Title: Non-convolutional graph neural networks.
Venue: NeurIPS 2024
Accepted: yes
Keywords: Graph neural networks
Abstract: Rethink convolution-based graph neural networks (GNN)---they characteristically suffer from limited expressiveness, over-smoothing, and over-squashing, and require specialized sparse kernels for efficient computation.
Here, we design a simple graph learning module entirely free of convolution operators, coined _random walk with unifying memory_ (RUM) neural network, where an RNN merges the topological and semantic graph features along the random walks terminating at each node.
Relating the rich literature on RNN behavior and graph topology, we theoretically show and experimentally verify that RUM attenuates the aforementioned symptoms and is more expressive than the Weisfeiler-Lehman (WL) isomorphism test.
On a variety of node- and graph-level classification and regression tasks, RUM not only achieves competitive performance, but is also robust, memory-efficient, scalable, and faster than the simplest convolutional GNNs.

Non-convolutional Graph Neural Networks

Abstract

Rethink convolution-based graph neural networks (GNN)-they characteristically suffer from limited expressiveness, over-smoothing, and over-squashing, and require specialized sparse kernels for efficient computation. Here, we design a simple graph learning module entirely free of convolution operators, coined random walk with unifying memory (RUM) neural network, where an RNN merges the topological and semantic graph features along the random walks terminating at each node. Relating the rich literature on RNN behavior and graph topology, we theoretically show and experimentally verify that RUM attenuates the aforementioned symptoms and is more expressive than the Weisfeiler-Lehman (WL) isomorphism test. On a variety of node- and graph-level classification and regression tasks, RUM not only achieves competitive performance, but is also robust, memory-efficient, scalable, and faster than the simplest convolutional GNNs.

1 Introduction: Convolutions in GNNs

Graph neural networks (GNNs) [1, 2, 3, 4, 5]-neural models operating on representations of nodes ( V ) and edges ( E ) in a graph (denoted by G = {V , E} , E ⊆ V × V , with structure represented by the adjacency matrix A ij = ✶ [( v i , v j ) ∈ E ] )-have shown promises in a wide array of social and physical modeling applications. Most GNNs follow a convolutional scheme, where the D -dimensional node representations X ∈ R | V |× D are aggregated based on the structure of local neighborhoods:

(1)

Here, ˆ A displays a unique duality-the input features doubles as a compute graph. The difference among convolutional GNN architectures, apart from the subsequent treatment of the resulting intermediate representation X ′ , typically amounts to the choices of transformations ( ˆ A ) of the original adjacency matrix ( A )-the normalized Laplacian for graph convolutional networks (GCN) [1], a learned, sparse stochastic matrix for graph attention networks (GAT) [6], powers of the graph Laplacian for simplifying graph networks (SGN) [7], and the matrix exponential thereof for graph neural diffusion (GRAND) [8], to name a few. For all such transformations, it is easy to verify that permutation equivariance (Equation 9) is trivially satisfied, laying the foundations of data-efficient graph learning. At the same time, this class of methods share common pathologies as well:

Limited expressiveness. (Figure 1) Xu et al. [2] groundbreakingly elucidates that GNNs cannot exceed the expressiveness of Weisfeiler-Lehman (WL) isomorphism test [9]. Worse still, when the support of neighborhood multiset is uncountable, no GNN with a single aggregation function can be

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as expressive as the WL test [10]. As such, crucial local properties of graphs meaningful in physical and social modeling, including cycle sizes (Example 8.1) and diameters (Example 8.2) [11], cannot be realized by convolution-based GNNs.

Over-smoothing. (Figure 2) As one repeats the convolution (or Laplacian smoothing) operation (Equation 1), sandwiched by linear and nonlinear transformations, the inter-node dissimilarity, measured by Dirichlet energy,

will decrease exponentially as the number of message-passing steps l increases [12, 13], E ( X ( l ) ) ≤ C 1 exp( -C 2 l ) with some constants C 1 , 2 , resulting in node representations only dependent upon the topology, but not the initial embedding.

Over-squashing. (Figure 5) As the number of Laplacian smoothing grows, the receptive field of GNNs increases exponentially, while the dimension of node representation, and thereby the possible combinations of neighborhood environment, stays unchanged [14]. Quantitatively, Topping et al. [15] quantifies this insight using the inter-node Jacobian and relates it to the powers of the adjacency matrix through sensitivity analysis :

where ϕ is the node-wise update function, whose Jacobian is typically diminishing. If this Equation 3 converges to zero, the node representation is agnostic to the changes happening l edges away, making convolutional GNNs difficult to learn long-range dependencies.

Main contributions: Non-convolutional GNNs as a joint remedy. In this paper, we propose a variant of GNN that does not engage the convolution operator (Equation 1) at all. 1 Specifically, we stochastically sample a random walk terminating at each node and use the anonymous experiment associated with the random walk to describe its topological environment. This is combined with the semantic representations along the walk and fed into a recurrent neural network layer [16] to form the node embedding, which we call the unifying memory . We theoretically (§ 4) and experimentally (§ 5) show that the resulting model, termed random walk with unifying memory (RUM) relieves the aforementioned symptoms and offers a compelling alternative to the popular convolution-based GNNs.

2 Related works: ways to walk on a graph

Walk-based GNNs. RAW-GNN ([17], compared and outperformed in Table 9) also proposes walk-based representations for representing node neighborhoods, which resembles our model without the anonymous experiment ( ω u in Equation 5). CRaWl ([18, 19], outperformed in Table 3) also incorporates a similar structural encoding for random walk-generated subgraphs to feed into an

iterative 1-dimensional convolutional network. AWE ([20], Table 3) and Wang et al. [21], like ours, use anonymous experiments for graph-level unsupervised learning and temporal graph learning, respectively. More elaborately, [22] and AgentNet ([23], Table 3) use agent-based learning on random walks and paths on graphs.

