SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\lambda_ {l _ {1}} (\mathbf {W}) \lambda_ {l _ {2}} (\mathbf {M}), \mathbf {e} _ {l _ {1}} \otimes \psi_ {l _ {2}} ^ {1} (\mathbf {M}), (\lambda_ {l _ {1}} (\mathbf {W})) ^ {- i + 1} \mathbf {e} _ {l _ {1}} \otimes \psi_ {l _ {2}} ^ {i} (\mathbf {M})
$$
for  $l_1 \in \{1, \ldots, K\}$ ,  $l_2 \in \{1, \ldots, m\}$  and  $i \in \{2, \ldots, k_l\}$ . Hence, by Lemma E.1, the solution of (14) is given by
$$
\operatorname {v e c} (\mathbf {x}) (t) = \sum_ {l _ {1} = 1} ^ {K} \sum_ {l _ {2} = 1} ^ {m} \exp \left(\lambda_ {l _ {2}} (\mathbf {M}) \lambda_ {l _ {1}} (\mathbf {W}) t\right) \sum_ {i = 1} ^ {k _ {l _ {2}}} c _ {l _ {1}, l _ {2}} ^ {i} \sum_ {j = 1} ^ {i} \frac {t ^ {i - j}}{(i - j) !} (\lambda_ {l _ {1}} (\mathbf {W})) ^ {1 - j} (\mathbf {e} _ {l _ {1}} \otimes \psi_ {l _ {2}} ^ {j} (\mathbf {L})) ,
$$
$$
\operatorname {v e c} \left(\mathbf {x} _ {0}\right) = \sum_ {l _ {1} = 1} ^ {K} \sum_ {l _ {2} = 1} ^ {m} \sum_ {i = 1} ^ {k _ {l _ {2}}} c _ {l _ {1}, l _ {2}} ^ {i} (\mathbf {I} \otimes \mathbf {P}) \tilde {\mathbf {P}} \left(\mathbf {e} _ {l _ {1}} \otimes \mathbf {e} _ {l _ {2}} ^ {i} (\mathbf {M})\right).
$$
E.1 Proof of Theorem 5.6
In the following, we reformulate and prove Theorem 5.6.
Corollary E.4. Let  $\mathcal{G}$  be a strongly connected directed graph with SNA  $\mathbf{L} \in \mathbb{R}^{N \times N}$ . Consider the initial value problem in (2) with diagonal channel mixing matrix  $\mathbf{W} \in \mathbb{R}^{K \times K}$  and  $\alpha = 1$ . Then, for almost all initial values  $\mathbf{x}_0 \in \mathbb{R}^{N \times K}$ , the solution to (2) is HFD if
$$
\lambda_ {K} (\mathbf {W}) \Re \lambda_ {1} (\mathbf {L}) <   \lambda_ {1} (\mathbf {W}) \lambda_ {N} (\mathbf {L})
$$
and  $\lambda_1(\mathbf{L})$  is the unique eigenvalue that minimizes the real part among all eigenvalues of  $\mathbf{L}$ . Otherwise, the solution is LFD.
Proof. Using the notation from Proposition E.3 and its proof, we can write the solution of the vectorized form of (2) as
$$
\operatorname {v e c} (\mathbf {x}) (t) = \sum_ {l _ {1} = 1} ^ {K} \sum_ {l _ {2} = 1} ^ {m} \exp \left(- \lambda_ {l _ {1}} (\mathbf {W}) \lambda_ {l _ {2}} (\mathbf {L}) t\right) \sum_ {i = 1} ^ {k _ {l _ {2}}} c _ {l _ {1}, l _ {2}} ^ {i} \sum_ {j = 1} ^ {i} \frac {t ^ {i - j}}{(i - j) !} (\lambda_ {l _ {1}} (\mathbf {W})) ^ {1 - j} (\mathbf {e} _ {l _ {1}} \otimes \psi_ {l _ {2}} ^ {j} (\mathbf {L})).
