SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof. Figure 1 provides a pair of graphs that 1-FWL with 0-RWSE can differentiate, while 0-RWSE cannot. Therefore, 1-FWL with 0-RWSE initialization is strictly more powerful than 0-RWSE. However, as shown in Theorem C.1,  $\exists t > 0$ ,  $c^{(t)}(uu,G) \to RWSE_u$ . Therefore,
$$
\begin{array}{l} c ^ {(t + 1)} (u v, G) \rightarrow c ^ {(0)} (u v, G), \{(c ^ {(t)} (u w, G), c ^ {(t)} (u v, G)) | w \in V \} (37) \\ \rightarrow c ^ {(0)} (u v, G), \left\{c ^ {(t)} (u w, G) | w \in V \right\}, \left\{c ^ {(t)} (w v, G) | w \in V \right\} (38) \\ \rightarrow c ^ {(0)} (u v, G), c ^ {(t)} (u u, G), c ^ {(t)} (v v, G) (39) \\ \rightarrow c ^ {(0)} (u v, G), R W S E _ {u}, R W S E _ {v} (40) \\ \rightarrow c ^ {(0)} (u v, G ^ {R W S E}). (41) \\ \end{array}
$$
where  $G^{RWSE}$  is the graph with the 0-RWSE feature. In other words, 2-FWL can first capture RWSE with a few iterations and then simulate 1-FWL on the graph with RWSE.
Despite its upper bound of 2-FWL, 0-RWSE is capable of distinguishing some cases where 1-FWL fails. For example, the non-isomorphic graph pair shown in Figure 2 is a well-known case that 1-FWL does not distinguish. However, 0-RWSE can easily distinguish them, since a 0-random walk on graph (b) can only visit three nodes within the same triangle, thus having a larger return probability in the third step (and some subsequent steps) than on graph (a). 0-RWSE can also greatly enhance the performance of GNNs in real-world tasks, as shown in experiments.
For the same reason, we can prove that the newly proposed RRWP [33] SE is also strictly less powerful than 2-FWL. RRWP is a natural extension of RWSE, which not only uses return probabilities  $\mathbf{P}_{ii}^{t}$ , but also uses non-diagonal elements  $\mathbf{P}_{ij}^{t}$  as augmented edge features. However, it is still based on node-level random walk and can be completely simulated by 2-FWL. Combining our new results and the results from [51], we provide the following summary for the expressive power of node-level PE/SEs:
- RWSE is strictly less powerful than 2-FWL.
- SPDPE (shortest path distance) is strictly less powerful than 2-FWL.
- RDPE (resistance distance) is strictly less powerful than 2-FWL, but more powerful than SPDPE in distinguishing non-isomorphic distance-regular graphs.
- RRWP is strictly less powerful than 2-FWL, but it is strictly more powerful than SPDPE.
- HK (heat kernel) is strictly less powerful than 2-FWL.
C.1.2 Theoretical analysis of Hodge 0 spectra
As we will analyze Hodge 1-Laplacians in the following text in detail, Hodge 0-Laplacians that have been widely studied are not our main concentrations. In this section, we briefly summarize the properties, aiming to compare with Hodge 1-Laplacians.
Theorem C.3. Hodge 0-isospectra is incomparable with 1-FWL and not more powerful than 2-FWL.
Proof. Rattan and Seppelt [39] has already proved that Hodge 0-isospectra is upper-bounded by 2-FWL. Here we show that graphs in Figure 2 which are indistinguishable by 1-FWL can be distinguished by Hodge 0-isospectra. The characteristic polynomial for  $\mathbf{L}_1$  of graph (a) is
$$
\det  \left(\lambda \mathbf {I} - \mathbf {L} _ {0} (a)\right) = (\lambda - 4) (\lambda - 3) ^ {2} (\lambda - 1) ^ {2} \lambda \tag {42}
$$
The characteristic polynomial for  $\mathbf{L}_1$  of graph (b) is
$$
\det  \left(\lambda \mathbf {I} - \mathbf {L} _ {0} (b)\right) = (\lambda - 8) ^ {9} (\lambda - 4) ^ {6} \lambda \tag {43}
$$
Hence they are distinguishable by Hodge 0 eigenvalues.
