SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} 0 = \mathcal {E} (\mathbf {x}) = \frac {1}{4} \sum_ {i, j = 1} ^ {N} a _ {i, j} \left| \frac {x _ {i}}{\sqrt {d _ {i} ^ {\mathrm {l n}}}} - \frac {x _ {j}}{\sqrt {d _ {j} ^ {\mathrm {o u t}}}} \right| ^ {2} \\ \geq \frac {1}{4} \sum_ {i = 1} ^ {N} \frac {1}{d _ {i} ^ {\mathrm {i n}}} \left(\sum_ {j = 1} ^ {N} a _ {i, j} \left| \frac {x _ {i}}{\sqrt {d _ {i} ^ {\mathrm {i n}}}} - \frac {x _ {j}}{\sqrt {d _ {j} ^ {\mathrm {o u t}}}} \right| ^ {2}\right) \left(\sum_ {j = 1} ^ {N} a _ {i, j}\right) \\ \geq \frac {1}{4} \sum_ {i = 1} ^ {N} \frac {1}{d _ {i} ^ {\text {i n}}} \left(\sum_ {j = 1} ^ {N} a _ {i, j} \left| \frac {x _ {i}}{\sqrt {d _ {i} ^ {\text {i n}}}} - \frac {x _ {j}}{\sqrt {d _ {j} ^ {\text {o u t}}}} \right|\right) ^ {2} \\ \geq \frac {1}{4} \sum_ {i = 1} ^ {N} \frac {1}{d _ {i} ^ {\mathrm {i n}}} \left| \sum_ {j = 1} ^ {N} a _ {i, j} \left(\frac {x _ {i}}{\sqrt {d _ {i} ^ {\mathrm {i n}}}} - \frac {x _ {j}}{\sqrt {d _ {j} ^ {\mathrm {o u t}}}}\right) \right| ^ {2} \\ > 0, \\ \end{array}
$$
where we used Cauchy-Schwartz and triangle inequalities.
We give the following simple corollary.
Corollary B.1. For every  $\mathbf{x} \in \mathbb{R}^{N \times K}$ , it holds  $\mathcal{E}(\mathbf{x}) = \frac{1}{2} \Re \left( \mathrm{vec}(\mathbf{x})^{\mathrm{H}} (\mathbf{I} \otimes (\mathbf{I} - \mathbf{L})) \mathrm{vec}(\mathbf{x}) \right)$ .
C Appendix for Section 4
In this section, we provide some properties about FGLs. The first statement shows that the FGL of a normal SNA  $\mathbf{L}$  only changes the magnitude of the eigenvalues of  $\mathbf{L}$ .
Lemma C.1. Let  $\mathbf{M}$  be a normal matrix with eigenvalues  $\lambda_1, \ldots, \lambda_N$  and corresponding eigenvectors  $\mathbf{v}_1, \ldots, \mathbf{v}_N$ . Suppose  $\mathbf{M} = \mathbf{L}\Sigma \mathbf{R}^{\mathrm{H}}$  is its singular value decomposition. Then it holds
$$
\pmb {\Sigma} = | \pmb {\Lambda} |, \mathbf {L} = \mathbf {V}, \mathbf {R} = \mathbf {V} \exp {(i \pmb {\Theta})}, \pmb {\Theta} = \mathrm {d i a g} \left(\{\theta_ {i} \} _ {i = 1} ^ {N}\right), \theta_ {i} = \mathrm {a t a n 2} \left(\Re \lambda_ {i}, \Im \lambda_ {i}\right).
$$
Proof. By hypothesis, there exist a unitary matrix  $\mathbf{V}$  such that  $\mathbf{M} = \mathbf{V}\mathbf{\Lambda}\mathbf{V}^{\mathrm{H}}$ , then
$$
\mathbf {M} ^ {\mathrm {H}} \mathbf {M} = \mathbf {V} \boldsymbol {\Lambda} ^ {*} \mathbf {V} ^ {\mathrm {H}} \mathbf {V} \boldsymbol {\Lambda} \mathbf {V} ^ {\mathrm {H}} = \mathbf {V} | \boldsymbol {\Lambda} | ^ {2} \mathbf {V} ^ {\mathrm {H}},
$$
$$
\mathbf {M} ^ {\mathrm {H}} \mathbf {M} = \mathbf {R} \boldsymbol {\Sigma} \mathbf {L} ^ {\mathrm {H}} \mathbf {L} \boldsymbol {\Sigma} \mathbf {R} ^ {\mathrm {H}} = \mathbf {L} \boldsymbol {\Sigma} ^ {2} \mathbf {L} ^ {\mathrm {H}}.