$$
As done extensively, we separate the terms corresponding to the frequency with minimal real part. This frequency dominates as the exponential converges faster than polynomials for  $t$  going to infinity. Consider the case  $\lambda_K(\mathbf{W})\Re (\lambda_1(\mathbf{L})) < \lambda_1(\mathbf{W})\Re (\lambda_N(\mathbf{L}))$ . As  $\lambda_{1}(\mathbf{L})$  is unique, the product  $\lambda_{K}(\mathbf{W})\Re (\lambda_{1}(\mathbf{L}))$  is the unique most negative frequency. Assume without loss of generality that  $\lambda_K(\mathbf{W})$  has multiplicity one. The argument does not change for higher multiplicities as the corresponding eigenvectors are orthogonal since  $\mathbf{W}$  is diagonal. Then,  $\lambda_{K}(\mathbf{W})\lambda_{1}(\mathbf{L})$  has multiplicity one, and we calculate  $\operatorname {vec}(\mathbf{x})(t)$  as
$$
\begin{array}{l} \sum_ {l _ {1} = 1} ^ {K} \sum_ {l _ {2} = 1} ^ {m} \exp \left(- \lambda_ {l _ {1}} (\mathbf {W}) \lambda_ {l _ {2}} (\mathbf {L}) t\right) \sum_ {i = 1} ^ {k _ {l _ {2}}} c _ {l _ {1}, l _ {2}} ^ {i} \sum_ {j = 1} ^ {i} \frac {t ^ {i - j}}{(i - j) !} \left(\lambda_ {l _ {1}} (\mathbf {W})\right) ^ {1 - j} \left(\mathbf {e} _ {l _ {1}} \otimes \psi_ {l _ {2}} ^ {j} (\mathbf {L})\right) \\ = c _ {K, 1} ^ {k _ {1}} \exp \left(- t \lambda_ {K} (\mathbf {W}) \lambda_ {1} (\mathbf {L})\right) \frac {t ^ {k _ {1} - 1}}{(k _ {1} - 1) !} (\mathbf {e} _ {K} \otimes \psi_ {1} ^ {1} (\mathbf {L})) \\ + c _ {K, 1} ^ {k _ {1}} \exp (- t \lambda_ {K} (\mathbf {W}) \lambda_ {1} (\mathbf {L})) \sum_ {j = 2} ^ {k _ {1}} \frac {t ^ {k _ {1} - j}}{(k _ {1} - j) !} (\lambda_ {K} (\mathbf {W})) ^ {1 - j} (\mathbf {e} _ {K} \otimes \psi_ {1} ^ {j} (\mathbf {L})) \\ + \exp \left(- t \lambda_ {K} (\mathbf {W}) \lambda_ {1} (\mathbf {L})\right) \sum_ {i = 1} ^ {k _ {1} - 1} c _ {K, 1} ^ {i} \sum_ {j = 1} ^ {i} \frac {t ^ {i - j}}{(i - j) !} \left(\lambda_ {K} (\mathbf {W})\right) ^ {1 - j} \left(\mathbf {e} _ {K} \otimes \psi_ {1} ^ {j} (\mathbf {L})\right) \\ + \sum_ {l _ {1} = 1} ^ {K} \sum_ {l _ {2} = 2} ^ {m} \exp \left(- \lambda_ {l _ {1}} (\mathbf {W}) \lambda_ {l _ {2}} (\mathbf {L}) t\right) \sum_ {i = 1} ^ {k _ {l _ {2}}} c _ {l _ {1}, l _ {2}} ^ {i} \sum_ {j = 1} ^ {i} \frac {t ^ {i - j}}{(i - j) !} \left(\lambda_ {l _ {1}} (\mathbf {W})\right) ^ {1 - j} \left(\mathbf {e} _ {l _ {1}} \otimes \psi_ {l _ {2}} ^ {j} (\mathbf {L})\right) \\ = \exp \left(- t \lambda_ {K} (\mathbf {W}) \lambda_ {1} (\mathbf {L})\right) t ^ {k _ {1} - 1} \left(c _ {K, 1} ^ {k _ {1}} \frac {1}{(k _ {1} - 1) !} \left(\mathbf {e} _ {K} \otimes \psi_ {1} ^ {1} (\mathbf {L})\right) \right. \\ + c _ {K, 1} ^ {k _ {1}} \sum_ {j = 2} ^ {k _ {1}} \frac {t ^ {1 - j}}{(k _ {1} - j) !} (\lambda_ {K} (\mathbf {W})) ^ {1 - j} (\mathbf {e} _ {K} \otimes \psi_ {1} ^ {j} (\mathbf {L})) \\ + \sum_ {i = 1} ^ {k _ {1} - 1} c _ {K, 1} ^ {i} \sum_ {j = 1} ^ {i} \frac {1}{(i - j) !} t ^ {i - j - k _ {1} + 1} \left(\lambda_ {K} (\mathbf {W})\right) ^ {1 - j} \left(\mathbf {e} _ {K} \otimes \psi_ {1} ^ {j} (\mathbf {L})\right) \\ + \sum_ {l _ {1} = 1} ^ {K} \sum_ {l _ {2} = 2} ^ {m} \exp \left(- t (\lambda_ {l _ {1}} (\mathbf {W}) \lambda_ {l _ {2}} (\mathbf {L}) - \lambda_ {K} (\mathbf {W}) \lambda_ {1} (\mathbf {L}))\right) \sum_ {i = 1} ^ {k _ {l _ {2}}} c _ {l _ {1}, l _ {2}} ^ {i} \\ \left. \cdot \sum_ {j = 1} ^ {i} \frac {t ^ {i - j - k _ {1} + 1}}{(i - j) !} \left(\lambda_ {l _ {1}} (\mathbf {W})\right) ^ {1 - j} \left(\mathbf {e} _ {l _ {1}} \otimes \psi_ {l _ {2}} ^ {j} (\mathbf {L})\right)\right). \\ \end{array}