Random walk kernel GNNs. In a closely related line of research, RWK ([24], Table 3) employs the reproducing kernel Hilbert space representations in a neural network for graph modeling and GSN ([25], Table 3) counts the explicitly enumerated subgraphs to represent graphs. The subgraph counting techniques intrinsically require prior knowledge about the input graph of a predefined set of node and edge sets. For these works, superior expressiveness has been routinely argued, though usually limited to a few special cases where the WL test fails whereas they do not, and often within the unlabelled graphs only.

More importantly, focusing mostly on expressiveness, no aforementioned walk-based or random walk kernel -based GNNs address the issue of over-smoothing and over-squashing in GNNs. Some of these works are also de facto convolutional, as the random walks are only incorporated as features in the message-passing scheme. Interestingly, most of these works are either not applicable to, or have not been tested on, node-level tasks. In the experimental § 5, we show that RUM not only outperforms these baselines on most graph-level tasks (Table 3) but also competes with a wide array of state-of-the-art convolutional GNNs on node-level tests (Table 2). Moreover, random walk kernelbased architectures, which explicitly enumerate random walk-generated subgraphs are typically much slower than WL-equivalent convolutional GNNs, whereas RUM is faster than even the simplest variation of convolutional GNN (Figure 4).

Graph transformers. Graph transformers [26, 27]-neural models that perform attention among all pairs of nodes and encode graph structure via positional encoding-are well-known solutions that are not locally convolutional. Their inductive biases determine that over-smoothing and over-squashing among local neighborhoods are, like RUM, also not prominent.

Because of its all-to-all nature, the runtime complexity of graph transformers, just like that of almost all transformers, contains a quadratic term, w.r.t. the size of the system (number of nodes, edges, or subgraphs). This makes it prohibitively expensive and memory intensive on large social graphs (such as that used in Table 8 with millions of nodes). On smaller social graphs, we show in Tables 9, 4 that graph transformers are experimentally outperformed by RUM.

State-of-the-art methods to alleviate oversmoothing. Stochastic regularization. DropEdge ([28], Figure 2) regularizes the smoothing of the node representations by randomly disconnecting edges. Its associated Dirichlet energy indeed decreases slower, though eventually still diminishes as the number of layers increases. Graph rewiring. [29, 30] and GPR-GNN ([31], Appendix Table 9) rewire the graph using personalized page rank algorithm [32] and generalized page rank on graphs, respectively. Similar to JKNet [33], they mitigate over-smoothing by allowing direct message passing between faraway nodes. Constant-energy methods. Zhao and Akoglu [34], Rusch et al. [35] constrain the pair-wise distance or Dirichlet energy among graphs to be constant. Nevertheless, the non-decreasing energy does not necessarily translate to better performance, as they sometimes come with the sacrifice of expressiveness, as argued in Rusch et al. [13]. Residual GNNs. Residual connection [36] can naturally be added to the GNN architecture, such as GCNII ([37], Table 2), to restraint activation to be similar to the input to allow deep networks. They however can make the model less robust to perturbations in the input. In sum, these works have similar, if not compromised expressiveness compared to a barebone GCN.

3 Architecture: combining topologic and semantic trajectories of walks

Random walks on graphs. An unbiased random walk w on a graph G is a sequence of nodes w = ( v 0 , v 1 , . . . ) with landing probability:

P ( v j | ( v 0 , . . . , v i -1 )) = ✶ [( v i , v j ) ∈ E G ] /D ( v i ) , (4) where D ( v i ) = ∑ A ij is the degree of the node v i . Walks originating from or terminating at any given node v can thus be easily generated using this Markov chain. We record the trajectory of embeddings associated with the walk as ω x ( w ) = ( X i ) = ( X 0 , X 1 , . . . , X l ) . In this paper, we only

̸

consider finite-long l -step random walk | w | = l ∈ Z + . In our implementation, the random walks are sampled ad hoc during each training and inference step directly on GPU using Deep Graph Library [38] (see Appendix § A). Moreover, the walk considered here is not necessarily a path , as repeated traversal of the same node v i = v j , i = j is not only permitted, but also crucial to effective topological representation, as discussed below.

Anonymous experiment. We use a function describing the topological environment of a walk, termed anonymous experiment [39], ω u ( w ) : R l → R l that records the first unique occurrence of a node in a walk (Appendix Algorithm C). To put it plainly, we label a node as the number of unique nodes insofar traversed in a walk if the node has not been traversed, and reuse the label otherwise. Practically, this can be implemented using any tensor-accelerating framework in one line ( w is the node sequence of a walk) and trivially parallelized 2 : (1*(w[..., :,None]==w[..., None,:])).argmax(-1)

Unifying memory: combining semantic and topological representations. Given any walk w , we now have two sequences ω x ( w ) and ω u ( w ) describing the semantic and topological (as we shall show in the following sections) features of the walk. We project such sequential representations onto a latent dimension to combine them (illustrated in Table 1):

where ϕ x : R l × D → R D x maps the sequence of semantic embeddings generated by a l -step walk to a fixed D x -dimensional latent space, ϕ u : R l → R D u maps the indicies sequence to another latent space D u , and f : R D x ⊕ R D u → R D combines them. We call Equation 5 the unifying memory of a random walk. Subsequently, the node representations can also be formed as the average representations of l -step ( l being a hyperparameter) walks terminating (for the sake of gradient aggregation) at that node:

which can be stochastically sampled with unbiased Monte Carlo gradient and used in downstream tasks as such node classification and regression. We note that this is the only time we perform SUM or MEAN operations. Unlike other GNNs incorporating random walk-generated features (which are sometimes still convolutional and iterative), we do not iteratively pool representations within local neighborhoods. The likelihood of the data written as:

The node representation can be summed

to form global representations for graph classification and regression. We call ψ in Equation 6 and Ψ in Equation 8 the node and graph output representations of RUM.