$$
Therefore,  $\Sigma = |\Lambda|$  and  $\mathbf{L} = \mathbf{V}$
$$
\mathbf {M} = \mathbf {R} \left| \boldsymbol {\Lambda} \right| \mathbf {V} ^ {\mathsf {H}}
$$
Finally, we note that it must hold  $\mathbf{R} = \mathbf{V}\exp (i\Theta)$  where  $\Theta = \mathrm{diag}\left(\{\mathrm{atan2}(\Re \lambda_i,\Im \lambda_i)\}_{i = 1}^N\right)$  and  $\mathrm{atan2}$  is the 2-argument arctangent.
We proceed by proving Theorem 4.1, which follows the proof of a similar result given in (Benzi et al., 2020) for the fractional Laplacian defined in the spectral domain of an in-degree normalized graph Laplacian. However, our result also holds for directed graphs and in particular for fractional Laplacians that are defined via the SVD of a graph SNA.
Lemma C.2. Let  $\mathbf{M} \in \mathbb{R}^{n \times n}$  with singular values  $\sigma(\mathbf{M}) \subset [a, b]$ . For  $f: [a, b] \to \mathbb{R}$ , define  $f(\mathbf{M}) = \mathbf{U}f(\boldsymbol{\Sigma})\mathbf{V}^{\mathrm{H}}$ , where  $\mathbf{M} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{\mathrm{H}}$  is the singular value decomposition of  $\mathbf{M}$ . If  $f$  has modulus of continuity  $\omega$  and  $d(i, j) \geq 2$ , it holds
$$
\left| f (\mathbf {M}) \right| _ {i, j} \leq \left(1 + \frac {\pi^ {2}}{2}\right) \omega \left(\frac {b - a}{2} \left| d (i, j) - 1 \right| ^ {- 1}\right).
$$
Proof. Let  $g:[a,b]\to \mathbb{R}$  be any function, then
$$
\begin{array}{l} \left\| f (\mathbf {M}) - g (\mathbf {M}) \right\| _ {2} = \left\| \mathbf {U} f (\boldsymbol {\Sigma}) \mathbf {V} ^ {\mathsf {H}} - \mathbf {U} g (\boldsymbol {\Sigma}) \mathbf {V} ^ {\mathsf {H}} \right\| _ {2} \\ = \| f (\boldsymbol {\Sigma}) - g (\boldsymbol {\Sigma}) \| _ {2} \\ = \| f (\lambda) - g (\lambda) \| _ {\infty , \sigma (M)}. \\ \end{array}
$$
The second equation holds since the 2-norm is invariant under unitary transformations. By Jackson's Theorem, there exists for every  $m \geq 1$  a polynomial  $p_m$  of order  $m$  such that
$$
\left\| f (\mathbf {M}) - p _ {m} (\mathbf {M}) \right\| _ {2} \leq \left\| f - p _ {m} \right\| _ {\infty , [ a, b ]} \leq \left(1 + \frac {\pi^ {2}}{2}\right) \omega \left(\frac {b - a}{2 m}\right).
$$
Fix  $i,j\in \{1,\ldots ,n\}$ . If  $d(i,j) = m + 1$ , then any power of  $\mathbf{M}$  up to order  $m$  has a zero entry in  $(i,j)$ , i.e.,  $(\mathbf{M}^m)_{i,j} = 0$ . Hence,  $f(\mathbf{M})_{i,j} = f(\mathbf{M})_{i,j} - p_m(\mathbf{M})_{i,j}$ , and we get
$$
| f (\mathbf {M}) _ {i, j} | \leq \| f (\mathbf {M}) - g (\mathbf {M}) \| _ {2} \leq \omega \left(1 + \frac {\pi^ {2}}{2}\right) \left(\frac {b - a}{2 m}\right) = \left(1 + \frac {\pi^ {2}}{2}\right) \omega \left(\frac {b - a}{2} | d (i, j) - 1 | ^ {- 1}\right)
$$
from which the thesis follows.
Finally, we give a proof of Theorem 4.1, which is a consequence of the previous statement.
Proof of Theorem 4.1. The eigenvalues of  $\mathbf{L}$  are in the unit circle, i.e.,  $\| \mathbf{L}\| \leq 1$ . Hence,  $\| \mathbf{LL}^{\mathsf{H}}\| \leq 1$  and the singular values of  $\mathbf{L}$  are in [0, 1]. By Lemma C.2 and the fact that  $f(x) = x^{\alpha}$  has modulus of continuity  $\omega (t) = t^{\alpha}$  the thesis follows.