$$
We can then write the normalized solution as
$$
\begin{array}{l} \left(\frac {c _ {K , 1} ^ {k _ {1}}}{(k _ {1} - 1) !} (\mathbf {e} _ {K} \otimes \psi_ {1} ^ {1} (\mathbf {L})) + c _ {K, 1} ^ {k _ {1}} \sum_ {j = 2} ^ {k _ {1}} \frac {t ^ {1 - j}}{(k _ {1} - j) !} (\lambda_ {K} (\mathbf {W})) ^ {1 - j} (\mathbf {e} _ {K} \otimes \psi_ {1} ^ {j} (\mathbf {L})) \right. \\ + \sum_ {i = 1} ^ {k _ {1} - 1} c _ {K, 1} ^ {i} \sum_ {j = 1} ^ {i} \frac {t ^ {i - j - k _ {1} + 1}}{(i - j) !} (\lambda_ {K} (\mathbf {W})) ^ {1 - j} (\mathbf {e} _ {K} \otimes \psi_ {1} ^ {j} (\mathbf {L})) \\ \left. + \sum_ {l _ {1} = 1} ^ {K} \sum_ {l _ {2} = 2} ^ {m} e ^ {- t (\lambda_ {K} (\mathbf {W}) \lambda_ {l _ {2}} (\mathbf {L}) - \lambda_ {K} (\mathbf {W}) \lambda_ {1} (\mathbf {L}))} \sum_ {i = 1} ^ {k _ {l _ {2}}} c _ {l _ {1}, l _ {2}} ^ {i} \sum_ {j = 1} ^ {i} \frac {t ^ {i - j - k _ {1}}}{(i - j) !} (\lambda_ {l _ {1}} (\mathbf {W})) ^ {1 - j} (\mathbf {e} _ {l _ {1}} \otimes \psi_ {l _ {2}} ^ {j} (\mathbf {L}))\right) \\ \cdot \left\| \frac {c _ {K , 1} ^ {k _ {1}}}{(k _ {1} - 1) !} \left(\mathbf {e} _ {K} \otimes \psi_ {1} ^ {1} (\mathbf {L})\right) + c _ {K, 1} ^ {k _ {1}} \sum_ {j = 2} ^ {k _ {1}} \frac {t ^ {1 - j}}{(k _ {1} - j) !} \left(\lambda_ {K} (\mathbf {W})\right) ^ {1 - j} \left(\mathbf {e} _ {K} \otimes \psi_ {1} ^ {j} (\mathbf {L})\right) \right. \\ + \sum_ {i = 1} ^ {k _ {1} - 1} c _ {K, 1} ^ {i} \sum_ {j = 1} ^ {i} \frac {t ^ {i - j - k _ {1} + 1}}{(i - j) !} (\lambda_ {l _ {1}} (\mathbf {W})) ^ {1 - j} (\mathbf {e} _ {K} \otimes \psi_ {1} ^ {j} (\mathbf {L})) \\ + \sum_ {l _ {1} = 1} ^ {K} \sum_ {l _ {2} = 2} ^ {m} \exp \left(- t (\lambda_ {l _ {1}} (\mathbf {W}) \lambda_ {l _ {2}} (\mathbf {L}) - \lambda_ {K} (\mathbf {W}) \lambda_ {1} (\mathbf {L}))\right) \\ \cdot \sum_ {i = 1} ^ {k _ {l _ {2}}} c _ {l _ {1}, l _ {2}} ^ {i} \sum_ {j = 1} ^ {i} \frac {t ^ {i - j - k _ {1}}}{(i - j) !} (\lambda_ {l _ {1}} (\mathbf {W})) ^ {1 - j} (\mathbf {e} _ {l _ {1}} \otimes \psi_ {l _ {2}} ^ {j} (\mathbf {L})) \Bigg \| _ {2} ^ {- 1}. \\ \end{array}
$$
All summands, except the first, converge to zero for  $t$  going to infinity. Hence,
$$
\frac {\operatorname {v e c} (\mathbf {x}) (t)}{\| \operatorname {v e c} (\mathbf {x}) (t) \| _ {2}} \xrightarrow {t \to \infty} \left\| \frac {c _ {K , 1} ^ {k _ {1}}}{(k _ {1} - 1) !} \left(\mathbf {e} _ {K} \otimes \psi_ {1} ^ {1} (\mathbf {L})\right) \right\| _ {2} ^ {- 1} \left(\frac {c _ {K , 1} ^ {k _ {1}}}{(k _ {1} - 1) !} \left(\mathbf {e} _ {K} \otimes \psi_ {1} ^ {1} (\mathbf {L})\right)\right).
$$
We apply Lemma D.2 to finish the proof for the HFD case. Note that  $\psi_1^1 (\mathbf{L})$  is an eigenvector corresponding to  $\lambda_{1}(\mathbf{L})$ . The LFD case is equivalent. By Perron-Frobenius for irreducible nonnegative matrices, there is no other eigenvalue with the same real part as  $1 - \lambda_{N}(\mathbf{L}) = \lambda_{1}(\mathbf{I} - \mathbf{L})$ .