However, the graph pair in Figure 4 that can be distinguished by 1-FWL cannot be distinguished by Hodge 0 eigenvalues, since the characteristic polynomial are both
$$
\det  \left(\lambda \mathbf {I} - \mathbf {L} _ {0} (a)\right) = \det  \left(\lambda \mathbf {I} - \mathbf {L} _ {0} (b)\right) = \lambda (\lambda - 3) (\lambda - 1) ^ {2} \left(\lambda^ {3} - 9 \lambda^ {2} + 2 1 \lambda - 7\right) \tag {44}
$$
Therefore, there exist some 1-FWL distinguishable graph pairs that cannot be distinguished by Hodge-0 isospectra and vice versa.
C.2 Random walk on 1-simplices
C.2.1 Theoretical analysis of 1-RWSE
We start by proving the expressive power of 1-RWSE (Full EdgeRWSE and its variations).
Theorem C.4. (First part of Theorem 5.1 in the main text). Full EdgeRWSE can distinguish some non-isomorphic graphs that are indistinguishable by 2-FWL.
Proof. We will show that a pair of strongly regular graphs parameterized by (16, 6, 2, 2),  $4 \times 4$  Rook's graph and Shrikhande's graph (shown in Figure 3), are indistinguishable by 2-FWL but distinguishable by 1-RWSE. 2-FWL fails to distinguish any pair of strongly regular graphs with same parameters, which is a classical result, hence we refer readers to [8] for a complete proof and skip here.
Now we show that the full 1-RWSE is able to distinguish these two graphs. Only the 1-up random walk (i.e. walk towards those 1-up neighbors via shared 2-faces) needs to be considered, since the 1-down random walks on two graphs are identical (see Theorem C.5). For 1-up random walk, note that all edges in the same graph share the same status, thus we only need to analyze the random walk rooted at one arbitrary edge in two graphs, respectively. All edges in two graphs are contained in two different 2-simplices (triangles). However, the two 2-faces in (a)  $4 \times 4$  Rook's graph belong to the same 3-simplex or 4-clique, but there are no 4-cliques in (b) Shrikhande's graph. Note that all 2-simplices in graph (a) are only connected to the 2-simplices that are in the same 3-simplex; consequently, an 1-up random walk on graph can only visit 4 different 2-simplices (which are all faces of the 3-simplex or 4-clique the source edge is in), and can only visit 6 different edges in the 4-clique including the source itself. But the 2-simplices in graph (b) are all connected, so the 1-up random walk has the probability of visiting all triangles and edges in the graph. Therefore, as time tends to infinity, the limit return probability of a 1-up random walk on  $4 \times 4$  Rook's graph is
$$
\lim  _ {t \rightarrow \infty} p _ {1, u p} ^ {\mathrm {R}} (t) = \frac {1}{6} \tag {45}
$$
The limit return probability of a 1-up random walk on Shrikhande's graph is
$$
\lim  _ {t \rightarrow \infty} p _ {1, u p} ^ {\mathrm {S}} (t) = \frac {1}{4 8} \tag {46}
$$
Also, one can verify the difference on two walks via the return probabilities at the third step (or any other following steps, since the return probability on Rook's graph is always larger than that of Shrikhande's graph when  $t \geq 3$ ):
$$
p _ {1, u p} ^ {\mathrm {R}} (3) = \frac {1}{8}, p _ {1, u p} ^ {\mathrm {S}} (3) = \frac {1}{1 6} \tag {47}
$$
Therefore, we have shown 1-up random walk can distinguish these two graphs, so can 1-random walk. Therefore, 1-RWSE can distinguish this pair indistinguishable from 2-FWL.
Note that the 1-down RWSE is unable to distinguish  $4 \times 4$  Rook's graph and Shrikhande's graph, since it does not consider any 2-simplicial faces. Similarly, 0-RWSE is also unable to distinguish them, which is consistent with our previous results. Actually, both 1-down RWSE and 0-RWSE are upper bounded by 2-FWL, see below.
