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A Discrete Hodge-deRham theory of abstract simplicial complex
Inspired by differential geometry and algebraic topology, this work investigates how random walk on simplicial complexes can facilitate graph and simplicial learning. In our main text, we merely introduce the Hodge Laplacian (in Hilbert space) due to limited space. In this section, we give a complete background on discrete Hodge-deRham theory of abstract simplicial complexes to help readers better understand relevant concepts.
An abstract simplicial complex  $\mathcal{K}$  in a finite set  $V$  is a collection of subsets of  $V$  that is closed under inclusion. In our paper,  $V$  will be a vertex set  $[n] = \{1,2,\dots,n\}$  if without special statement. An element of cardinality  $k + 1$  is called a  $k$ -face or  $k$ -simplex of  $\mathcal{K}$ . For instance, 0-faces are usually called vertices, 1-faces are directed edges and 2-faces are 3-cliques (triangles) with an orientation. We denote the collection of all  $k$ -faces of  $\mathcal{K}$  as  $S_{k}(\mathcal{K})$ . The dimension of a  $k$ -face is  $k$  and the dimension of a complex  $\mathcal{K}$  is defined as the maximum dimension of faces in  $\mathcal{K}$ .
The definition of neighbors of simplices is crucial in this paper. Two  $k + 1$ -simplices sharing a collective  $k$ -face are called  $k$ -down neighbors, and two  $k$ -simplices sharing a collective  $k + 1$ -simplex are called  $k + 1$ -up neighbors. Generally, a face  $F$  is chosen an ordering on its vertices and is said to be oriented, denoted by  $[F]$ . For any permutation element  $\sigma \in \mathcal{G}_{k + 1}$  where  $\mathcal{G}_{k + 1}$  is the symmetric group of permutations on  $\{0, \dots, k\}$ , two orderings of vertices transformed by  $\sigma$  are said to determine the same orientation if  $\sigma$  is an even permutation and opposite if  $\sigma$  is odd. In addition, a  $k$ -cochain or  $k$ -form is a function defined on  $\mathcal{K}_{k + 1}$ ,  $f: V \times \dots \times V \to \mathbb{R}$  that satisfies the following.
$$
f \left(i _ {\sigma (0)}, \dots , i _ {\sigma (k)}\right) = \operatorname {s g n} (\sigma) f \left(i _ {0}, \dots , i _ {k}\right) \tag {8}
$$
for all  $\{i_0,\ldots ,i_k\} \in \mathcal{K}_{k + 1}$  and all  $\sigma \in \mathcal{G}_{k + 1}$ . Specifically,  $f(i_0,\dots ,i_k) = 0$  if  $\{i_0,\dots ,i_k\} \notin \mathcal{K}_{k + 1}$ . Although they have the structure of vector spaces, the vector spaces are usually called cochain groups  $\mathcal{C}^k (\mathcal{K},\mathbb{R})$ . Chain groups  $\mathcal{C}_k(\mathcal{K},\mathbb{R})$  are defined as duals of co-chain groups. In addition, we define the simplicial coboundary maps  $\delta_{k}:\mathcal{C}^{k}(\mathcal{K},\mathbb{R})\to \mathcal{C}^{k + 1}(\mathcal{K},\mathbb{R})$ :
$$
\left(\delta_ {k} f\right) \left([ v _ {0}, \dots , v _ {i + 1} ]\right) = \sum_ {j = 0} ^ {k + 1} (- 1) ^ {j} f \left([ v _ {0}, \dots , \hat {v} _ {j}, \dots , v _ {k + 1} ]\right) \tag {9}
$$
where  $\hat{v}_j$  suggests that the vertex  $v_j$  is omitted. One can view  $\delta_k$  as the dual of the boundary map  $\partial_{k + 1}$ , which connects the cochain complex of  $\mathcal{K}$  with coefficients in  $\mathbb{R}$ . Further, we can define the adjoint of coboundary operator:  $\delta_k^*:\mathcal{C}^{k + 1}(\mathcal{K},\mathbb{R})\to \mathcal{C}^k (\mathcal{K},\mathbb{R})$ . Therefore, we have the following connection.
