac {\left\langle p \boldsymbol {\mu} + q \boldsymbol {\nu} , \tilde {\mathbf {v}} \right\rangle}{p + q} (1 + o (1)) + O \left(\sqrt {\frac {\log n}{d n (p + q) \alpha}}\right) + \tilde {b} \quad (\text {B y}) \\ = \left\langle \frac {(p - q) (\boldsymbol {\mu} - \boldsymbol {\nu})}{2 (p + q)}, \tilde {\mathbf {v}} \right\rangle (1 + o (1)) + O \left(\sqrt {\frac {\log n}{d n (p + q) \alpha}}\right) \\ = - \gamma \Gamma (p, q) (1 + o (1)) + O \left(\sqrt {\frac {\log n}{d n (p + q) \alpha}}\right) \\ = - \gamma \Gamma (p, q) (1 + o (1)) + o (\gamma) \quad (\alpha \in \omega \left(\frac {\log n}{d n (p + q) \gamma^ {2}}\right)) \\ <   0. \quad (\text {b y}) \\ \end{array}
$$
Similarly, for all  $i\in C_1(\alpha)$  , we have
$$
\begin{array}{l} \langle \tilde {\mathbf {x}} _ {i}, \tilde {\mathbf {v}} \rangle + \tilde {b} \\ = \frac {\left\langle q \boldsymbol {\mu} + p \boldsymbol {\nu} , \tilde {\mathbf {v}} \right\rangle}{p + q} (1 + o (1)) + O \left(\sqrt {\frac {\log n}{d n (p + q) \alpha}}\right) + \tilde {b} \quad (\text {B y}) \\ = \left\langle \frac {(p - q) (\boldsymbol {\nu} - \boldsymbol {\mu})}{2 (p + q)}, \tilde {\mathbf {v}} \right\rangle (1 + o (1)) + O \left(\sqrt {\frac {\log n}{d n (p + q) \alpha}}\right) \\ = \gamma \Gamma (p, q) (1 + o (1)) + O \left(\sqrt {\frac {\log n}{d n (p + q) \alpha}}\right) \\ = \gamma \Gamma (p, q) (1 + o (1)) + o (\gamma) \quad (\alpha \in \omega \left(\frac {\log n}{d n (p + q) \gamma^ {2}}\right)) \\ > 0. \quad (\text {b y}) \\ \end{array}
$$
The above two inequalities imply the linear separability of  $\{\tilde{\mathbf{x}}_i,i\in \mathcal{V}_\alpha \}$ , which completes the proof.
Table 3: Description of parameters used in the controlled experiments.
Parameter Name	Description
n	Number of vertices in the graph.
cluster size slope	the slope of cluster sizes when ordered by size.
feature dimension	the number of dimensions of node features.
feature center distance	distance between feature cluster centers.
p/q ratio	the ratio of intra-class edge probability to inter-class edge probability.
average degree	the average expected degrees of nodes.
power exponent	the value of power-law exponent used to generate expected node degrees.
feature cluster variance	variance of feature clusters around their centers.
Table 4: Remaining parameters used in experiments. One of the parameters is manipulated to generate synthetic datasets with varying data properties indicated in the first column.
Experiments	p/q ratio	Average Degree	Power Exponent	Feature Cluster Variance
Gini-Degree	3	20	[1.5, 2, 2.5, 3, 5]	0.25
Average Degree	3	[10, 20, 30, 40, 50]	2	0.25
Edge Homogeneity	[1,2,3,5,10]	20	2	0.1
In-Feature Similarity	2	20	2	[2, 1, 0.5, 0.2, 0.1]
Feature Angular SNR	2	20	2	[2, 1, 0.5, 0.2, 0.1]
D Controlled Experiments of Identified Salient Factors
From Section 3.3, we discover six prominent dataset properties that correlate with some or all of the GNN models' performance. In Section 5, we have presented controlled experiments for Gini-Degree to verify its relationship to GNNs' performance (Table 2). In this section, we further conduct controlled experiments for all the remaining identified salient factors, except for Pseudo Diameter, which is hard to control via manipulating explicit parameters provided by GraphWorld.
Across all experiments, we fix the number of nodes  $n = 5000$ , cluster size slope as 0.0, the number of clusters as 4, feature dimension as 16, and feature center distance as 0.05. For each of the experiments, we will keep most of the remaining GraphWorld parameters the same and only vary one of the parameters. The remaining parameters that we will manipulate are the  $p / q$  ratio, average degree, power exponent, and feature cluster variance. We give a short description of all the parameters in Table 3. For completeness, we summarize the value of the remaining parameters used in all four experiments in Table 4.
Table 5, 6, and 7 show the results of the four controlled experiments, correspondingly. Note that varying feature cluster variance can manipulate In-Feature Similarity and Feature Angular SNR simultaneously (Table 7). In general, all the results closely follow the regression results indicated in Table 1 and the discussion in Section 3.3.
Table 5: Controlled experiment results for varying Average Degree. Standard deviations are derived from 5 independent runs. The performances of all models except for GAT, MixHop, and MLP have an evident positive correlation with Average Degree.