Theorem C.5. (Second part of Theorem 5.1 in the main text.) 1-down RWSE is not more powerful than 2-FWL.
Proof. When the graph has no triangle, the edge level random walk matrix is
$$
P _ {1} = B _ {1} ^ {T} D _ {1} ^ {- 1} B _ {1} \tag {48}
$$
while the node level random walk matrix is
$$
P _ {0} = D _ {1} ^ {- 1} B _ {1} B _ {1} ^ {T} \tag {49}
$$
Therefore,  $\forall l,k$ $tr(P_1^{kl}) = tr(P_0^{kl})$ , and thus we can determine the multiset of  $\{(P_1^l)_{[u,v],[u,v]}|[u,v]\in E\}$ .
Now we give more discussion on random walks on 1-simplices. We start by further interpretation of the normalized Hodge 1-Laplacian and the edge-level random walk that corresponds. The transition matrix defined by  $-\frac{1}{2}\tilde{\mathbf{L}}_{\mathbf{1}}\mathbf{V}^{T} = \mathbf{V}^{T}\hat{\mathbf{P}}$  (see Section 5.1 in the main text) can be interpreted as follows:  $\hat{\mathbf{P}} = \frac{1}{2}\mathbf{P}_{down} + \frac{1}{2}\mathbf{P}_{up}$ , where  $\mathbf{P}_{down}$  and  $\mathbf{P}_{up}$  are the transition matrices determined by a lower-adjacent and upper-adjacent random walk. Specifically, in each step with probability of 0.5 each, we take a step toward either upper or lower adjacent edges. (1) If the step is taken towards the upper adjacent face (2-simplex or oriented triangle), there are two cases: (1a) the edge has an upper adjacent face, then the edge uniformly transits to an upper adjacent edge with different orientation relative to the shared face. (2a) the edge does not have upper adjacent faces, then the walk will keep the same orientation of this edge or change the orientation with equal probability 0.5 each. (2) If the step is taken toward the lower adjacent face (0-simplex or node), the walk transits along or against the edge direction to the lower adjacent nodes with each probability 0.5. Then from the selected node, the walk further transmits to the target edges connected with the node with probability proportional to the upper degrees or the weights of the target edges.
One can easily verify that the limit distribution of the directed 1-down random walk (case 1) is the uniform distribution at all edges, independent of the starting position. In comparison, the limit distribution of the undirected 1-down random walk (case 2) is proportional to the number of 1-down neighbors of the edges (i.e., the number of edges that share one node with the interested edge), thus providing no information except the number of neighboring edges. However, the initial steps of both two types of edge-level 1-down random walk are capable of providing rich information on the graph structure.
Here we further explain why simplified 1-down EdgeRWSE variations are appropriate in the cases where the SCs contain few 2-simplices (triangles). Recall that the full edge-level 1-random walk will have probability 0.5 to go through the upper adjacent 2-faces, but stays at the same edge (with the same probabilities to keep or change direction) if the edge is not a face of the 2-complex. In this case, the full edge-level random walk described by  $\hat{\mathbf{P}}$  will almost have probability 0.5 to stay on the same edge (or the edge with an inverse direction if the graph is directed). This results in the walk having larger probability to stay still, or the walk becomes 'lazy' and provides less information about
surrounding structures. In comparison, the simplified walk variations neglect the upper adjacent 2-faces, thus having a greater probability of moving out of the source edge and exploring more about structure information. By reducing  $\hat{\mathbf{P}}$  to  $\mathbf{P}_{\text{down}}$ , the simplified walk has more probability of moving away from the source, encouraging exploration of more information about the structure.
C.2.2 Theoretical analysis of Hodge1Lap
We start analyzing the spectral properties of Hodge 1-Laplacians with proving the theorems in the main text. The following theorem discusses the relationship between spectra of Hodge 0-Laplacian  $\mathbf{L}_0$  and Hodge 1-Laplacian  $\mathbf{L}_1$ .