D Appendix for Section 5
In this section, we provide the appendix for Section 5. We begin by analyzing the solution of linear matrix ODEs. For this, let  $\mathbf{M} \in \mathbb{C}^{N \times N}$ . For  $\mathbf{x}_0 \in \mathbb{C}^N$ , consider the initial value problem
$$
\mathbf {x} ^ {\prime} (t) = - \mathbf {M} \mathbf {x} (t), \quad \mathbf {x} (0) = \mathbf {x} _ {0}. \tag {5}
$$
Theorem D.1 (Existence and uniqueness of linear ODE solution). The initial value problem given by (5) has a unique solution  $\mathbf{x}(t)\in \mathbb{C}^N$  for any initial condition  $\mathbf{x}_0\in \mathbb{C}^N$ .
The solution of (5) can be expressed using matrix exponentials, even if  $\mathbf{M}$  is not symmetric. The matrix exponential is defined as:
$$
\exp (- \mathbf {M} t) = \sum_ {k = 0} ^ {\infty} \frac {(- \mathbf {M}) ^ {k} t ^ {k}}{k !},
$$
where  $\mathbf{M}^k$  is the  $k$ -th power of the matrix  $\mathbf{M}$ . The solution of (5) can then be written as
$$
\mathbf {x} (t) = \exp (- \mathbf {M} t) \mathbf {x} _ {0}. \tag {6}
$$
D.1 Appendix for Section 5.1
In this section, we analyze the solution to (2) and (3). We further provide a proof for Theorem 5.3. We begin by considering the solution to the fractional heat equation (2). The analysis for the Schrödinger equation (3) follows analogously.
The fractional heat equation  $\mathbf{x}'(t) = -\mathbf{L}^{\alpha}\mathbf{x}\mathbf{W}$  can be vectorized and rewritten via the Kronecker product as
$$
\operatorname {v e c} (\mathbf {x}) ^ {\prime} (t) = - \mathbf {W} \otimes \mathbf {L} ^ {\alpha} \operatorname {v e c} (\mathbf {x}) (t). \tag {7}
$$
In the undirected case  $\mathbf{L}$  and  $\mathbf{I} - \mathbf{L}$  are both symmetric, and the eigenvalues satisfy the relation  $\lambda_{i}(\mathbf{I} - \mathbf{L}) = 1 - \lambda_{i}(\mathbf{L})$ . The corresponding eigenvectors  $\psi_{i}(\mathbf{L})$  and  $\psi_{i}(\mathbf{I} - \mathbf{L})$  can be chosen to be the same for  $\mathbf{L}$  and  $\mathbf{I} - \mathbf{L}$ . In the following, we assume that these eigenvectors are orthonormalized.
If  $\mathbf{L}$  is symmetric, we can decompose it via the spectral theorem into  $\mathbf{L} = \mathbf{U}\mathbf{D}\mathbf{U}^T$ , where  $\mathbf{U} = [\psi_1(\mathbf{L}),\dots,\psi_N(\mathbf{L})]$  is an orthogonal matrix containing the eigenvectors of  $\mathbf{L}$ , and  $\mathbf{D}$  is the diagonal matrix of eigenvalues.
Due to Lemma C.1, the fractional Laplacian  $\mathbf{L}^{\alpha}$  can be written as  $\mathbf{L}^{\alpha} = \mathbf{U}\mathrm{f}_{\alpha}(\mathbf{D})\mathbf{U}^{T}$ , where  $\mathrm{f}_{\alpha}:\mathbb{R}\to \mathbb{R}$  is the map  $x\mapsto \operatorname {sign}(x)|x|^{\alpha}$  and is applied element-wise. Clearly, the eigendecomposition of  $\mathbf{L}^{\alpha}$  is given by the eigenvalues  $\{\mathrm{f}_{\alpha}(\lambda_1(\mathbf{L})),\ldots ,\mathrm{f}_{\alpha}(\lambda_N(\mathbf{L}))\}$  and the corresponding eigenvectors  $\{\psi_{1}(\mathbf{L}),\dots,\psi_{N}(\mathbf{L})\}$ .