$$
\mathcal {C} ^ {k + 1} (\mathcal {K}, \mathbb {R}) \xrightarrow [ \delta_ {k} ^ {*} ]{\delta_ {k}} \mathcal {C} ^ {k} (\mathcal {K}, \mathbb {R}) \xleftarrow [ \delta_ {k - 1} ^ {*} ]{\delta_ {k - 1}} \mathcal {C} ^ {k - 1} (\mathcal {K}, \mathbb {R}) \tag {10}
$$
After determining the inner products  $<, >$ , we can define the adjoint of the coboundary operator  $\delta_k^*:\mathcal{C}^{k + 1}(\mathcal{K},\mathbb{R})\to \mathcal{C}^k (\mathcal{K},\mathbb{R})$
$$
<   \delta_ {k} \mathbf {f} _ {1}, \mathbf {f} _ {2} > _ {\mathcal {C} ^ {k + 1}} = <   \mathbf {f} _ {1}, \delta_ {k} ^ {*} \mathbf {f} _ {2} > _ {\mathcal {C} ^ {k}} \tag {11}
$$
where  $\mathbf{f}_1\in \mathcal{C}^k (\mathcal{K},\mathbb{R})$ $\mathbf{f}_2\in \mathcal{C}^{k + 1}(\mathcal{K},\mathbb{R})$  are arbitrary. Specifically, in the Hilbert space  $L^2$  , the matrix representations of  $\delta_{k}$  and  $\delta_k^*$  are adjoint matrix  $\mathbf{B}_{k + 1}^{*}$  and  $\mathbf{B}_{k + 1}$  as described in the main texts.
Define the Hodge  $k$ -Laplacian operator (also called combinatorial Laplace operator):
$$
\mathbf {L} _ {k} = \mathbf {L} _ {k, d o w n} + \mathbf {L} _ {k, u p} = \delta_ {k - 1} \delta_ {k - 1} ^ {*} + \delta_ {k} ^ {*} \delta_ {k} \tag {12}
$$
where we omit the notation  $\mathcal{K}$  for simplicity. It is easy to verify by definition that all three operators  $\mathbf{L}_k, \mathbf{L}_{k,up}, \mathbf{L}_{k,down}$  are self-adjoint, nonnegative and compact.
In the Hilbert space, the matrix representations for boundary and co-boundary operators are adjacent matrices of the  $k$  and  $k + 1$  order simplices. In order to keep coordinate with most existing literature,
we write the matrix representation for  $\delta_k^*$  as  $\mathbf{B}_{k + 1}\in \mathbb{R}^{|S_k|\times |S_{k + 1}|}$  (one can view it as the adjacent matrix of  $k$ -th and  $k + 1$ -th simplices); therefore, we have the definition in our main text:
$$
\mathbf {L} _ {k} = \mathbf {B} _ {k} ^ {*} \mathbf {B} _ {k} + \mathbf {B} _ {k + 1} \mathbf {B} _ {k + 1} ^ {*} \tag {13}
$$
where  $\mathbf{B}_k^* = \mathbf{B}_k^T$  is the adjoint of  $\mathbf{B}_k$  and is equivalent to the transpose of  $\mathbf{B}_k$  in the Hilbert space. It is remarkable that when  $k = 0$ ,  $\mathbf{L}_0$  is exactly the graph Laplacian  $\mathbf{L}_0 = \mathbf{D} - \mathbf{A}$ . In our paper, we make use of higher-order Hodge Laplacians such as  $\mathbf{L}_1$  rather than previously used  $\mathbf{L}_0$  alone.
Note that the following equation always holds:
$$
\delta_ {k} \delta_ {k - 1} = 0 \tag {14}
$$
This result is sometimes called the fundamental theorem of topology, which can be intuitively interpreted as "the coboundary of a coboundary is zero".
The kernel space of  $\mathbf{L}_k$  is called the  $k$ -th cohomology group:
$$
\tilde {\mathcal {H}} ^ {k} (\mathcal {K}, \mathbb {R}) := \ker \left(\delta_ {k}\right) / \operatorname {i m} \left(\delta_ {k - 1}\right) \cong \ker \left(\delta_ {k}\right) \cap \ker \left(\delta_ {k - 1} ^ {*}\right) = \ker \left(\mathbf {L} _ {k}\right) \tag {15}
$$
We will write  $\tilde{\mathcal{H}}^k (\mathcal{K},\mathbb{R})$  simply as  $\tilde{\mathcal{H}}^k$  without causing confusion. The kernel space is closely associated with harmonic functions and will play an important role in our following analysis.