Layer choices. Obvious choices to model f include a feed-forward neural network after concatenation, and ϕ x , ϕ u recurrent neural networks (RNNs). This implies that, different from most convolutional GNNs, parameter sharing is natural and the number of parameters is going to stay constant as the model incorporates a larger neighborhood. Compared to dot product-based, transformer-like modules [40], RNNs not only have linear complexity (see detailed discussions below) w.r.t. the sequence length but also naturally encodes the inductive bias that nodes closer to the origin have stronger impact on the representation. The gated recurrent unit (GRU)[16] variant is used everywhere in this paper. Additional regularizations are described in Appendix § B.1.

Runtime complexity. To generate random walks for one node has the runtime complexity of O (1) , and for a graph O ( |V| ) , where |V| is the number of nodes in a graph G = {V , E} . To calculate the anonymous experiment , as shown in Appendix Algorithm C, has O (1) complexity (also see Footnote 2). If we use linear-complexity models, such as RNNs, to model ϕ x , ϕ u , the overall complexity is O ( |V| lkD ) where l is the length of the random walk, k the samples used to estimate Equation 6, and D the latent size of the model (assumed uniform). Note that different from convolutional GNNs, RUM does not depend on the number of edges |E| (which is usually much larger than |V| ) for runtime complexity, and is therefore agnostic to the sparsity of the graph. See Figure 4 for an empirical verification of the time complexity. In Appendix Table 8, we show, on a large graph, the overhead introduced by generating random walks and computing anonymous experiments accounts for roughly 1 / 1500 of the memory footprint and 1 / 8 of the wall time.

Mini-batches. RUM is naturally compatible with mini-batching. For convolutional GNNs, large graphs that do not fit into the GPU memory have traditionally been a challenge, as all neighbors are required to be present and boundary conditions are hard to define [41]. RUM, on the other hand, can be inherently applied on subsets of nodes of a large graph without any alteration in the algorithm-the random walks can be generated on a per-node basis, and the FOR loop in Algorithm C can be executed sequentially, in parallel, or on subsets. Empirically, in Appendix Table 8, RUM can be naturally scaled to huge graphs.

4 Theory: RUM as a joint remedy.

We have insofar designed a new graph learning framework-convolution-free graph neural networks (GNNs) that cleanly represent the semantic ( ω x ) and topological ( ω u ) features of graph-structured data before unifying them. First, we state that RUM is permutation equivariant,

Remark 1 (Permutation equivariance) . For any permutation matrix P , we have

which sets the foundation for the data-efficient modeling of graphs. Next, we theoretically demonstrate that this formulation jointly remedies the common pathologies of the convolution-based GNNs by showing that: (a) the topological representation ω u is more expressive than convolutional-GNNs in distinguishing non-isomorphic graphs; (b) the semantic representation ω x no longer suffers from over-smoothing and over-squashing.

4.1 RUMis more expressive than convolutional GNNs.

For the sake of theoretical arguments in this section, we assume that in Equation 5:

Assumption 2. ϕ x , ϕ u , f are universal and injective.

Assumption 3. Graph G discussed in this section is always connected, unweighted, and undirected.

̸

Assumption 2 is easy to satisfy for feed-forward neural networks [42] and RNNs [43]. Note that RUM can be easily extended to weighted graphs by sampling a biased random walk with edge weights w ij and keeping rest of the algorithm the same: P ( v j | ( v 0 , ..., v i -1 )) ∝ I [( v i , v j ) ∈ E G ] ∗ w ij /D ( v i ) . Composing injective functions, we remark that h ( w ) is also injective w.r.t. ω x ( w ) and ω u ( w ) ; despite of Assumption 3, our analysis can be extended to disjointed graphs by restricting the analysis to the connected regions in a graph. Under such assumptions, we show, in Remark 8 (deferred to the Appendix), that ψ is injective , meaning that nodes with different random walks will have different distributions of representations ψ ( v 1 ) = ψ ( v 2 ) . We also refer the readers to the Theorem 1 in Micali and Zhu [39] for a discussion on the representation power of anonymous experiments on unlabelled graphs. Combining with the semantic representations and promoting the argument from a node level to a graph level, we arrive at:

Theorem 4 (RUMcan distinguish non-isomorphic graphs) . Up to the Reconstruction Conjecture [44], RUM with sufficiently long l -step random walks can distinguish non-isomorphic graphs satisfying Assumption 3.

The main idea of the proof of Theorem 4 (in Appendix § D.2) involves explicitly enumerating all possible non-isomorphic structures for graphs with 3 nodes and showing, by induction, that if

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the theorem stands for graph of N -1 size it also holds for N -sized graphs. We also show in Appendix § D.1 that a number of key graph properties such as cycle size (Example 8.1) and radius (Example 8.2) that convolutional GNNs struggle [45, 46] to learn can be analytically expressed using ω u . As these are solely functions of ω x , they can be approximated arbitrarily well by universal approximators. These examples are special cases of the finding that RUM is stricly more expressive than Weisfeiler-Lehman isomorphism test [9]:

Corollary 4.1 (RUM is more expressive than WL-test) . Up to the Reconstruction Conjecture, two graphs with G 1 , G 2 labeled as non-isomorphic by the k -dimensional Weisfeiler-Lehman ( k -WL) isomorphism test, is the necessary, but not sufficient condition that the representations resulting from RUM with walk length k are also different.