Now, by well-known properties of the Kronecker product, one can write the eigendecomposition of  $\mathbf{W} \otimes \mathbf{L}^{\alpha}$  as
$$
\{\lambda_ {r} (\mathbf {W}) \mathrm {f} _ {\alpha} (\lambda_ {l} (\mathbf {L})) \} _ {r \in \{1, \dots , K \}, l \in \{1, \dots , N \}}, \{\psi_ {r} (\mathbf {W}) \otimes \psi_ {l} (\mathbf {L}) \} _ {r \in \{1, \dots , K \}, l \in \{1, \dots , N \}}.
$$
Note that  $1 \in \lambda(\mathbf{L})$  and, since trace  $(\mathbf{L}) = 0$ , the SNA has at least one negative eigenvalue. This property is useful since it allows to retrieve of the indices  $(r,l)$  corresponding to eigenvalues with minimal real (or imaginary) parts in a simple way.
The initial condition  $\operatorname{vec}(\mathbf{x}_0)$  can be decomposed as
$$
\operatorname {v e c} (\mathbf {x} _ {0}) = \sum_ {r = 1} ^ {K} \sum_ {l = 1} ^ {N} c _ {r, l} \psi_ {r} (\mathbf {W}) \otimes \psi_ {l} (\mathbf {L}), c _ {r, l} = \left\langle \operatorname {v e c} (\mathbf {x} _ {0}), \psi_ {r} (\mathbf {W}) \otimes \psi_ {l} (\mathbf {W}) \right\rangle .
$$
Then, the solution  $\operatorname{vec}(\mathbf{x})(t)$  of (7) can be written as
$$
\operatorname {v e c} (\mathbf {x}) (t) = \sum_ {r = 1} ^ {K} \sum_ {l = 1} ^ {N} c _ {r, l} \exp \left(- t \lambda_ {r} (\mathbf {W}) f _ {\alpha} (\lambda_ {l} (\mathbf {L}))\right) \psi_ {r} (\mathbf {W}) \otimes \psi_ {l} (\mathbf {L}). \tag {8}
$$
The following result shows the relationship between the frequencies of  $\mathbf{I} - \mathbf{L}$  and the Dirichlet energy and serves as a basis for the following proofs.
Lemma D.2. Let  $\mathcal{G}$  be a graph with SNA L. Consider  $\mathbf{x}(t)\in \mathbb{C}^{N\times K}$  such that there exists  $\varphi \in \mathbb{C}^{N\times K}\setminus \{0\}$  with
$$
\frac {\operatorname {v e c} (\mathbf {x}) (t)}{\| \operatorname {v e c} (\mathbf {x}) (t) \| _ {2}} \xrightarrow {t \to \infty} \operatorname {v e c} (\boldsymbol {\varphi}),
$$
and  $(\mathbf{I}\otimes (\mathbf{I} - \mathbf{L}))\mathrm{vec}(\pmb {\varphi}) = \lambda \mathrm{vec}(\pmb {\varphi})$  . Then,
$$
\mathscr {E} \left(\frac {\mathbf {x} (t)}{\| \mathbf {x} (t) \| _ {2}}\right) \xrightarrow {t \to \infty} \frac {\Re (\lambda)}{2}.
$$
Proof. As  $\operatorname{vec}(\varphi)$  is the limit of unit vectors,  $\operatorname{vec}(\varphi)$  is a unit vector itself. We calculate its Dirichlet energy,
$$
\mathcal {E} (\mathrm {v e c} (\pmb {\varphi})) = \frac {1}{2} \Re \left(\mathrm {v e c} (\pmb {\varphi}) ^ {\mathrm {H}} (\mathbf {I} \otimes (\mathbf {I} - \mathbf {L})) \mathrm {v e c} (\pmb {\varphi})\right) = \frac {1}{2} \Re \left(\lambda \mathrm {v e c} (\pmb {\varphi}) ^ {\mathrm {H}} \mathrm {v e c} (\pmb {\varphi})\right) = \frac {1}{2} \Re \left(\lambda\right).
$$
Since  $\mathbf{x} \mapsto \mathcal{E}(\mathbf{x})$  is continuous, the thesis follows.
Another useful result that will be extensively used in proving Theorem 5.3 is presented next.