Proof. We already have  $\delta_k\delta_{k - 1} = 0$  and thus  $\delta_{k - 1}^*\delta_k^* = 0$ . Then,
$$
\operatorname {i m} \left(\mathbf {L} _ {\mathrm {k}, \text {d o w n}}\right) \subset \ker \left(\mathbf {L} _ {\mathrm {k}, \text {u p}}\right) \operatorname {i m} \left(\mathbf {L} _ {\mathrm {k}, \text {u p}}\right) \subset \ker \left(\mathbf {L} _ {\mathrm {k}, \text {d o w n}}\right) \tag {16}
$$
$$
\begin{array}{l} \ker \left(\mathbf {L} _ {k}\right) = \ker \left(\delta_ {k} ^ {*} \delta_ {k}\right) \cap \ker \left(\delta_ {k - 1} \delta_ {k - 1} ^ {*}\right) (17) \\ = \ker (\delta_ {k}) \cap \ker (\delta_ {k - 1} ^ {*}) (18) \\ = \ker \left(\delta_ {k}\right) \cap \left(\operatorname {i m} \left(\delta_ {k - 1}\right)\right) ^ {\perp} (19) \\ \cong \tilde {\mathcal {H}} ^ {k} (20) \\ \end{array}
$$
Equation (16) reveals an important conclusion:  $\lambda > 0$  is a nonzero eigenvalue of  $\mathbf{L}_k$  if and only if it is a nonzero eigenvalue of  $\mathbf{L}_{k,up}$  and  $\mathbf{L}_{k,down}$ . Let the sorted eigenvalues of operator  $\mathbf{A}$  be  $\mathbf{s}(\mathbf{A}) = (\lambda_0,\dots ,\lambda_m)$ , where the eigenvalues are in a weakly increasing rearrangement. Denote  $\mathbf{s}(\mathbf{A})\doteq \mathbf{s}(\mathbf{B})$  if the multisets of  $\mathbf{s}(\mathbf{A})$  and  $\mathbf{s}(\mathbf{B})$  have exactly the same nonzero eigenvalues (or they only differ in the multiplicities of zero). Then we have
$$
\mathbf {s} \left(\mathbf {L} _ {k}\right) \doteq \mathbf {s} \left(\mathbf {L} _ {k, u p}\right) \cup \mathbf {s} \left(\mathbf {L} _ {k, d o w n}\right) \tag {21}
$$
Further using  $\mathbf{s}(AB) \doteq \mathbf{s}(BA)$ , we have an important property that bridges the up and down Hodge Laplacians of the adjacent dimension:
$$
\mathbf {s} \left(\mathbf {L} _ {k, u p}\right) \dot {=} \mathbf {s} \left(\mathbf {L} _ {k + 1, d o w n}\right) \tag {22}
$$
Now we turn our attention to the zero eigenvalues and eigenvectors. The multiplicity of zero eigenvalues of  $\mathbf{L}_k$ , or the dimension of null space of Hodge  $k$ -Laplacian  $\ker(\mathbf{L}_k)$ , is called the  $k$ -th Betti number  $\beta_k$ . This is exactly the number of cycles composed of  $k$ -simplicials that are not induced by a  $k$ -boundary, or intuitively,  $k$ -dimensional "holes" in the simplicial complex  $\mathcal{K}$ . For example, zero eigenvalues and their eigenvectors of  $\mathbf{L}_0$  are associated with 0-th homology group of the graph,
corresponding to the connected components in the graph. The zero eigenvalues and eigenvectors of  $\mathbf{L}_1$  are associated with cycles (in the usual sense), and those of  $\mathbf{L}_2$  correspond to cavities.
Also note that the following relation always holds.
$$
\operatorname {i m} \left(\mathbf {L} _ {k}\right) = \operatorname {i m} \left(\delta_ {k} ^ {*}\right) \oplus \operatorname {i m} \left(\delta_ {k - 1}\right) \tag {23}
$$
Finally, we present the central theorem of Hodge theory, known as Hodge decomposition. It states that  $\mathcal{C}^k$  can be decomposed into three orthogonal subspaces:
$$
\mathcal {C} ^ {k} (\mathcal {K}, \mathbb {R}) = \overbrace {\operatorname {i m} \left(\delta_ {k} ^ {*}\right) \oplus \underbrace {\ker \left(\mathbf {L} _ {k}\right)} _ {\ker \left(\delta_ {k}\right)} \oplus \operatorname {i m} \left(\delta_ {k - 1}\right)} ^ {\ker \left(\delta_ {k - 1} ^ {*}\right)} \tag {24}
$$
For example, for a vector field  $\mathcal{C}^1$ , the above equation can be interpreted as follows. Any edge flow can be decomposed into three orthogonal components: a solenoidal (divergence-free) component, a harmonic (both divergence-free and curl-free) component, and a gradient (curl-free) component. In a discrete graph, a divergence of a vertex (0-simplex) is defined as the excess of outgoing over incoming weights, while the curl in a directed triangle (2-simplex) is defined as the edge sum around the triangle along the positive direction.