̸

Thus, due to Xu et al. [2], RUM is also more expressive than convolutional GNNs in distinguishing non-isomorphic graphs. This also confirms the intuition that RUM with longer walks are more expressive. (See Figure 3 on an empirical evaluation.) The proof of this corollary is straightforward to sketch-even if we only employ the embedding trajectory ω x , it would have the effect of performing the function ϕ x in Equation 5 on each traversal of the WL expanded trees.

4.2 RUMalleviates over-smoothing and over-squashing

Over-smoothing refers to the phenomenon where the node dissimilarity (e.g., measured by Dirichlet energy in Equation 2) decreases exponentially and approaches zero with the repeated rounds of message passing. Cai and Wang [12] relates Dirichlet energy directly with the convolutional operator:

Lemma 3.1 from Cai and Wang [12].

where λ is the smallest non-zero eigenvalue of ˜ ∆ , the normalized Laplacian of a graph.

Free of convolution operators, it seems only natural that RUM does not suffer from this symptom (Figure 2). We now formalize this intuition by first restricting ourselves to a class of non-contractive mappings for f in Definition 5.

Definition 5. A map f is non-contractive on region Ω if ∃ α ∈ [1 , + ∞ ) such that | f ( x ) -f ( y ) | ≥ α || x -y || , ∀ x, y ∈ Ω .

A line of fruitful research has been focusing on designing non-contractive RNNs [48, 49], and to be non-contractive is intimately linked with desirable properties such as preserving the long-range information content and non-vanishing gradients. From this definition, it is easy to show that, for each sampled random walk in Equation 5, the Dirichlet energy is greater than its input. One only needs to verify that the integration in Equation 6 does not change this to arrive at:

Lemma 6 (RUM alleviates over-smoothing.) . If ϕ x , f are non-contractive w.r.t. all elements in the sequence, the expected Dirichlet energy of the corresponding RUM node representation in Equation 6 is greater than its initial value

This implies that the expectation of Dirichlet energy does not diminish even when l → + ∞ , as it is bounded by the Dirichlet energy of the initial node representation, which is consistent with the trend shown in Figure 2, although the GRU is used out-of-box without constraining it to be explicitly non-contractive.

RUM alleviates over-squashing is deferred to Appenxix § B.2, where we verify that the internode Jacobian | ∂ X ( l +1) v /∂ X (0) u | decays slower as the distance between u, v grows vis-` a-vis the convolutional counterparts. Briefly, although RUM does not address the information bottleneck with exponentially growing receptive field (the 1 / ( ˆ A l +1 ) uv term in Equation 16), it nevertheless can have a non-vanishing (nor exploding) gradient from the aggregation function ( |∇ ϕ x | ).

5 Experiments

On a wide array of real-world node- and graph-level tasks, we benchmark the performance of RUM to show its utility in social and physical modeling. Next, to thoroughly examine the performance of RUM, we challenge it with carefully designed illustrative experiments. Specifically, we ask the following questions in this section, with Q1 , Q2 , and Q3 already theoretically answered in § 4: Q1: Is RUM more expressive than convolutional GNNs? Q2: Does RUM alleviate over-smoothing? Q3: Does RUM alleviate over-squashing? Q4: Is RUM slower with convolutional GNNs? Q5: Is RUM robust? Q6: How does RUM scale up to huge graphs? Q7: What components of RUM are contributing most to the performance of RUM?

Real-world benchmark performance. For node classification, we benchmark our model on the popular Planetoid citation datasets [47], as well as the coauthor [50] and co-purchase [51] datasets common in social modeling. Additionally, we hypothesize that RUM, without the smoothing operator, will perform competitively on heterophilic datasets [52]-we test this hypothesis. For graph classification, we benchmark on the popular TU dataset [53]. We also test the graph regression performance on molecular datasets in MoleculeNet [54] and Open Graph Benchmark [55]. In sum, RUMalmost always outperforms, is within the standard deviation of, the state-of-the-art architectures, as shown in Tables 2, 3, 4, 5, as well as in Tables 6, 7, 8, 9 moved to the Appendix due to space constraint.

On sparsity: the subpar performance on the Computer dataset. The most noticeable exception to the good performance of RUM is that on the Computer co-purchase [51] dataset, where RUM is outperformed even by GCN and GAT. This dataset is very dense with an average node degree ( |E| / |V| ) of 18 . 36 , the highest among all datasets used in this paper. As the variance of the node embedding (Equation 6) scales with the average degree, we hypothesize that dense graphs with very high average node degrees would have high-variance representations from RUM.

On the other hand, RUM outperforms all models surveyed in two large-scale benchmark studies on molecular learning, GAUCHE [56] and MoleculeNet [54]. The atoms in the molecules always have a degree of 2 ∼ 4 with intricate subgraph structures like small rings. This suggests the utility of unifying memory in chemical graphs and furthermore chemical and physical modeling.

Graph isomorphism testing (Q1). Having illustrated in § 4 that RUM can distinguish nonisomorphic graphs, we experimentally test this insight on the popular Circular Skip Link dataset [60, 61]. Containing 4-regular graph with edges connected to form a cycle and containing skip-links between nodes, this dataset is artificially synthesized in Murphy et al. [60] to create an especially challenging task for GNNs. As shown in Appendix Table 7, all convolutional GNNs fail to perform better than a constant baseline (there are 10 classes uniformly distributed). 3WLGNN [62], a higherorder GNN of at least O (2) complexity that operates on explicitly enumerated triplets of graphs, can distinguish these highly similar 4-regular graphs by comparing subgraphs. RUM, with O ( |N| ) linear complexity, achieves similarly high accuracy. One can think of RUM as a stochastic approximation

of the higher-order GNN, with all of its explicitly enumerated subgraphs being identified by RUM with a probability that decreases with the complexity of the subgraph.