Theorem C.6. (Theorem 5.2 in the main text.) The number of non-zero eigenvalues of Hodge 1-Laplacian  $\mathbf{L}_1$  is not less than the number of non-zero eigenvalues of Hodge 0-Laplacian  $\mathbf{L}_0$ .
Proof. This is a direct conclusion from Equation (21) and Equation (22), which we have proved in Appendix A.
A direct conclusion is that graph isomorphism based on Hodge 1-Laplacian isospectra (we only discuss the multiset of eigenvalues) is strictly more powerful than Hodge 0-Laplacian. The precondition here is that the two graphs have the same number of nodes and edges, or we can construct counter examples that two non-isomorphic graphs have the same  $\mathbf{L}_1$  but have different numbers of nodes and different  $\mathbf{L}_0$  (their  $\mathbf{L}_0$  may have different dimensions and different multiplicities of zero eigenvalues if two graphs only differ from some extra isolated nodes). Under this condition, Equation (21) and Equation (22) state that if we can distinguish two graphs by their eigenvalues of Hodge 0-Laplacians, we can also do so by eigenvalues of Hodge 1-Laplacians. For the strictness, the graph pair in Figure 3 can be distinguished by Hodge 1 isospectra but not by Hodge 0 isospectra, see below.
Theorem C.7. (Theorem 5.3 in main text.)  $\mathbf{L}_1$  isospectra is incomparable with 1-FWL and 2-FWL.
Proof. This theorem compares the expressive power of  $\mathbf{L}_1$  isospectra with the traditional FWL hierarchy. First, we show that there are some non-isomorphic graph pairs distinguishable by 1-FWL and 2-FWL cannot be distinguished by  $\mathbf{L}_1$  isospectra, see the example in Figure 4. 1-FWL can distinguish them for the following reasons. The first iteration of 1-FWL (equivalently, 1-WL) can hash the nodes of different degrees into different colors. Then in the second iteration, in graph (b) there is one node with degree 3 that owns two neighbors with degree 1. However, there is no such node in graph (a); thus, the two graphs have different multisets of colors after 2 iterations of 1-WL.
Therefore, both 1-FWL and 2-FWL can distinguish them. However, we do not distinguish them by the multisets of eigenvalues of  $\mathbf{L}_1$ . Both of their characteristic polynomials of  $\mathbf{L}_1$  are:
$$
\det  \left(\lambda \mathbf {I} - \mathbf {L} _ {1} (a)\right) = \det  \left(\lambda \mathbf {I} - \mathbf {L} _ {1} (b)\right) = (\lambda - 3) ^ {2} (\lambda - 1) ^ {2} \left(\lambda^ {3} - 9 \lambda^ {2} + 2 1 \lambda - 7\right) \tag {50}
$$
Thus, they have identical multisets of eigenvalues of  $\mathbf{L}_1$ .
On the other hand, we show that  $\mathbf{L}_1$  isospectra can distinguish strongly regular graphs in Figure 3, while 2-FWL and 1-FWL are known to fail on them. The characteristic polynomial for Hodge 1 Laplacian of  $4\times 4$  Rook's graph (which we denote as  $\mathbf{L}_1(R)$ ) is
$$
\det  \left(\lambda \mathbf {I} - \mathbf {L} _ {1} (R)\right) = (\lambda - 8) ^ {9} (\lambda - 4) ^ {3 0} \lambda^ {9} \tag {51}
$$
The characteristic polynomial of Shrikhande's graph's Hodge 1-Laplacian  $\mathbf{L}_1(S)$  is
$$
\det  \left(\lambda \mathbf {I} - \mathbf {L} _ {1} (S)\right) = (\lambda - 8) ^ {9} (\lambda - 6) (\lambda - 3 - \sqrt {5}) ^ {6} (\lambda - 4) ^ {1 5} (\lambda - 2) ^ {9} (\lambda - 3 + \sqrt {5}) ^ {6} \lambda^ {2} \tag {52}
$$
Obviously, their multisets of eigenvalues are different, hence Hodge 1-isospectra can distinguish some non-isomorphic graph pairs that even 2-FWL fail.