The Hodge decomposition theorem provides insight into our methods. For instance, the kernel space of  $\mathbf{L}_1$  (which plays an important role in our Hodge1Lap) is both divergence-free and curl-free, thus only occurring along the cycles in the graph. Moreover, the Hodge decomposition holds for any order of cochains  $\mathcal{C}^k$ , enriching the physical insights of our PE and SE based on Hodge  $k$ -Laplacians.
B Classical results of node-level random walk
Here are some classical conclusions about the random walk at node level.
Let  $X_{k}$  be the node we are at step  $k$ , and  $S_{k}(x)$  be the number of times that the random walk visits  $x$  during first  $t$  steps, then we have
Theorem B.1.  $\lim_{k\to \infty}\mathbb{E}[\frac{S_k(x)}{k}] = \frac{d(x)}{2m}$ .  $\frac{S_k(x)}{k}$  tends to stationary probability  $\pi$  in probability as  $k\to \infty$ .
Denote the mean hitting time of  $y$  from  $x$  by  $H_{(}x,y)$ , then clearly
$$
H (x, y) = \sum_ {k = 1} ^ {\infty} k \mathbb {P} \left(X _ {k} = y, X _ {k} \neq y, 1 \leq i <   k \mid X _ {0} = x\right) \tag {25}
$$
Generally,  $H(x,y)$  is not symmetric. Specifically,  $H(x,x)$  is the mean return time from  $x$  to itself. We have
$$
H (x, x) = 1 + \sum_ {z \in V} P _ {x z} H (z, x) = 1 + \frac {1}{d (x)} \sum_ {z \in \mathcal {N} (x)} H (z, x) \tag {26}
$$
In terms of connected graph, we have the following conclusion.
Theorem B.2. If the graph  $G$  is connected, then  $H(x,x) = \frac{2m}{d(x)}$ .
A slight variation of the conclusion states,
Theorem B.3.  $\frac{\mathbb{E}S_k(yx)}{k} = \frac{\mathbb{E}S_k(y)}{kd(y)}\to \frac{1}{2m}$
Theorem B.4.  $G$  is a connected graph, then a simple random walk satisfies  $\frac{1}{2m}\sum_{x\in V(G)}\sum_{y\in \mathcal{N}(x)}H(x,y) = n - 1.$
We have a reformulation for the above result if we define the mean commute time as  $C(x,y) = H(x,y) + H(y,x)$ , then
$$
\frac {1}{2 m} \sum_ {(x, y) \in E (G)} C (x, y) = n - 1 \tag {27}
$$
There is an interesting and well-known connection between mean commute time and resistance between nodes  $r_{xy}$ , see [2, 51].
Theorem B.5.  $C(x,y) = 2mr_{xy}$
In addition, the cover time is defined as  $C \coloneqq \max \{C_v : v \in V\}$ , where  $C_v$  is the expected number of steps taken by a random walk that starts from  $v$  to hit every vertex in the graph. A bound is obtained by Aleiunas et al. (1979),
Corollary B.6. The cover time  $C$  is at most  $2m(n - 1)$
These classical conclusions in node-level random walk provide insights into the design of PE/SE methods such as resistance distance (RD). However, since they have been widely studied, we will not restate them.
C Theoretical analysis and proof
In this section, we provide proofs, details and examples for Section 4, Section 5 and Section 6 in the main text. Note that all the following results are novel and are of great significance in theoretically understanding our methods.
C.1 Random walk on 0-simplices
C.1.1 Theoretical analysis of 0-RWSE
We start with proving the theoretical expressive power of PE/SE based on spatical domain of random walk on 0-simplices. In order to distinguish RWSE for different orders of simplices, we denote the normal node-level RWSE as 0-RWSE, our proposed EdgeRWSE as 1-RWSE, and RWSE based on random walk on  $k$ -simplices as  $k$ -RWSE.