Effects of walk lengths and number of samples on performance (Q2, Q3). Having studied the relationship between inference speed and the walk lengths and number of samples, we furthermore study its impact on performance. Using Cora [47] citation graph and vary the walk lengths and number of samples from 1 to 9 , where the performance of RUM improves as more samples are taken and longer walks are employed, though more than 4 samples and walks longer than L > 4 yield qualitatively satisfactory results; this empirical finding has guided our hyperparameter design. In Figure 2, we also compare the Dirichlet energy of RUM-generated layer representations with not only baselines GCN [1] and GAT [6], but also strategies to alleviate over-smoothing discussed in § 2, namely residual connection and stochastic regularization [37, 28, 63], and show that when L gets large, only RUM can maintain Dirichlet energy. Traditionally, since Kipf and Welling [1] (see its Figure 5 compared to Figure 3), the best performance on Cora graph was found with 2 or 3 message-passing rounds, since mostly local interactions are dominating the classification, and more rounds of message-passing almost always lead to worse performance. As theoretically demonstrated in § 4.2, RUM is not as affected by these symptoms. Thus, RUM is especially appropriate for modeling long-range interactions in graphs without sacrificing local representation power.

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Long-range neighborhood matching (Q3). To verify that RUM indeed alleviates over-squashing (§ B.2), in Figure 5, we adopt the tree neighborhood matching synthetic task in Alon and Yahav [14] where binary tree graphs are proposed with the label and the attributes of the root matching a faraway leave. The full discussion is moved to the Appendix § B.3.

Speed (Q4). Though both have linear runtime complexity (See § 3), intuitively, it might seem that RUM would be slower than convolution-based GCN due to the requirement of multiple random walk samples. This is indeed true for CPU implementations shown in Figure 4 left. When equipped with GPUs (specs in Appendix § A), however, RUM is significantly faster than even the simplest convolutional GNN-GCN [1]. It is worth mentioning that the GCN and GAT [6] results were harvested using the heavily optimized Deep Graph Library [38] sparse implementation whereas RUM is implemented na¨ ıvely in PyTorch [64], though the popular GRU component [16] have already undergone CUDA-specific optimization.

Robustness to attacks (Q5). With the stochasticity afforded by the random walks, it is natural to suspect RUM to be robust. We adopt the robustness test from Feng et al. [65] and attack by randomly adding fake edges to the Cora [47] graph and record the test set performance in Figure 6. Indeed, RUM is much more robust than traditional convolutional GNNs including GCN [1] and GAT [6] and is even slightly more robust than the convolutional GNN specially designed for robustness [65], with the performance only decreased less than 10% with 10% fake edges added.

Scaling to huge graphs (Q6) and Ablation study (Q7) are deferred to Appendix § B.5.

6 Conclusions

We design an innovative GNN that uses an RNN to unify the semantic and topological representations along stochastically sampled random walks, termed random walk with unifying memory (RUM) neural networks. Free of the convolutional operators, our methodology does not suffer from symptoms characteristic of Laplacian smoothing, including limited expressiveness, over-smoothing, and oversquashing. Most notably, our method is more expressive than the Weisfeiler-Lehman isomorphism test and can distinguish all non-isomorphic graphs up to the reconstruction conjecture. Thus, it is more expressive than all of convolutional GNNs equivalent to the WL test, as we demonstrate theoretically in § 4. RUM is significantly faster on GPUs than even the simplest convolutional GNNs (§ 5) and shows superior performance across a plethora of node- and graph-level benchmarks.

Limitations. Very dense graphs. As evidenced by the underwhelming performance of the Computer [51] dataset and discussed in § 5, RUM might suffer from high variance with very dense graphs (average degree over 15). Tottering. In our implementation, we have not ruled out the 2-cycles from the random walks, as that would require specialized implementation for walk generation. This, however, would reduce the average information content in a fixed-length random walk (known as tottering [66]). We plan to characterize the effect of excluding these walks. Biased walk. Here, we have only considered unbiased random walk, whereas biased random walk might display more intriguing properties as they tend to explore faraway neighborhoods more effectively [17]. Directed graphs. Since we have only developed RUM for undirected graph (random walk up to a random length is not guaranteed to exist for directed graphs), we would have to symmetrize the graph to perform on directed graphs (such as the heterophilic datasets [52]); this create additional information loss and complexity.

Future directions. Theoretical. We plan to expand our theoretical framework to account for the change in layer width and depth to derive analytical estimates for realizing key graph properties. Applications. Random walks are intrinsically applicable to uncertainty-aware learning. We plan to incorporate the uncertainty naturally afforded by the model to design active learning models. On the other hand, the physics-based graph modeling field is also heavily dominated by convolutional GNNs. Inspired by the superior performance of RUM on chemical datasets, we plan to apply our method in drug discovery settings [67, 68, 69, 70, 71] and furthermore on the equivariant modeling of n -body physical systems [72].

Impact statement. We here present a powerful, robust, and efficient learning algorithm on graphs. Used appropriately, this algorithm might advance the modeling of social [73] and physical [74] systems, which can oftentimes modeled as graphs. As with all graph machine learning methods, negative implications may be possible if used in the design of explosives, toxins, chemical weapons, and overly addictive recreational narcotics.

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Disclosures. YWhas limited financial interests in Flagship Pioneering, Inc. and its subsidiaries.

Funding. YWacknowledges support from the Schmidt Science Fellowship, in partnership with the Rhodes Trust, and the Simons Center for Computational Physical Chemistry at New York University.

A Experimental details

Code availability. All architectures, as well as scripts to execute the experiment, are distributed open-source under MIT license at https://anonymous.4open.science/r/rum-834D/ . Core dependencies of our package include PyTorch [64] and Deep Graph Library [38].