Now we turn our attention to the physical meanings and insights of the spectra of Hodge 1-Laplacians. Recall that zero eigenvalues and corresponding eigenvectors are associated with harmonic functions, and Hodge1Lap is able to figure out the cycles in the graph by projecting a unit vector into the subspace spanned by zero-eigenvalue eigenvectors (or equivalently, the kernel space of  $\mathbf{L}_1$ ). Furthermore, the near-zero eigenvalues and their eigenvectors are associated with near-harmonic functions. The eigenvectors of large eigenvalues are non-homology generators for local structures such as clusters, stars, chains, and so on.
The following theorems state the properties of the Hodge 1-Laplacian and its harmonic space.
Theorem C.8. In the Hilbert space, the  $\mathbf{L}_1$  of a directed graph without multiple edges or self-loops is a  $m\times m$  matrix, where  $m$  is the number of directed edges. The elements of  $\mathbf{L}_1$  satisfy: (1) the  $i$ -th diagonal element corresponding to the edge  $e_i$  is  $2 + |\delta_1(e_i)|$ , where  $|\delta_{1}(e_{i})|$  is the number of 2-simplices  $e_i$  is in; (2) element  $\mathbf{L}_1[i,j] = 1$  if  $e_i$  and  $e_j$  are 1-down adjacent (share one node but not in the same 2-simplex) and two edges have the directions starting from the shared node; (3)  $\mathbf{L}_1[i,j] = -1$  if  $e_i$  and  $e_j$  are 1-down adjacent but have different directions starting from the shared node; (4)  $\mathbf{L}_1[i,j] = 0$  if  $e_i$  and  $e_j$  are 1-up adjacent (in the same 2-simplex).
Proof. This can be directly verified by the definition of  $\mathbf{L}_1 = \mathbf{B}_k^*\mathbf{B}_k + \mathbf{B}_{k + 1}\mathbf{B}_{k + 1}^*$ .
We take Figure 5 (a) as an illustrative example of calculating  $\mathbf{L}_1$ . This is a simplicial complex of order  $K = 2$ . When writing matrices, we follow the order  $a, b, c, d$  for 0-simplices (nodes) and the order  $1, 2, 3, 4$  for 1-simplices (directed edges). The node-to-edge incidence matrix is
$$
\mathbf {B} _ {1} = \left[ \begin{array}{c c c c} - 1 & 0 & 0 & 0 \\ + 1 & + 1 & 0 & - 1 \\ 0 & - 1 & - 1 & 0 \\ 0 & 0 & + 1 & + 1 \end{array} \right] \tag {53}
$$
Therefore the Hodge-0 Laplacian is (recall that  $\mathbf{B}_0$  is not defined and is omitted in calculating  $\mathbf{L}_0$ )
$$
\mathbf {L} _ {0} = \mathbf {B} _ {1} \mathbf {B} _ {1} ^ {*} = \left[ \begin{array}{r r r r} + 1 & - 1 & 0 & 0 \\ - 1 & + 3 & - 1 & - 1 \\ 0 & - 1 & + 2 & - 1 \\ 0 & - 1 & - 1 & + 2 \end{array} \right] = \mathbf {D} - \mathbf {A} \tag {54}
$$
where  $\mathbf{D}$  is the diagonal matrix with node degrees as diagonal elements, and  $\mathbf{A}$  is the adjacency matrix (in the normal sense). This is completely identical to the widely used graph Laplacian.