Remark E.5. If the hypotheses are met, the convergence result also holds for  $\mathbf{L}^{\alpha}$ . With the same reasoning, we can prove that the normalized solution converges to the eigenvector corresponding to the eigenvalue of  $\mathbf{L}^{\alpha}$  with minimal real part. It suffices to consider the eigenvalues and generalized eigenvectors of  $\mathbf{L}^{\alpha}$ . However, we do not know the relationship between the singular values of  $\mathbf{L}^{\alpha}$ , where we defined the fractional Laplacian, and the eigenvalues of  $\mathbf{L}$ . Hence, it is much more challenging to draw conclusions on the Dirichlet energy.
E.2 Explicit Euler
In this subsection, we show that the convergence properties of the Dirichlet energy from Theorem 5.6 are also satisfied when (2) is approximated via an explicit Euler scheme.
As noted in (11), the vectorized solution to (2) can be written as
$$
\operatorname {v e c} (\mathbf {x}) (n h) = (\mathbf {I} - h (\mathbf {W} \otimes \mathbf {L})) ^ {n} \operatorname {v e c} (\mathbf {x} _ {0}),
$$
when  $\alpha = 1$ . We thus aim to analyze the Jordan decomposition of  $\mathbf{L}^n$  for  $\mathbf{L} \in \mathbb{C}^{n \times n}$  and  $n \in \mathbb{N}$ . Let  $\mathbf{L} = \mathbf{P}\mathbf{J}\mathbf{P}^{-1}$ , where  $\mathbf{J}$  is the Jordan form, and  $\mathbf{P}$  is a invertible matrix of generalized eigenvectors.
Consider a Jordan block  $\mathbf{J}_i$  associated with the eigenvalue  $\lambda_i(\mathbf{M})$ . For a positive integer  $n$ , the  $n$ -th power of the Jordan block can be computed as:
$$
\mathbf {J} _ {l} ^ {n} = \lambda_ {l} (\mathbf {L}) ^ {n} \left[ \begin{array}{c c c c c} 1 & \binom {n} {1} \lambda_ {l} (\mathbf {L}) ^ {- 1} & \binom {n} {2} \lambda_ {l} (\mathbf {L}) ^ {- 2} & \dots & \binom {n} {k _ {l} - 1} \lambda_ {l} (\mathbf {L}) ^ {- k _ {l} + 1} \\ & 1 & \binom {n} {1} \lambda_ {l} (\mathbf {L}) ^ {- 1} & & \binom {n} {k _ {l} - 2} \lambda_ {l} (\mathbf {L}) ^ {- k _ {l} + 2} \\ & & 1 & & \vdots \\ & & & \ddots & \binom {n} {1} \lambda_ {l} (\mathbf {L}) ^ {- 1} \\ & & & & 1 \end{array} \right]
$$
We compute the  $n$ -th power of  $\mathbf{L}$  as  $\mathbf{L}^n = (\mathbf{P}\mathbf{J}\mathbf{P}^{-1})^n = \mathbf{P}\mathbf{J}^n\mathbf{P}^{-1}$ , and we expand  $\mathbf{x}_0$  as
where  $\{\mathbf{e}_l^i : i \in \{1, \dots, k_l\}, l \in \{1, \dots, m\}\}$  is the standard basis and  $\mathbf{Pe}_l^i = \psi_l^i(\mathbf{L})$  are the generalized eigenvectors of  $\mathbf{L}$ . It is easy to see that
$$
\mathbf {L} ^ {n} \mathbf {x} _ {0} = \mathbf {P J} ^ {n} \mathbf {P} ^ {- 1} \left(\sum_ {l = 1} ^ {m} \sum_ {i = 1} ^ {k _ {l}} c _ {l} ^ {i} \mathbf {P e} _ {l} ^ {i}\right) = \mathbf {P J} ^ {n} \left(\sum_ {l = 1} ^ {m} \sum_ {i = 1} ^ {k _ {l}} c _ {l} ^ {i} \mathbf {e} _ {l} ^ {i}\right).