Average Degree	GCN	GAT	GraphSAGE	MoNet	MixHop	LINKX	MLP
10	0.71±0.018	0.67±0.009	0.725±0.002	0.556±0.024	0.696±0.001	0.693±0.006	0.632±0.004
20	0.823±0.001	0.734±0.013	0.797±0.006	0.593±0.012	0.806±0.003	0.825±0.001	0.54±0.024
30	0.839±0.005	0.722±0.017	0.801±0.002	0.761±0.005	0.756±0.002	0.852±0.003	0.653±0.004
40	0.876±0.003	0.742±0.006	0.825±0.001	0.795±0.002	0.794±0.003	0.876±0.002	0.648±0.003
50	0.9±0.004	0.734±0.019	0.86±0.002	0.814±0.003	0.788±0.011	0.89±0.005	0.651±0.002
Table 6: Controlled experiment results for varying Edge Homogeneity. Standard deviations are derived from 5 independent runs. The performances of all models except for MixHop and MLP have an evident positive correlation with Edge Homogeneity.
Edge Homogeneity	GCN	GAT	GraphSAGE	MoNet	MixHop	LINKX	MLP
0.249	0.737±0.004	0.565±0.009	0.732±0.005	0.515±0.004	0.836±0.002	0.823±0.005	0.744±0.033
0.375	0.873±0.002	0.825±0.011	0.847±0.003	0.57±0.009	0.945±0.002	0.93±0.003	0.93±0.001
0.452	0.917±0.002	0.887±0.004	0.896±0.007	0.598±0.005	0.947±0.001	0.949±0.002	0.784±0.09
0.559	0.925±0.002	0.89±0.004	0.925±0.004	0.678±0.003	0.913±0.005	0.943±0.005	0.9±0.004
0.702	0.946±0.004	0.933±0.004	0.953±0.001	0.802±0.003	0.942±0.001	0.959±0.001	0.865±0.004
Table 7: Controlled experiment results for varying In-Feature Similarity / Feature Angular SNR. Standard deviations are derived from 5 independent runs. The performances of all models except for MoNet have an evident positive correlation with In-Feature Similarity / Feature Angular SNR.
In-Feature Similarity	Feature Angular SNR	GCN	GAT	GraphSAGE	MoNet	MixHop	LINKX	MLP
0.506	1.009	0.478±0.016	0.412±0.016	0.446±0.005	0.562±0.021	0.433±0.001	0.598±0.002	0.402±0.002
0.516	1.022	0.563±0.004	0.47±0.006	0.517±0.008	0.615±0.002	0.531±0.003	0.661±0.004	0.47±0.001
0.527	1.039	0.717±0.008	0.507±0.006	0.6±0.006	0.555±0.021	0.621±0.007	0.737±0.001	0.486±0.003
0.582	1.101	0.784±0.011	0.599±0.014	0.74±0.01	0.533±0.01	0.848±0.001	0.854±0.001	0.611±0.003
0.602	1.154	0.887±0.006	0.791±0.004	0.825±0.006	0.627±0.004	0.924±0.004	0.913±0.006	0.915±0.002
E Robustness of the Sparse Regression Analysis
To demonstrate the robustness of the identified salient factors (defined in Section 3.3), we expand our sparse regression analysis to include five additional models that are more recent and popular. The models are TAGCN [10], GATv2 [5], SGC [40], APPNP [12] and GCNII [7]. For the additional experiments, we adopt the same configuration as in Appendix B. The updated analysis result is presented in Table 8.
We can observe that the widely influential factors and the narrowly influential factors all remain salient after incorporating the five additional models. The results show that our proposed multivariate regression analysis is robust with respect to our diverse choice of GNN models.
Table 8: The estimated coefficient matrix  $\mathbf{B}$  of the multivariate sparse regression analysis with five additional GNN models. These five models are indicated in blue. Each entry indicates the strength (magnitude) and direction (+, -) of the relationship between a graph data property and the performance of a GNN model. The six most salient data properties are indicated in bold. Notice that these factors are the same as the ones we presented in Section 3.3.
Graph Data Property	GCN	GAT	GraphSAGE	MoNet	MixHop	LINKX	MLP	TAGCN	GATv2	SGC	APPNP	GCNII
Edge Density	0	0	0	0	0	0.0279	0.0937	0	0	0	0	0
Average Degree	0.2136	0	0.098	0.1047	0	0.3362	0	0.173	0	0.4588	0	0
Pseudo Diameter	0	-0.3824	-0.1608	-0.0173	-0.4915	-0.3937	-0.6191	-0.2514	-0.1428	-0.0816	-0.401	-0.2962
Degree Assortativity	0	0	0	-0.0587	0	0	0	0	0	0	0	0
RSLCC	0.1014	0	0	0.0673	0	0.1312	0	0.0333	0	0	0	0
ACC	0	0	0	0	0	0	-0.0523	-0.1139	0.0276	0	0	0
Transitivity	0	-0.0458	0	-0.148	0	0.2168	0	0	0	-0.0795	0	-0.0315
Degeneracy	0	0	0	0	0	0	-0.1555	0	-0.0652	-0.3099	-0.0276	0
Gini-Degree	-0.4437	-0.2955	-0.3313	-0.292	-0.4269	-0.3681	-0.1993	-0.3838	-0.2043	-0.1907	-0.3021	-0.33
Edge Homogeneity	0.714	0.4197	0.7241	0.8108	0.6396	0.2017	0.4777	0.7147	0.7007	0.2817	0.7962	0.7184
In-Feature Similarity	0.3103	0.0926	0.1878	0.0989	0.4576	0.6406	0.2421	0.4394	0.0359	0	0	0.0255
Out-Feature Similarity	0	0	0	0	0	0	0	0	0	0	0	0
Feature Angular SNR	0.2492	0.0393	0.2455	0	0.2355	0.3564	0.3682	0.1354	0.3308	-0.0997	0.2733	0.359
Homophily Hat	0	0.4569	0	0	0	0	0	0	0	0.4795	0	0
Attribute Assortativity	0	0	0	0	0	0	0	0	0	0	0	0