Lemma D.3. Suppose  $\mathbf{x}(t)$  can be expressed as
$$
\mathbf {x} (t) = \sum_ {k = 1} ^ {K} \sum_ {n = 1} ^ {N} c _ {k, n} \exp \left(- t \lambda_ {k, n}\right) \mathbf {v} _ {k} \otimes \mathbf {w} _ {n},
$$
for some choice of  $c_{k,n}$ ,  $\lambda_{k,n}$ ,  $\{\mathbf{v}_k\}$ ,  $\{\mathbf{w}_n\}$ . Let  $(a,b)$  be the unique index of  $\lambda_{k,n}$  with minimal real part and corresponding non-null coefficient  $c_{k,n}$ , i.e.
$$
(a, b) := \underset {(k, n) \in [ K ] \times [ N ]} {\arg \min } \left\{\Re \left(\lambda_ {k, n}\right): c _ {k, n} \neq 0 \right\}.
$$
$$
\frac {\mathbf {x} (t)}{\| \mathbf {x} (t) \| _ {2}} \xrightarrow {t \to \infty} \frac {c _ {a , b} \mathbf {v} _ {a} \otimes \mathbf {w} _ {b}}{\| c _ {a , b} \mathbf {v} _ {a} \otimes \mathbf {w} _ {b} \| _ {2}}.
$$
Proof. The key insight is to separate the addend with index  $(a, b)$ . It holds
$$
\begin{array}{l} \mathbf {x} (t) = \sum_ {k = 1} ^ {K} \sum_ {n = 1} ^ {N} c _ {k, n} \exp (- t \lambda_ {k, n}) \mathbf {v} _ {n} \otimes \mathbf {w} _ {m} \\ = \exp \left(-t\lambda_{a,b}\right)\left(c_{a,b}\mathbf{v}_{a}\otimes \mathbf{w}_{b} + \sum_{\substack{(k,n)\in [K]\times [N]\\ (k,n)\neq (a,b)}}c_{k,n}\exp \left(-t\left(\lambda_{k,n} - \lambda_{a,b}\right)\right)\mathbf{v}_{k}\otimes \mathbf{w}_{n}\right). \\ \end{array}
$$
We note that
$$
\begin{array}{l} \lim _ {t \rightarrow \infty} | \exp (- t (\lambda_ {k, n} - \lambda_ {a, b})) | = \lim _ {t \rightarrow \infty} | \exp (- t \Re (\lambda_ {k, n} - \lambda_ {a, b})) \exp (- i t \Im (\lambda_ {k, n} - \lambda_ {a, b})) | \\ = \lim  _ {t \rightarrow \infty} \exp \left(- t \Re \left(\lambda_ {k, n} - \lambda_ {a, b}\right)\right) \\ = 0, \\ \end{array}
$$
for all  $(k,n)\neq (a,b)$  , since  $\Re (\lambda_{k,n} - \lambda_{a,b}) > 0$  . Therefore, one gets
$$
\frac {\mathbf {x} (t)}{\| \mathbf {x} (t) \| _ {2}} \xrightarrow {t \to \infty} \frac {c _ {a , b} \mathbf {v} _ {a} \otimes \mathbf {w} _ {b}}{\| c _ {a , b} \mathbf {v} _ {a} \otimes \mathbf {w} _ {b} \| _ {2}},
$$
where the normalization removes the dependency on  $\exp (-t\lambda_{a,b})$
When  $\lambda_{a,b}$  is not unique, it is still possible to derive a convergence result. In this case,  $\mathbf{x}$  will converge to an element in the span generated by vectors corresponding to  $\lambda_{a,b}$ , i.e.,
$$
\frac {\mathbf {x} (t)}{\| \mathbf {x} (t) \| _ {2}} \xrightarrow {t \to \infty} \frac {\sum_ {(a , b) \in \mathcal {A}} c _ {a , b} \mathbf {v} _ {a} \otimes \mathbf {w} _ {b}}{\| \sum_ {(a , b) \in \mathcal {A}} c _ {a , b} \mathbf {v} _ {a} \otimes \mathbf {w} _ {b} \| _ {2}},
$$
where  $\mathcal{A} \coloneqq \{(k,n) : \Re(\lambda_{k,n}) = \Re(\lambda_{a,b}), c_{k,n} \neq 0\}$ .
A similar result to Lemma D.3 holds for a slightly different representation of  $\mathbf{x}(t)$ .