Now we come to compute  $\mathbf{L}_1$ . There is one 2-simplex in the simplicial complex, and we take the direction  $c \to b \to d$  as the positive orientation of the 2-simplex, then
$$
\mathbf {B} _ {2} = \left[ \begin{array}{l} 0 \\ + 1 \\ - 1 \\ + 1 \end{array} \right] \tag {55}
$$
and we have the Hodge-1 Laplacian
$$
\mathbf {L} _ {1} = \mathbf {B} _ {1} ^ {*} \mathbf {B} _ {1} + \mathbf {B} _ {2} \mathbf {B} _ {2} ^ {*} = \left[ \begin{array}{l l l l} + 2 & + 1 & 0 & - 1 \\ + 1 & + 2 & + 1 & - 1 \\ 0 & + 1 & + 2 & + 1 \\ - 1 & - 1 & + 1 & + 2 \end{array} \right] + \left[ \begin{array}{l l l l} 0 & 0 & 0 & 0 \\ 0 & + 1 & - 1 & + 1 \\ 0 & - 1 & + 1 & - 1 \\ 0 & + 1 & - 1 & + 1 \end{array} \right] = \left[ \begin{array}{l l l l} + 2 & + 1 & 0 & - 1 \\ + 1 & + 3 & 0 & 0 \\ 0 & 0 & + 3 & 0 \\ - 1 & 0 & 0 & + 3 \end{array} \right] \tag {56}
$$
which is identical to the description in Theorem C.8. It can be seen that  $\mathbf{B}_1^*\mathbf{B}_1$  contributes to (1), (2), (3) in Theorem C.8;  $\mathbf{B}_2\mathbf{B}_2^*$  contributes to (1) and (4) in Theorem C.8. In some sense, if  $e_i$  and  $e_j$  are both 1-down adjacent and 1-up adjacent, the elements  $e_{ij}$  or  $e_{ji}$  in  $\mathbf{B}_1^*\mathbf{B}_1$  and  $\mathbf{B}_2\mathbf{B}_2^*$  cancel out each other, resulting  $e_{ij} = e_{ji} = 0$  if  $e_i$  and  $e_j$  belong to the same 2-simplex.
Next, we declare some properties about the kernel space of  $\mathbf{L}_1$  (i.e., the 1-cohomology group). Note that the following cycles are that of length larger than 4 since directed triangles are considered as 2-simplices, and in the sense of the undirected graph induced by the directed graph (i.e., we do not care about the directions of edges in the cycle, since any changes of direction are equivalent multiplying  $-1$  on the original basis).
Theorem C.9. For a directed graph without multiple edges, self-loops or 2-simplices that contains  $c$  cycles, and there are no cycles that share edges, then a group of linearly independent and orthogonal basis of the kernel space of  $\mathbf{L}_1$  is  $\mathbf{u}_i$ ,  $i = 1,\dots ,c$ , where elements in  $\mathbf{u}_i$  corresponding to the edges on the  $i$ -th cycle is  $\pm \frac{1}{\sqrt{l(i)}}$  and 0 otherwise, where  $l(i)$  is the length of the  $i$ -th cycle.
Proof. We have already stated that the direction of edges only affects the sign of corresponding element in eigenvectors, WLOG, we suppose that all edges in the  $i$ -th cycle follow the anticlockwise direction. Then all nonzero elements on the same basis  $\mathbf{u}_i$  have identical signs. We are going to prove this.
Obviously,  $\mathbf{u}_i, i = 1, \dots, c$  are linearly independent and orthogonal, since the cycles do not share edges, the nonzero elements in  $\mathbf{u}_i$  do not overlap consequently. Therefore, we only need to prove that  $\mathbf{L}_1\mathbf{u}_i = 0$ . (1) For the  $j$ -th row where edge  $e(j)$  is not in the  $i$ -th cycle, there are two cases: (1a)  $e(j)$  is not connected with any edges in the  $i$ -th cycle, then obviously the inner product of  $j$ -th row of  $\mathbf{L}_1$  and  $\mathbf{u}_i$  is zero, since all nonzero elements in the  $j$ -th row of  $\mathbf{L}_1$  corresponds to zero elements in  $\mathbf{u}_i$ ; (1b)  $e(j)$  is connected with some edges in the cycle, then the only situation is that  $e(j)$  connects with two edges  $e(j_1), e(j_2)$ , where the shared node of these three edges is the source node of  $e(j_1)$  and is the target node of  $e(j_2)$ , and the  $j$ -th row of  $\mathbf{L}_1[j]$  will have  $-1$  in element  $j_1$ ,  $+1$  in element  $j_2$ . Therefore, the inner product  $\mathbf{L}_1[j]^T \cdot \mathbf{u}_i = 0 + 2 \times 0 + (-1) \times \frac{1}{\sqrt{l(i)}} + 1 \times \frac{1}{\sqrt{l(i)}} = 0$ . (2) If  $e(j)$  is in the cycle  $i$ , then similar to (1b), the non-zero index in both  $\mathbf{L}_1[j]^T$  and  $\mathbf{u}_i$  is the index of two neighboring nodes of  $e(j)$ , hence  $\mathbf{L}_1[j]^T \cdot \mathbf{u}_i = 0 + 2 \times \frac{1}{\sqrt{l(i)}} + 2 \times (-1) \times \frac{1}{\sqrt{l(i)}} = 0$ . Putting all the pieces together, we have proved  $\mathbf{L}_1\mathbf{u}_i = 0$ .