Hyperparameters. All models are optimized using Adam [75] optimizer and SiLU [76] activation functions. 4 random walk samples are drawn everywhere unless specified. Other hyperparameterslearning rate ( 10 -5 ∼ 10 -2 ), hidden dimension ( 32 ∼ 64 ), L2 regularization strength ( 10 -8 ∼ 10 -2 ), walk length ( 3 ∼ 16 ), temperature for L consistency ( 0 ∼ 1 ), coefficient for L consistency ( 0 ∼ 1 ), coefficient for L self , and dropout probability-are tuned using the Ray platform [77] with the default Ax [78] search algorithm with 1000 trails or 24 hours tuning budget on a Nvidia A100 ® GPU.

B Additional technical details

B.1 Self-supervised regularization.

The stochasticity encoded in our model naturally affords it with some level of regularization. Apart from using the consistency loss ( L consistency ) used in Feng et al. [65] for classifications, we further regularize the model by using the RNNs in ϕ x to predict the semantic representation of the next node on the walk given ω u and jointly maximize this likelihood:

where g ( ·| θ ) is modeled as the sequence output of the RNN ϕ x in Equation 5. The total loss is modeled as a combination of:

B.2 RUMattenuates over-squashing

Similarly, we can show that if the composing neural networks defy the vanishing gradient problem (w.r.t. the input) [79], the sensitivity analysis in Equation 3 [15] has a lower bound for RUM.

Lemma 7 (RUM attenuates over-squashing) . If ϕ x , f have lower-bounded derivatives, the inter-node Jacobian for nodes u, v separated by a shortest path of length l , RUM with walk length l also has a lower bound:

where ˆ A ij = A ij / ∑ j A ij is the degree-normalized adjacency matrix.

Like the upper bound in Equation 3, this lower bound is also controlled by the power of the (normalzied) adjacency matrix, albeit the absence of self-loop will result in a slightly looser bottleneck. The term ( ˆ A l ) uv corresponds to the probability of the shortest path among all possible walks as a product of inverse node degrees (see Equation 4). There is no denying that the lower bound is still controlled by the power of the adjacency matrix, which corresponds to the exponentially growing receptive fields. One can also argue that, without prior knowledge, the contribution of the sensitivity analysis by the power of the graph adjacency matrix can never be alleviated, since there are always roughly 1 / ( ˆ A l +1 ) uv (assuming uniform node degree) structurally equivalent nodes. Nevertheless, since ϕ x is not necessarily an iterative function, we alleviate the over-squashing problem by eliminating the power of the update function gradient term.

Now, we plug in the layer choices of ϕ x -a GRU [16] unit. Its success, just like that of long shortterm memory (LSTM) [80], can be attributed to the near-linear functional form of the long-range gradient. The term |∇ ϕ x | is controlled by a sequence of sigmoidal update gates, which can be optimized to approach 1 (fully open). If we ignore the gradient contribution of X 0 u to these gates, the non-linear activation function has only been applied exactly once on X 0 u ; therefore, the gradient | ∂ X ( l +1) v /∂ X (0) u | is neither rapidly vanishing nor exploding.

B.3 Long-range neighborhood matching (Q3).

Once identifying the target leaf, this task seems trivial; nonetheless, this piece of information needs to be passed through layers of aggregation and non-linear update and is usually lost in the convolution. Since, on this binary tree, the receptive field grows exponentially, Alon and Yahav [14] argues that there is a theoretical lower boundary for the layer width D for the convolutional GNN to be able to encode all possible combinations of leaves, which is 2 32 D for single-precision floating point ( float32 ). This corresponds to the structural ˆ A l term in Equation 3 and Equation 16. Evidently, when D = 32 , as is the adopted experimental setting in Figure 5, this limit is far from being hit. So we hypothesize that the reason why convolutional GNNs cannot overfit the training set is because of the limitation of the functional forms, which are remedied by RUM, which shows 100% accuracy up to tree depth or problem radius r = 5 , and a relatively moderate decrease afterward. Note that when the problem radius exceeds r = 7 , RUM's performance is not significantly different from the convolutional counterparts.

B.4 Scalaing to large graphs (Q6).

In Appendix Table 8, we apply RUM on an ultra-large graph OGB-PRODUCTS , that cannot fit easily on a single GPU, and compare RUM with architectures specifically designed for large graphs [41].

B.5 Ablation study (Q7).

In Table 6, we conduct a brief ablation study where we delete, one by one, the components introduced in § 3. ω u = 0 and ω x = 0 refer to the deletion of the topological and semantic representations of walks, respectively. Neglecting topological information results in a moderate decrease in performance, whereas neglecting semantic representation is more detrimental. The ω u = 0 also resembles Jin et al. [17] albeit with different walk-wise aggregation. This offers a qualitative comparison between our work and Jin et al. [17] as no source code was released for this package so no rigorous comparison was possible. We also see that the regularization methods are helpful to the performance, with self-supervision being more crucial. We attribute this effect to firstly the relative simplicity of the Cora classification task, and secondly the flexibility of the (overparametrized) RNNs.

C Additional results

Algorithm 1 anonymous experiment

̸

Remark 8 (Inequality in distribution) . For two nodes v 1 , v 2 with distribution of random walks terminating at v 1 , v 2 not equal in distribution p ( ω u ( w 1 )) = p ( ω u ( w 2 )) or p ( ω x ( w 1 )) = p ( ω x ( w 2 )) , the node representations in Equation 6 are also different ψ ( v 1 ) = ψ ( v 2 ) .

̸

̸

̸

One way to construct h function in Equation 5 is to have h ( w ) positive only where p ( ω u ( w 1 )) > p ( ω u ( w 2 )) ; the same thing can be argued for ω x . In other words, one only needs to prove two walks terminating at two nodes p ( ω u ( w 1 )) = p ( ω u ( w 2 )) or p ( ω x ( w 1 )) = p ( ω x ( w 2 )) are not equal in distribution to verify that RUM can distinguish two nodes. Conversely, we can also show that p ( ω u ( w 1 )) = p ( ω u ( w 2 )) implies that v 1 , v 2 are isomorphic without labels- we refer the readers to the Theorem 1 in Micali and Zhu [39] for a discussion on reconstructing unlabelled graphs using anonymous experiments.