Lemma D.4. Suppose  $\mathbf{x}(t)$  can be expressed as
$$
\mathbf {x} (t) = \sum_ {k = 1} ^ {K} \sum_ {n = 1} ^ {N} c _ {k, n} \exp \left(i t \lambda_ {k, n}\right) \mathbf {v} _ {k} \otimes \mathbf {w} _ {n},
$$
for some choice of  $c_{k,n}$ ,  $\lambda_{k,n}$ ,  $\{\mathbf{v}_k\}$ ,  $\{\mathbf{w}_n\}$ . Let  $(a,b)$  be the unique index of  $\lambda_{k,n}$  with minimal imaginary part and corresponding non-null coefficient  $c_{k,n}$ , i.e.
$$
(a, b) := \underset {(k, n) \in [ K ] \times [ N ]} {\arg \min } \left\{\Im \left(\lambda_ {k, n}\right): c _ {k, n} \neq 0 \right\}.
$$
Proof. The proof follows the same reasoning as in the proof of Lemma D.3. The difference is that the dominating frequency is the one with the minimal imaginary part, since
$$
\Re (i \lambda_ {k, n}) = - \Im (\lambda_ {k, n}),
$$
and, consequently,
$$
\operatorname * {a r g   m a x} _ {(k, n) \in [ K ] \times [ N ]} \left\{\mathfrak {R} \left(i   \lambda_ {k, n}\right) \right\} = \operatorname * {a r g   m i n} _ {(k, n) \in \in [ K ] \times [ N ]} \left\{\mathfrak {I} \left(\lambda_ {k, n}\right) \right\}.
$$
D.1.1 Proof of Theorem 5.3
We denote the eigenvalues of  $\mathbf{L}$  closest to 0 from above and below as
$$
\lambda_ {+} (\mathbf {L}) := \underset {l} {\arg \min } \left\{\lambda_ {l} (\mathbf {L}): \lambda_ {l} (\mathbf {L}) > 0 \right\},
$$
$$
\lambda_ {-} (\mathbf {L}) := \underset {l} {\arg \max } \left\{\lambda_ {l} (\mathbf {L}): \lambda_ {l} (\mathbf {L}) <   0 \right\}. \tag {9}
$$
We assume that the channel mixing  $\mathbf{W} \in \mathbb{R}^{K \times K}$  and the graph Laplacians  $\mathbf{L}, \mathbf{I} - \mathbf{L} \in \mathbb{R}^{N \times N}$  are real matrices. Finally, we suppose the eigenvalues of a generic matrix  $\mathbf{M}$  are sorted in ascending order, i.e.,  $\lambda_{i}(\mathbf{M}) \leq \lambda_{j}(\mathbf{M}), i < j$ .
We now reformulate Theorem 5.3 for the fractional heat equation (2) and provide its full proof, which follows a similar frequency analysis to the one in (Di Giovanni et al., 2023, Theorem B.3)
Theorem D.5. Let  $\mathcal{G}$  be an undirected graph with SNA L. Consider the initial value problem in (2) with channel mixing matrix  $\mathbf{W} \in \mathbb{R}^{K \times K}$  and  $\alpha \in \mathbb{R}$ . Then, for almost all initial conditions  $\mathbf{x}_0 \in \mathbb{R}^{N \times K}$  the following is satisfied.
$(\alpha >0)$  The solution to (2) is HFD if
$$
\lambda_ {K} (\mathbf {W}) f _ {\alpha} \left(\lambda_ {1} (\mathbf {L})\right) <   \lambda_ {1} (\mathbf {W}),
$$
and LFD otherwise.
$(\alpha < 0)$  The solution to (2) is  $(1 - \lambda_{-}(\mathbf{L}))$  FD if
$$
\lambda_ {K} (\mathbf {W}) f _ {\alpha} (\lambda_ {-} (\mathbf {L})) <   \lambda_ {1} (\mathbf {W}) f _ {\alpha} (\lambda_ {+} (\mathbf {L})) ,
$$
and  $(1 - \lambda_{+}(\mathbf{L}))$ -FD otherwise.
Proof of  $(\alpha > 0)$ . As derived in (8), the solution of (2) with initial condition  $\mathbf{x}_0$  can be written in a vectorized form as
$$
\begin{array}{l} \operatorname {v e c} (\mathbf {x}) (t) = \exp \left(- t \mathbf {W} ^ {\top} \otimes \mathbf {L} ^ {\alpha}\right) \operatorname {v e c} (\mathbf {x} _ {0}) \\ = \sum_ {r = 1} ^ {K} \sum_ {l = 1} ^ {N} c _ {r, l} \exp \left(- t \lambda_ {r} (\mathbf {W}) \mathrm {f} _ {\alpha} \left(\lambda_ {l} (\mathbf {L})\right)\right) \psi_ {r} (\mathbf {W}) \otimes \psi_ {l} (\mathbf {L}), \\ \end{array}
$$
where  $\lambda_r(\mathbf{W})$  are the eigenvalues of  $\mathbf{W}$  with corresponding eigenvectors  $\psi_r(\mathbf{W})$ , and  $\lambda_l(\mathbf{L})$  are the eigenvalues of  $\mathbf{L}$  with corresponding eigenvectors  $\psi_l(\mathbf{L})$ . The coefficients  $c_{r,l}$  are the Fourier coefficients of  $\mathbf{x}_0$ , i.e.,
$$
c _ {r, l} := \left\langle \operatorname {v e c} \left(\mathbf {x} _ {0}\right), \psi_ {r} (\mathbf {W}) \otimes \psi_ {l} (\mathbf {L}) \right\rangle .