We first introduce 2-FWL, a powerful graph isomorphism test. It assigns colors to all 2-tuples of nodes and iteratively updates them. The initial color  $c_2^0(\pmb{v}, G)$  of tuple  $\pmb{v} \in V(G)^2$  is determined by the isomorphism type of tuple  $\pmb{v}$  [34]. At the  $t$ -th iteration, the color updating scheme is
$$
c _ {2} ^ {t} (\boldsymbol {v}, G) = \operatorname {H a s h} \left(c _ {2} ^ {t - 1} (\boldsymbol {v}, G), \{\} \right.
$$
$$
\left.\left(c _ {2} ^ {t - 1} \left(\psi_ {i} (\boldsymbol {v}, u), G\right) | i \in [ k ]\right) | u \in V (G) \} \right\}\left. \right), \tag {28}
$$
where  $\psi_i(\pmb{v}, u)$  means replacing the  $i$ -th element in  $\pmb{v}$  with  $u$ . The color of  $\pmb{v}$  is updated by its original color and the color of its high-order neighbors  $\psi_i(\pmb{v}, u)$ . The color of the whole graph is the multiset of all tuple colors,
$$
c _ {2} ^ {t} (G) = \operatorname {H a s h} \left(\left\{\left\{c _ {2} ^ {t} (\boldsymbol {v}, G) \mid \boldsymbol {v} \in V (G) ^ {2} \right\} \right\}\right). \tag {29}
$$
Given two function  $f, g$ ,  $f$  can be expressed by  $g$  means that there exists a function  $\phi \circ g = f$ , which is equivalent to given arbitrary input  $H, G$ ,  $f(H) = f(G) \Rightarrow g(H) = g(G)$ . We use  $f \to g$  to denote that  $f$  can be expressed with  $g$ . If both  $f \to g$  and  $g \to f$ , there exists a bijective mapping between the output of  $f$  to the output of  $g$ , denoted as  $f \leftrightarrow g$ .
Theorem C.1. (Theorem 4.1. in main text.) 0-RWSE is strictly less powerful than 2-FWL.
Proof. First, we will prove that 2-FWL can express 0-RWSE. Specifically, we will prove that  $\forall u\in V,c_2^{(t)}(uv,G)\to [(D^{-1}A)_{uv}^k |k = 1,2,\dots,t]\forall t = 1,2,\dots,$ . We prove this by induction on  $t$ .
1. First, 2-FWL can capture degree information in one iteration.
$$
\begin{array}{l} c _ {2} ^ {(1)} (u v, G) \rightarrow c _ {2} ^ {(0)} (u v, G), \left\{\left\{\left(c ^ {(0)} (u w, G), c ^ {(0)} (w v, G)\right) \mid w \in V \right\}\right\} (30) \\ \rightarrow c _ {2} ^ {(0)} (u v, G), \left\{\left\{c ^ {(0)} (u w, G) \mid w \in G \right\}\right\} \quad \rightarrow A _ {u v}, D _ {u} (31) \\ \rightarrow \left(D ^ {- 1} A\right) _ {u v} (32) \\ \end{array}