̸

D Missing mathematical arguments.

D.1 Examples of Theorem 4

Example 8.1 (Cycle detection.) . A k -cycle C k is a subgraph of G consisting of k nodes, each with degree two. The existence of k -cycle can be determined by:

̸

✶ ( C k ⊆ G ) = ✶ [ P ( ω x j +1 = ω x 0 , ω x i = ω x j , ∀ i < j ) > 0] (17) Example 8.2 (Diameter.) . The diameter, δ G of graph G which equals the length of the longest shortes path in G , can be expressed as

̸

̸

D.2 Proof of Theorem 4

Proof. First, we enumerate all possible unlabelled graphs with three nodes satisfying Assumption 3one with two edges, one with three edges. (Note that there is only one non-isomorphic graph with

two nodes.) Now we consider random walks of length l = 3 , where

only stands for the graph with three edges, but not with two edges, just like Example 8.1.

Furthermore, we can also distinguish between the 2-degree node and the 1-degree node in the graph with 2 edges and 3 nodes simply by verifying that

̸

only stands when v 2 is the 2-degree node.

Moving on to the labeled 3-node graph case, we can reduce the problem to investigate whether RUM can distinguish 3-node graphs that are isomorphic when unlabeled, but non-isomorphic when labeled. For the three-edged graph, ω x uniformly samples the labels of three nodes. For the two-edged graph, suppose the node labels are A,B,C, and we start from the node with 2 degrees B (with nodes bearing A and C labels locally, structurally isomorphic),

If graphs have the same ω x , they have the same B and the same or swapped A,C . As such, we have proven Theorem 4 for graphs with 3 nodes.

Now we prove that Theorem 4 stand for graphs of N nodes, they also stand for graphs of N +1 nodes, the Reconstruction Conjecture [44]

Suppose we have two non-isomorphic graphs with N + 1 nodes G 1 and G 2 with the same RUM embedding Ψ 1 = Ψ 2 . We enumerate all N +1 subgraphs with each node deleted for each of these two graphs. By the Reconstruction Conjecture, at least one pair of subgraphs are non-isomorphic. For this pair, suppose the deleted vertex is v (ruling out the trivial case where the label or connectivity of v is different for these two graphs), and two remaining subgraphs G \ v 1 , G \ v 2 ; since ϕ x , ϕ u , f are injective, Ψ 1 = Ψ 2 implies ω u , ω x are equal in distribution for G 1 , G 2 . As such, the walk distribution

̸

̸

are also equal in distribution for G 1 , G 2 .

If there is a link between v and the nodes in G 1 , G 2 , we assign a new label to contain both the old label and the connection. As such, if the second term is not equal in distribution for G 1 , G 2 , we would have

̸

which is in conflict with the assumption that Theorem 4 stands for graphs with N nodes.

D.3 Proof of Lemma 6

Proof. By the definition of Dirichlet energy (Equation 2) and non-contractive mappings (Definition 5),

The last inequality is due to the fact that ϕ x is non-contractive w.r.t. all, and therefore, the last element of the sequence, which are always X u and X v .

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: We have claimed the theoretical and experimental advantage of RUM, which have been illustrated in § 4 and § 5, respectively.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: In the Limitations paragraph in § 6.

Guidelines:

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a random walk-based graph neural network, where an RNN is used to combine the topological and semantic graph features along the walks. The proposed RUM model is free of convolution operators and does not suffer from the limited expressiveness, over-smoothing, and over-squashing, which are commonly faced issues in convolution-based GNNs.\n\nSTRENGTHS:\nThis paper proposes a non-convolutional GNN, named RUM. The benefits of this model over convolutional GNNs are elaborated both theoretically and empirically.\n\nWEAKNESSES:\nEmpirically, the gains with RUM are marginal and in some cases, RUM is worse than the baselines.\n\nThe organization and the presentation of the paper can be further improved.\n1. There are too many hyper-references. Most of the hyper-references in Section 1 and Section 2 are unnecessary, especially when reading the paper for the first time.\n2. There are too many supplementary demonstrations (...) Some additional information could be helpful, but too many will hinder the reading. It would be better to reorganize the sentences to make the presentation smoother.\n3. There are some grammar mistakes, like in line 244. Proofreading is needed.\n\nQUESTIONS:\n1. In Lemma 6, the author shows that RUM increases Dirichlet energy. Why would this be helpful intuitively? As for homophily graphs, GNNs should decrease the Dirichlet energy. This phenomenon is also analyzed in [1,2]. It seems RUM is not just alleviating over-smoothing, but is anti-smoothing.\n2. In line 243, the author claims that RUM almost always outperforms SOTA architectures, this is not true according to the results.\n\n[1] Interpreting and unifying graph neural networks with an optimization framework\n\n[2] A Unified View on Graph Neural Networks as Graph Signal Denoising\n\nLIMITATIONS:\nThe authors adequately discussed the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces Random Walk with Unifying Memory (RUM), a non-convolutional graph neural network (GNN) that addresses limitations like limited expressiveness, over-smoothing, and over-squashing typically associated with convolution-based GNNs. RUM leverages random walks and recurrent neural networks (RNNs) to merge topological and semantic graph features, theoretically showing and experimentally verifying its enhanced expressiveness over traditional methods like the Weisfeiler-Lehman (WL) isomorphism test. It demonstrates competitive performance on node- and graph-level tasks while offering advantages in efficiency and scalability.\n\nSTRENGTHS:\n1. RUM introduces a new paradigm in GNNs by eliminating convolutions and leveraging random walks and RNNs for enhanced graph representation.\n\n2. The presentation of the paper is generally clear and coherent. The authors effectively contextualize their work relative to existing literature, highlighting the novelty of RUM and its contributions to the field of graph representation learning.\n\n3. The theoretical advancements (e.g., expressiveness proofs) and empirical results clearly demonstrate the advantages of RUM over traditional convolutional GNNs. The findings are valuable for advancing graph representation learning techniques and are likely to influence future research directions.\n\nWEAKNESSES:\nThe paper lacks detailed exploration on how varying random walk lengths impact RUM's performance across diverse tasks and whether optimal lengths are identified for different types of graphs or datasets. Additionally, while the paper highlights RUM's superiority over expressive GNNs in specific scenarios, it overlooks potential limitations when compared to these models, such as not thoroughly investigating upper bounds on the WL test, which could gauge RUM's discriminative power against more expressive GNNs.\n\nQUESTIONS:\n1. Can you elaborate on how the choice of random walk length $𝑙$ impacts the performance of RUM across different tasks? Are there optimal lengths identified for specific types of graphs or datasets?\n\n2. In practical applications, how does RUM handle graph datasets with varying degrees of sparsity? Are there specific strategies employed to ensure robust performance across different graph structures?\n\n3. While the paper demonstrates that RUM outperforms expressive GNNs in certain scenarios, it lacks exploration of scenarios where RUM might face challenges or limitations compared to expressive GNNs. For instance, it does not thoroughly investigate potential upper bounds on the WL test, which is a standard measure of graph isomorphism and could provide insights into the discriminative power of RUM compared to more expressive GNNs.\n\nLIMITATIONS:\nYes, The authors have addressed several limitations in the Limitations section.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-07"}