$$
The key insight is to separate the eigenprojection corresponding to the most negative frequency. By Lemma D.3, this frequency component dominates for  $t$  going to infinity.
$$
\lambda_ {K} (\mathbf {W}) f _ {\alpha} \left(\lambda_ {1} (\mathbf {L})\right) <   \lambda_ {1} (\mathbf {W}) f _ {\alpha} \left(\lambda_ {N} (\mathbf {L})\right) = \lambda_ {1} (\mathbf {W}).
$$
In this case,  $\lambda_K(\mathbf{W})\mathrm{f}_{\alpha}(\lambda_1(\mathbf{L}))$  is the most negative frequency. Assume for simplicity that  $\lambda_K(\mathbf{W})$  has multiplicity one; the argument can be applied even if this is not the case, since the corresponding eigenvectors are orthogonal for higher multiplicities.
For almost all initial conditions  $\mathbf{x}_0$ , the coefficient  $c_{K,1}$  is not null; hence
$$
\frac {\operatorname {v e c} (\mathbf {x}) (t)}{\| \operatorname {v e c} (\mathbf {x}) (t) \| _ {2}} \xrightarrow {t \to \infty} \frac {c _ {K , 1} \psi_ {K} (\mathbf {W}) \otimes \psi_ {1} (\mathbf {L})}{\| c _ {K , 1} \psi_ {K} (\mathbf {W}) \otimes \psi_ {1} (\mathbf {L}) \| _ {2}}.
$$
By standard properties of the Kronecker product, we have
$$
\left(\mathbf {I} \otimes \mathbf {L}\right) \left(\psi_ {K} (\mathbf {W}) \otimes \psi_ {1} (\mathbf {L})\right) = \left(\mathbf {I} \psi_ {K} (\mathbf {W})\right) \otimes \left(\mathbf {L} \psi_ {1} (\mathbf {L})\right) = \lambda_ {1} (\mathbf {L}) \psi_ {K} (\mathbf {W}) \otimes \psi_ {1} (\mathbf {L}), \tag {10}
$$
i.e.,  $\psi_K(\mathbf{W})\otimes \psi_1(\mathbf{L})$  is an eigenvector of  $\mathbf{I}\otimes \mathbf{L}$  corresponding to the eigenvalue  $\lambda_{1}(\mathbf{L})$ . Then, by Proposition 3.3,  $\psi_K(\mathbf{W})\otimes \psi_1(\mathbf{L})$  is also an eigenvector of  $\mathbf{I}\otimes \mathbf{I} - \mathbf{L}$  corresponding to the eigenvalue  $1 - \lambda_{1}(\mathbf{L}) = \lambda_{N}(\mathbf{I} - \mathbf{L})$ . An application of Lemma D.2 finishes the proof.
Similarly, we can show that if  $\alpha > 0$  and  $\lambda_K(\mathbf{W})\mathrm{f}_{\alpha}(\lambda_1(\mathbf{L})) > \lambda_1(\mathbf{W})$  the lowest frequency component  $\lambda_{1}(\mathbf{I} - \mathbf{L})$  is dominant.
Proof of  $(\alpha < 0)$ . In this case either  $\mathrm{f}_{\alpha}(\lambda_{+}(\mathbf{L}))\lambda_{1}(\mathbf{W})$  or  $\mathrm{f}_{\alpha}(\lambda_{-}(\mathbf{L}))\lambda_{K}(\mathbf{W})$  are the most negative frequency components. Hence, if  $\mathrm{f}_{\alpha}(\lambda_{-}(\mathbf{L}))\lambda_{K}(\mathbf{W}) > \mathrm{f}_{\alpha}(\lambda_{+}(\mathbf{L}))\lambda_{1}(\mathbf{W})$  the frequency  $\mathrm{f}_{\alpha}(\lambda_{+}(\mathbf{L}))\lambda_{1}(\mathbf{W})$  is dominating and otherwise the frequency  $\mathrm{f}_{\alpha}(\lambda_{-}(\mathbf{L}))\lambda_{K}(\mathbf{W})$ . We can see this by following the exact same reasoning of (i).