$$
As  $\mathbf{J}^n = \bigoplus_{l=1}^{m} \mathbf{J}_l^n$ , we can focus on a single Jordan block. Fix  $l \in \{1, \ldots, m\}$ , and compute
$$
\begin{array}{l} \mathbf {P J} _ {l} ^ {n} \left(\sum_ {i = 1} ^ {k _ {l}} c _ {l} ^ {i} \mathbf {e} _ {l} ^ {i}\right) = \mathbf {P} \left(\lambda_ {l} (\mathbf {M}) ^ {n} c _ {l} ^ {1} \mathbf {e} _ {l} ^ {1}\right) + \mathbf {P} \left(\binom {n} {1} \lambda_ {l} (\mathbf {M}) ^ {n - 1} c _ {l} ^ {1} \mathbf {e} _ {l} ^ {1} + \lambda_ {l} (\mathbf {M}) ^ {n} c _ {l} ^ {2} \mathbf {e} _ {l} ^ {2}\right) \\ + \mathbf {P} \left(\binom {n} {2} \lambda_ {l} (\mathbf {M}) ^ {n - 2} c _ {l} ^ {1} \mathbf {e} _ {l} ^ {1} + \binom {n} {1} \lambda_ {l} (\mathbf {L}) ^ {n - 1} c _ {l} ^ {2} \mathbf {e} _ {l} ^ {2} + \lambda_ {l} (\mathbf {M}) ^ {n} c _ {l} ^ {3} \mathbf {e} _ {l} ^ {3}\right) \\ + \dots . \\ \end{array}
$$
We can summarize our findings in the following lemma.
Lemma E.6. For any  $\mathbf{L} = \mathbf{P}\mathbf{J}\mathbf{P}^{-1} \in \mathbb{R}^{N \times N}$  and  $\mathbf{x}_0 = \sum_{l=1}^{m} \sum_{i=1}^{k_l} c_l^i \psi_l^i(\mathbf{L})$ , we have
$$
\mathbf {L} ^ {n} \mathbf {x} _ {0} = \sum_ {l = 1} ^ {m} \sum_ {i = 1} ^ {\min \{k _ {l}, n - 1 \}} \sum_ {j = 1} ^ {i} \binom {n} {i - j} \lambda_ {l} (\mathbf {L}) ^ {n - i + j} c _ {l} ^ {j} \psi_ {l} ^ {j} (\mathbf {L}).
$$
We proceed with the main result of this subsection.
Proposition E.7. Let  $\mathcal{G}$  be a strongly connected directed graph with SNA  $\mathbf{L} \in \mathbb{R}^{N \times N}$ . Consider the initial value problem in (2) with diagonal channel mixing matrix  $\mathbf{W} \in \mathbb{R}^{K \times K}$  and  $\alpha = 1$ . Approximate the solution to (2) with an explicit Euler scheme with a sufficiently small step size  $h$ . Then, for almost all initial values  $\mathbf{x}_0 \in \mathbb{C}^{N \times K}$  the following holds. If  $\lambda_1(\mathbf{L})$  is unique and
$$
\lambda_ {K} (\mathbf {W}) \Re \lambda_ {1} (\mathbf {L}) <   \lambda_ {1} (\mathbf {W}) \Re \lambda_ {N} (\mathbf {L}), \tag {15}
$$
the approximated solution is HFD. Otherwise, the solution is LFD.
Proof. As noted in (11), the vectorized solution to (2) with  $\alpha = 1$ , can be written as
$$
\operatorname {v e c} (\mathbf {x}) (n h) = (\mathbf {I} - h (\mathbf {W} \otimes \mathbf {L})) ^ {n} \operatorname {v e c} (\mathbf {x} _ {0}).