comments: SUMMARY:
The paper introduced a new graph neural network that is not based on convolution but on the random walk. Based on the results of the extensive experiment, the proposed architecture is effective for heterophilic graphs and long-range graphs.

STRENGTHS:
1. The experiment results include different tasks on different graphs, also, it is faster than baselines.
2. Theoretical analysis is conducted to show that the proposed architecture is more expressive.
3. Theoretical analysis shows that the proposed architecture is effective in overcoming over-squashing and over-smoothing issues.

WEAKNESSES:
Currently, the proposed architecture only works on unweighted and undirected graphs.

QUESTIONS:
Can the proposed method work well on directed and weighted graphs?

LIMITATIONS:
N/A
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-12

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors of the paper proposed a non-convolution based approach to solve graph learning tasks using random walks and recurrent neural networks (RNNs), namely a random walk neural network with unifying memory (RUM). Random walks, together with the “anonymous experiment” function, allow to extract topological and semantic information from the graph, while an RNN and a feed-forward neural network take care of “unifying” the features collected from the walks. Research questions on the ability of the model to be more expressive than classical GNNs and to reduce oversmoothing and oversquashing have been answered positively both theoretically and experimentally. The experimental section, along with the appendix section, presents several experiments, ablation studies, training details, and theorem proofs. The code is available.\n\nSTRENGTHS:\nOriginality: Novelty is fair/good. The proposed approach can be considered original by choosing to propose a model that does not use the graph convolution operator at all, which is the most widely used approach in the state of the art. In this way, it alleviates the problems associated with convolutions (e.g., oversmoothing, oversquashing) while maintaining or even improving performance. Random walks and RNNs (more specifically GRUs) are not new methodologies, but they are combined in an original way on graph learning.\n\nQuality: The quality of the proposed paper is good. The claims and solutions are technically sound. They present theoretical and experimental proofs of their proposed approach.\n\nClarity: The clarity of the paper is good. It is well written and organized. The best results in the tables should be in bold or more visible.\n\nSignificance: The significance of the paper is medium-high. They proposed an effective method to avoid convolutions in GNNs that effectively outperforms and competes with SOTA methodologies.\n\nWEAKNESSES:\nWeakness: The paper lacks a discussion on the proposed method and Graph Transformer architectures. Although they are present in the experimental comparisons on the node classification task in Table 9, Graph Transformer is actually a well-known solution that does not rely on convolutions and can mitigate the over-squashing and oversmoothing problems [1,2]; the discussion should also considering the different computational costs of the two approaches.\n\nRef.\n[1] \"Attending to Graph Transformers\", Luis Müller and Mikhail Galkin and Christopher Morris and Ladislav Rampášek, 2024, https://arxiv.org/abs/2302.04181\n[2] \"Do Transformers Really Perform Bad for Graph Representation?\", Chengxuan Ying et. al., 2021, https://arxiv.org/abs/2106.05234\n\nQUESTIONS:\n1) At Table.5, showing Graph Regression comparisons on ESOL, FreeSolv and Lipophilicity datasets, it seems to me that for the MoleculeNet [1] benchmark you used the results presented in their papers about the XGBoost model (best performing among \nconventional methods) with (0.99, 1.74, 0.799) RMSE scores, respectively to the three datasets, instead using them from the best performing graph-based methods which are (0.58,1.15) for ESOL and FreeSolv datasets by using the MPNN model and 0.655 for Lipophilicity dataset by using GC model, which are the best models in these datasets. This is in contrast on what you said in the caption of Table. 5, since those presented are not the best results and models of [1].\n\nRef.\n[1] \"MoleculeNet: a benchmark for molecular machine learning\", Wu, Zhenqin and et. al., \"Chem. Sci.\",2018, doi:10.1039/C7SC02664A, url:http://dx.doi.org/10.1039/C7SC02664A\n\nLIMITATIONS:\nThe authors have adequately discussed the potential negative societal impact and the current limitations of their work.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-28"}