Remark D.6. In the proof of  $(\alpha < 0)$ , we are tacitly assuming that  $\mathbf{L}$  has only non-zero eigenvalues. If not, we can truncate the SVD and remove all zeros singular values (which correspond to zeros eigenvalues). In doing so, we obtain the best invertible approximation of  $\mathbf{L}$  to which the theorem can be applied.
We now generalize the previous result to all directed graphs with normal SNA.
Theorem D.7. Let  $\mathcal{G}$  be a strongly connected directed graph with normal SNA  $\mathbf{L}$  such that  $\lambda_1(\mathbf{L})\in \mathbb{R}$ . Consider the initial value problem in (2) with channel mixing matrix  $\mathbf{W}\in \mathbb{R}^{K\times K}$  and  $\alpha >0$ . Then, for almost all initial values  $\mathbf{x}_0\in \mathbb{R}^{N\times K}$  the solution to (2) is HFD if
$$
\lambda_ {K} (\mathbf {W}) | \lambda_ {1} (\mathbf {L}) | ^ {\alpha} <   \lambda_ {1} (\mathbf {W}) | \lambda_ {N} (\mathbf {L}) | ^ {\alpha},
$$
Proof. Any normal matrix is unitary diagonalizable, i.e., there exist eigenvalues  $\lambda_1,\ldots ,\lambda_N$  and corresponding eigenvectors  $\mathbf{v}_1,\dots ,\mathbf{v}_N$  such that  $\mathbf{L} = \mathbf{V}\boldsymbol {\Lambda}\mathbf{V}^{\mathrm{H}}$ . Then, by Lemma C.1, the singular value decomposition of  $\mathbf{L}$  is given by  $\mathbf{L} = \mathbf{U}\boldsymbol {\Sigma}\mathbf{V}^{\mathrm{H}}$ , where
$$
\pmb {\Sigma} = | \pmb {\Lambda} |, \mathbf {U} = \mathbf {V} \exp (i \pmb {\Theta}), \pmb {\Theta} = \mathrm {d i a g} \left(\{\theta_ {i} \} _ {i = 1} ^ {N}\right), \theta_ {i} = \mathrm {a t a n 2} (\Re \lambda_ {i}, \Im \lambda_ {i}).
$$
$$
\mathbf {L} ^ {\alpha} = \mathbf {U} \boldsymbol {\Sigma} ^ {\alpha} \mathbf {V} ^ {\mathsf {H}} = \mathbf {V} | \boldsymbol {\Lambda} | ^ {\alpha} \exp (i \boldsymbol {\Theta}) \mathbf {V} ^ {\mathsf {H}}.
$$
Then, equivalent to the derivation of (8), the solution to the vectorized fractional heat equation
$$
\operatorname {v e c} (\mathbf {x}) ^ {\prime} (t) = - \mathbf {W} \otimes \mathbf {L} ^ {\alpha} \operatorname {v e c} (\mathbf {x}) (t)
$$
is given by
$$
\operatorname {v e c} (\mathbf {x}) (t) = \sum_ {r = 1} ^ {K} \sum_ {l = 1} ^ {N} c _ {r, l} \exp \left(- t \lambda_ {r} (\mathbf {W}) \mathrm {f} _ {\alpha} \left(\lambda_ {l} (\mathbf {L})\right)\right) \psi_ {r} (\mathbf {W}) \otimes \psi_ {l} (\mathbf {L}).
$$
$$
\mathrm {f} _ {\alpha} \left(\lambda_ {l} (\mathbf {L})\right) = \left| \lambda (\mathbf {L}) _ {l} \right| ^ {\alpha} \exp (i \theta_ {l}).
$$
Now, equivalent to the proof of Theorem 5.3, we apply Lemma D.3. Therefore, the dominating frequency is given by the eigenvalue of  $\mathbf{W} \otimes \mathbf{L}^{\alpha}$  with the most negative real part. The eigenvalues of  $\mathbf{W} \otimes \mathbf{L}^{\alpha}$  are given by  $\lambda_r(\mathbf{W})\mathrm{f}_\alpha (\lambda_l(\mathbf{L}))$  for  $r = 1,\dots ,K,l = 1,\dots ,N$ . The corresponding real parts are given by