$$
Consider the Jordan decomposition of  $\mathbf{L} = \mathbf{P}\mathbf{J}\mathbf{P}^{-1}$  and the Jordan decomposition of  $\mathbf{W} \otimes \mathbf{J} = \tilde{\mathbf{P}}\tilde{\mathbf{J}}\tilde{\mathbf{P}}^{-1}$ , where  $\tilde{\mathbf{J}}$  and  $\tilde{\mathbf{P}}$  are specified in Lemma E.2. Then,
$$
\begin{array}{l} \operatorname {v e c} (\mathbf {x}) (n h) = \left(\mathbf {I} + h \mathbf {W} \otimes \left(\mathbf {P J P} ^ {- 1}\right)\right) ^ {n} \operatorname {v e c} \left(\mathbf {x} _ {0}\right) \\ = (\mathbf {I} \otimes \mathbf {P}) (\mathbf {I} - h \mathbf {W} \otimes \mathbf {J}) ^ {n} (\mathbf {I} \otimes \mathbf {P}) ^ {- 1} \operatorname {v e c} (\mathbf {x} _ {0}) \\ \end{array}
$$
$$
\begin{array}{l} = (\mathbf {I} \otimes \mathbf {P}) (\mathbf {I} - h \tilde {\mathbf {P}} \tilde {\mathbf {J}} \tilde {\mathbf {P}} ^ {- 1}) ^ {n} (\mathbf {I} \otimes \mathbf {P}) ^ {- 1} \operatorname {v e c} (\mathbf {x} _ {0}) \\ = (\mathbf {I} \otimes \mathbf {P}) \tilde {\mathbf {P}} (\mathbf {I} - h \tilde {\mathbf {J}}) ^ {n} \tilde {\mathbf {P}} ^ {- 1} (\mathbf {I} \otimes \mathbf {P}) ^ {- 1} \mathrm {v e c} (\mathbf {x} _ {0}) \\ = (\mathbf {I} \otimes \mathbf {P}) \tilde {\mathbf {P}} (\mathbf {I} - h \tilde {\mathbf {J}}) ^ {n} ((\mathbf {I} \otimes \mathbf {P}) \tilde {\mathbf {P}}) ^ {- 1} \operatorname {v e c} (\mathbf {x} _ {0}). \\ \end{array}
$$
Now, decompose  $\mathbf{x}_0$  into the basis of generalized eigenvectors, i.e.,
$$
\operatorname {v e c} \left(\mathbf {x} _ {0}\right) = \sum_ {l _ {1} = 1} ^ {K} \sum_ {l _ {2} = 1} ^ {m} \sum_ {i = 1} ^ {k _ {l _ {2}}} c _ {l} ^ {i} \left(\left(\mathbf {I} \otimes \mathbf {P}\right) \tilde {\mathbf {P}}\right) \left(\mathbf {e} _ {l _ {1}} \otimes \mathbf {e} _ {l _ {2}} ^ {i} (\mathbf {L})\right).
$$
Then, by Lemma E.6, we have
$$
\begin{array}{l} \operatorname {vec}(\mathbf{x})(nh) = \sum_{l_{1} = 1}^{K}\sum_{l_{2} = 1}^{m}\sum_{i = 1}^{\min \left\{k_{l_{2}},t - 1\right\}}\sum_{j = 1}^{i}\binom {n}{i - j}\left(1 - h\lambda_{l_{1}}(\mathbf{W})\lambda_{l_{2}}(\mathbf{L})\right)^{n - i + j}c_{l_{1},l_{2}}^{j} \\ \left(\lambda_ {l _ {1}} (\mathbf {W})\right) ^ {1 - j} \psi_ {l _ {1}} (\mathbf {W}) \otimes \psi_ {l _ {2}} ^ {j} (\mathbf {L}). \\ \end{array}
$$
Now, consider the maximal frequency, i.e.,
$$
L _ {1}, L _ {2} = \underset {l _ {1}, l _ {2}} {\arg \max } \left\{\left| 1 - h \lambda_ {l _ {1}} (\mathbf {W}) \lambda_ {l _ {2}} (\mathbf {L}) \right| \right\}.
$$
Then, the solution  $\operatorname{vec}(\mathbf{x})(nh)$  can be written as
$$
\begin{array}{l} \sum_{l_{1} = 1}^{K}\sum_{l_{2} = 1}^{m}\sum_{i = 1}^{\min \{k_{l_{2}},n - 1\}}\sum_{j = 1}^{i}\binom {n}{i - j}\left(1 - h\lambda_{l_{1}}(\mathbf{W})\lambda_{l_{2}}(\mathbf{L})\right)^{n - i + j}c_{l_{1},l_{2}}^{j}\psi_{l_{1}}(\mathbf{W})\otimes \psi_{l_{2}}^{j}(\mathbf{L}) \\ = \left(1 - h \lambda_ {L _ {1}} (\mathbf {W}) \lambda_ {L _ {2}} (\mathbf {L})\right) ^ {n} \tag {16} \\ \end{array}
$$
$$
\cdot \sum_ {l _ {1} = 1} ^ {K} \sum_ {l _ {2} = 1} ^ {m} \sum_ {i = 1} ^ {\min \{k _ {l _ {2}}, t - 1 \}} \sum_ {j = 1} ^ {i} \binom {n} {i - j} \frac {(1 - h \lambda_ {l _ {1}} (\mathbf {W}) \lambda_ {l _ {2}} (\mathbf {L})) ^ {n - i + j}}{(1 - h \lambda_ {L _ {1}} (\mathbf {W}) \lambda_ {L _ {2}} (\mathbf {L})) ^ {n}} c _ {l _ {1}, l _ {2}} ^ {j} \psi_ {l _ {1}} (\mathbf {W}) \otimes \psi_ {l _ {2}} ^ {j} (\mathbf {L}).