Figure 5 (b) represents an example that satisfies the above conditions in the theorem. For illustration, we calculate its  $\mathbf{L}_1$
$$
\mathbf {L} _ {1} = \left[ \begin{array}{c c c c c} + 2 & - 1 & - 1 & 0 & 0 \\ - 1 & + 2 & 0 & + 1 & - 1 \\ - 1 & 0 & + 2 & + 1 & 0 \\ 0 & + 1 & + 1 & + 2 & - 1 \\ 0 & - 1 & 0 & - 1 & + 2 \end{array} \right] \tag {57}
$$
It is straightforward to verify that the eigenvector corresponding to the zero eigenvalue is
$$
\mathbf {u} _ {0} = \pm \frac {1}{2} [ 1, 1, 1, - 1, 0 ] ^ {\top} \tag {58}
$$
The elements corresponding to edges in a cycle (edge 1, 2, 3, 4) are non-zero, while other elements are strictly zero. Therefore, Hodge1Lap is capable of detecting cycles through the eigenvectors in  $\ker (\mathbf{L}_1)$ .
Corollary C.10. Suppose that the above conditions hold, denote  $\mathbf{U} = [\mathbf{u}_1,\dots ,\mathbf{u}_c]$ . Then for a vector  $\mathbf{e}$  of length  $m$  and all elements 1, then projection  $\left|\mathbf{UU}^T\right|_{\mathbf{e}}$  has  $j$ -th element 1 if  $e(j)$  is in a cycle and 0 otherwise. indicates taking the absolute values element-wise.
Proof. This is a direct conclusion of Theorem C.9. Note that  $|\mathbf{U}\mathbf{U}^T|$  has element  $|\mathbf{U}\mathbf{U}^T|[i,j] = \frac{1}{l(c_{ij})}$  if both edges  $i$  and  $j$  are in the cycle  $c_{ij}$ , and 0 otherwise. Hence the projection of  $\mathbf{e}$  in the kernel space has element 1 if the corresponding edge is in a cycle and 0 otherwise, which means that this projection can efficiently mark out cycles. Note that the projection matrix is actually independent of the choice of basis; hence the above conclusion is universal. For the basis matrix  $\mathbf{V}$  where the columns are independent but not orthogonal, the projection matrix should be calculated as  $\mathbf{V}(\mathbf{V}^T\mathbf{V})^{-1}\mathbf{V}^T$ .
Corollary C.11. Suppose that the above conditions hold, except that two cycles  $c_1, c_2$  share one edge  $e(j)$ . Then in every eigenvector  $\mathbf{u}$  in the kernel space of  $\mathbf{L}_1$ ,  $\mathbf{u}_j = \gamma_1 - \gamma_2$ , where  $\gamma_1, \gamma_2$  are the edge flow and are positive following the anticlockwise direction,
$$
\left| \mathbf {u} _ {i} (i \neq j) \right| = \left\{ \begin{array}{l} \gamma_ {1}, e (i) \in c _ {1} \\ \gamma_ {2}, e (i) \in c _ {2} \\ 0, e (i) \notin c _ {1}, e (i) \notin c _ {2} \end{array} \right. \tag {59}
$$
Further,  $(\mathbf{U}\mathbf{U}^T\mathbf{e})_j = 0$