$$
Therefore,  $c_2^{(1)}(uv,G)\to (D^{-1}A)_{uv}$
2. If  $c_2^{(t)}(uv, G) \to [(D^{-1}A)_{uv}^k]$ , at  $t + 1$ -th 2-FWL iteration.
$$
\begin{array}{l} c _ {2} ^ {(t + 1)} (u v, G) \rightarrow c _ {2} ^ {(t)} (u v, G), \left\{\left\{\left(c ^ {(t)} (u w, G), c ^ {(t)} (w v, G)\right) \mid w \in V \right\}\right\} (33) \\ \rightarrow \left[\left(D ^ {- 1} A\right) _ {u v} ^ {k} \mid k = 1, 2, \dots , t \right], \left\{\left\{\left(D ^ {- 1} A\right) _ {u w} ^ {k} \left(D ^ {- 1} A\right) _ {w v} ^ {k} \mid w \in V \right\}\right\} (34) \\ \rightarrow \left[ \right.\left(D ^ {- 1} A\right) _ {u v} ^ {k} \left. \right\rvert   k = 1, 2, \dots , t ], \sum_ {w} \left(D ^ {- 1} A\right) _ {u w} ^ {k} \left(D ^ {- 1} A\right) _ {w v} ^ {k} (35) \\ \rightarrow \left[ \right.\left(D ^ {- 1} A\right) _ {u v} ^ {k} \left. \right| k = 1, 2, \dots , t + 1 ] (36) \\ \end{array}
$$
Next we prove the strictness, i.e. there exists some non-isomorphic graph pairs distinguishable by 2-FWL but not by RWSE. We will show that the graph pair in Figure 1 satisfies this requirement. Both graphs consist of a square backbone with 4 nodes (hollow in the figure), and each backbone node connects with every node in either a copy of  $X$  or  $Y$ . The graph  $X$  consists of two triangles, while the graph  $Y$  consists of a six-ring. The difference of two graphs is that two backbone nodes connected with the same graph  $X$  or  $Y$  are adjacent in (a), but are not adjacent in (b). These two graphs are non-isomorphic and can be distinguished by 2-FWL, but not by RWSE.
The conclusion of 2-FWL can distinguish graphs (a) and (b) in Figure 1 can be proved by the 3-pebble bijective game [13] on these two graphs, see [39] for details.
Then we show that 0-RWSE fails to distinguish these two graphs step by step.
1. For the eight hollow nodes in the square backbones in (a) and (b), regardless of whether it is connected to a copy of  $X$  or  $Y$ , once we are at the other six nodes in the biclique  $K_{1,6}$  connected with it, the probability of returning to the backbone node for the first time within  $k$  step is  $P_k = \frac{1}{3}^k$ . This can be easily verified since every solid node in  $X$  and  $Y$  has degree 3, therefore the steps it takes to return to the backbone is a geometric distribution parameterized by  $\frac{1}{3}$ .
2. Consider random walks only within the 6 solid nodes in biclique  $K_{1,6}$  and not returning to the backbone. The probability distributions (or PMFs) for all nodes in all copies of  $X$  are the same (which we denote as  $p_{XX}$ ), and PMFs for all nodes in all copies of  $Y$  are the same (which we denote as  $p_{YY}$ ). The PMFs for nodes in  $X$  and  $Y$  are different ( $p_{XX} \neq p_{YY}$ ). This is also straightforward to verify, since nodes in  $X$  can only get to the other two solid nodes if they do not return to the
backbone, while nodes in  $Y$  can get to all nodes. Therefore, RWSE can distinguish subgraphs  $X$  and  $Y$  (not including the connection between these 6 nodes with backbone).
3. For any backbone node in two graphs, it will have probability  $\frac{3}{4}$  to walk toward the connected  $X$  or  $Y$  (but the walk cannot distinguish whether it is connected with  $X$  or  $Y$  according to 1.), and probability  $\frac{1}{4}$  to walk towards another backbone connected with  $X$  or  $Y$  (again, either case is the same since the returning PMF to backbone cannot distinguish  $X$  and  $Y$ ). Combining 1. and 3. and using induction, we can conclude that the returns through biclique or backbone are the same in two graphs, and thus the return probabilities of all eight backbone nodes are the same, which we denote as  $p_B$ .
4. For all nodes in any copies of  $X$ , in each step  $t$ , the return PMF to  $x \in X$  will have probability  $\frac{2}{3}$  depending on  $p_{XX}$ , and probability  $\frac{1}{3}$  depending on  $p_B$ . Once the connected backbone node is reached, the walk returns to the backbone node according to  $p_B$ , then returns to  $X$  with probability  $\frac{3}{4}$ . Therefore, the final return probabilities of  $x \in X$  are the same, depending on  $p_{XX}$  and  $p_B$ . Similarly, the return probabilities of  $y \in Y$  are the same, depending on  $p_{YY}$  and  $p_B$ .
5. Finally, for both two graphs, all backbone nodes are assigned the same 0-RWSE feature as  $\mathbf{b}$ , all  $x \in X$  are assigned the same  $\mathbf{x}$ , and all  $y \in Y$  are assigned  $\mathbf{y}$ . Therefore, the two graphs have  $4\mathbf{b}$ ,  $12\mathbf{x}$  and  $12\mathbf{y}$ , indicating that 0-RWSE fails to distinguish these two graphs.
Corollary C.2. 1-FWL with 0-RWSE initialization is strictly more powerful than 0-RWSE, but is not more powerful than 2-FWL.