$$
With a similar argument as in the proof of Theorem 5.6, we can then see that
$$
\frac {\operatorname {v e c} (\mathbf {x}) (n h)}{\| \operatorname {v e c} (\mathbf {x}) (n h) \| _ {2}} \xrightarrow {t \to \infty} \frac {c _ {L _ {1} , L _ {2}} ^ {1} \psi_ {L _ {1}} (\mathbf {W}) \otimes \psi_ {L _ {2}} ^ {1} (\mathbf {L})}{\| c _ {L _ {1} , L _ {2}} ^ {1} \psi_ {L _ {1}} (\mathbf {W}) \otimes \psi_ {L _ {2}} ^ {1} (\mathbf {L}) \| _ {2}},
$$
where  $\psi_{L_2}^1 (\mathbf{L})$  is the eigenvector corresponding to  $\lambda_{L_2}(\mathbf{L})$ . Note that for almost all  $\mathbf{x}$ , we have  $c_{L_1,L_2}^1\neq 0$ . Then  $\psi_{L_2}^1 (\mathbf{L})$  is also an eigenvector of  $\mathbf{I} - \mathbf{L}$  corresponding to the eigenvalue  $1 - \lambda_{L_2}(\mathbf{L})$ . By Lemma D.2, we have that the approximated solution is  $(1 - \lambda_{L_2}(\mathbf{L}))$ -FD.
We finish the proof by showing that  $L_{2} = 1$  if (15) is satisfied, and  $L_{2} = N$  otherwise. First, note that either  $\lambda_{K}(\mathbf{W})\Re \lambda_{1}(\mathbf{L})$  or  $\lambda_{1}(\mathbf{W})\Re \lambda_{N}(\mathbf{L})$  are the most negative real parts among all  $\{\lambda_l(\mathbf{W})\Re \lambda_r(\mathbf{L})\}_{l\in \{1,\ldots ,K\}, r\in \{1,\ldots ,N\}}$ . Assume first that  $\lambda_{K}(\mathbf{W})\Re \lambda_{1}(\mathbf{L})$  has the most negative real part, i.e., (15) holds. Then, define
$$
\varepsilon := \max _ {l, r} \left| \lambda_ {K} (\mathbf {W}) \Re \lambda_ {1} (\mathbf {L}) - \lambda_ {l} (\mathbf {W}) \Re \lambda_ {r} (\mathbf {L}) \right|,
$$
and assume  $h < \varepsilon \| \mathbf{W}\|^2$ . Now it is easy to see that
$$
2 \lambda_ {K} (\mathbf {W}) \Re \lambda_ {1} (\mathbf {L}) - h \lambda_ {K} (\mathbf {W}) ^ {2} | \lambda_ {1} (\mathbf {L}) | ^ {2} <   2 \lambda_ {l} (\mathbf {W}) \Re \lambda_ {r} (\mathbf {L}) - h \lambda_ {l} (\mathbf {W}) ^ {2} | \lambda_ {r} (\mathbf {L}) | ^ {2},
$$
which is equivalent to  $(K,1) = (L_1,L_2)$ . Hence, the dynamics are  $(1 - \lambda_{1}(\mathbf{L}))$ -FD. As  $(1 - \lambda_{1}(\mathbf{L}))$  is highest frequency of  $\mathbf{I} - \mathbf{L}$ , we get HFD dynamics. Similar, we can show that if  $\lambda_{1}(\mathbf{W})\Re \lambda_{N}(\mathbf{L})$  is the most negative frequency, we get LFD dynamics. Note that for the HFD argument, we must assume that  $\lambda_{1}(\mathbf{L})$  is the unique eigenvalue with the smallest real part. For the LFD argument, it is already given that  $\lambda_{N}(\mathbf{L})$  has multiplicity one by Perron-Frobenius Theorem.
E.3 GCN oversmooths
Proposition E.8. Let  $\mathcal{G}$  be a strongly connected and aperiodic directed graph with SNA  $\mathbf{L} \in \mathbb{R}^{N \times N}$ . A GCN with the update rule
$$
\mathbf {x} _ {t + 1} = \mathbf {L} \mathbf {x} _ {t} \mathbf {W},
$$
where  $\mathbf{x}_0\in \mathbb{R}^{N\times K}$  are the input node features, always oversmooths.
Proof. The proof follows similarly to the proof of Proposition E.7. The difference is that instead of (16), we can write the node features after  $t$  layers as
$$
\operatorname {v e c}(\mathbf{x}_{t}) = \sum_{l_{1} = 1}^{K}\sum_{l_{2} = 1}^{m}\sum_{i = 1}^{\min \{k_{l_{2}},t - 1\}}\sum_{j = 1}^{i}\binom {t}{i - j}\bigl(\lambda_{l_{1}}(\mathbf{W})\lambda_{l_{2}}(\mathbf{L})\bigr)^{t - i + j}c_{l_{1},l_{2}}^{j}\psi_{l_{1}}(\mathbf{W})\otimes \psi_{l_{2}}^{j}(\mathbf{L}).
$$
Now note that by Perron-Frobenius the eigenvalue  $\lambda_{N}(\mathbf{L})$  with the largest absolute value is real and has multiplicity one. Then,  $\max_{l_1,l_2}|\lambda_{l_1}(\mathbf{W})\lambda_{l_2}(\mathbf{L})|$  is attained at either  $\lambda_{1}(\mathbf{W})\lambda_{N}(\mathbf{L})$  or  $\lambda_{K}(\mathbf{W})\lambda_{N}(\mathbf{L})$ . Equivalently to the proof of Proposition E.7, we can show that the corresponding GCN is  $1 - \lambda_{N}(\mathbf{L})$ -FD. Now  $1 - \lambda_{N}(\mathbf{L}) = \lambda_{1}(\mathbf{I} - \mathbf{L})$ , and  $\lambda_{1}(\mathbf{I} - \mathbf{L})$ -FD corresponds to LFD, hence the GCN oversmooths.
F Appendix for the Cycle Graph Example
Consider the cycle graph with  $N$  nodes numbered from 0 to  $N - 1$ . Since each node has degree 2, the SNA  $\mathbf{L} = \mathbf{A} / 2$  is a circulant matrix produced by the vector  $\mathbf{v} = (\mathbf{e}_1 + \mathbf{e}_{N - 1}) / 2$ . Denote  $\omega = \exp (2\pi i / N)$ , the eigenvectors can be computed as
$$
\mathbf {v} _ {j} = \frac {1}{\sqrt {N}} \left(1, \omega^ {j}, \omega^ {2 j}, \dots , \omega^ {(N - 1) j}\right)
$$
associated to the eigenvalue  $\lambda_{j} = \cos (2\pi j / N)$ . First, we can note that  $\lambda_{j} = \lambda_{N - j}$  for all  $j\in$ $\{1,\dots ,N / 2\}$ ; therefore, the multiplicity of each eigenvalue is 2 except  $\lambda_0$  and, if  $N$  is even,  $\lambda_{N / 2}$ . Since the original matrix is symmetric, there exists a basis of real eigenvectors. A simple calculation
$$
\mathbf {L} \Re \mathbf {v} _ {j} + i \mathbf {L} \Im \mathbf {v} _ {j} = \mathbf {L} \mathbf {v} _ {j} = \lambda_ {j} \mathbf {v} _ {j} = \lambda_ {j} \Re \mathbf {v} _ {j} + i \lambda_ {j} \Im \mathbf {v} _ {j}
$$
shows that  $\Re \mathbf{v}_j$  and  $\Im \mathbf{v}_j$ , defined as
$$
\Re \mathbf {v} _ {j} = \frac {1}{\sqrt {N}} \left(\cos \left(\frac {2 \pi j n}{N}\right)\right) _ {n = 0} ^ {N - 1}, \Im \mathbf {v} _ {j} = \frac {1}{\sqrt {N}} \left(\sin \left(\frac {2 \pi j n}{N}\right)\right) _ {n = 0} ^ {N - 1}
$$
are two eigenvectors of the same eigenvalue  $\lambda_{j}$ . To show that they are linearly independent, we compute under which conditions
$$
0 = a \Re \mathbf {v} _ {j} + b \Im \mathbf {v} _ {j}.
$$
We note that the previous condition implies that for all  $n \notin \{0, N/2\}$
$$
\begin{array}{l} 0 = a \cos \left(\frac {2 \pi j n}{N}\right) + b \sin \left(\frac {2 \pi j n}{N}\right) \\ = \sqrt {a ^ {2} + b ^ {2}} \sin \left(\frac {2 \pi j n}{N} + \arctan \left(\frac {b}{a}\right)\right) \\ \end{array}
$$
Suppose  $a, b \neq 0$ , then it must be
$$
\frac {2 \pi j n}{N} + \arctan \left(\frac {b}{a}\right) = k \pi , k \in \mathbb {Z}
$$
$$
2 j n = \left(k - \frac {\arctan \left(\frac {b}{a}\right)}{\pi}\right) N, k \in \mathbb {Z}
$$
The left-hand side is always an integer, while the right-hand side is an integer if and only if  $b = 0$ . This reduces the conditions to
$$
\left\{ \begin{array}{l} a \cos \left(\frac {2 \pi j n}{N}\right) = 0 \\ | a | \sin \left(\frac {2 \pi j n}{N}\right) = 0 \end{array} \right.
$$
which is true if and only if  $a = 0$ . Consider now an even number of nodes  $N$ ; the eigenspace of  $\lambda_{N/2} = -1$  is
$$
\mathbf {v} _ {N / 2} = \frac {1}{\sqrt {N}} \left((- 1) ^ {n}\right) _ {n = 0} ^ {N - 1}
$$
hence, the maximal eigenvector of  $\mathbf{I} - \mathbf{L}$  guarantees homophily 0. Consider now a number of nodes  $N$  divisible by 4; the eigenspace of  $\lambda_{N / 4} = 0$  has basis