Title: AH-UGC: $\underline{\text{A}}$daptive and $\underline{\text{H}}$eterogeneous-$\underline{\text{U}}$niversal $\underline{\text{G}}$raph $\underline{\text{C}}$oarsening
Venue: NeurIPS 2025
Accepted: no
Keywords: Graph Coarsening

Graph Neural Networks

Scaling Graph Learning

Hashing
Abstract: **Graph Coarsening (GC)** is a prominent graph reduction technique that compresses large graphs to enable efficient learning and inference. However, existing GC methods generate only one coarsened graph per run and must recompute from scratch for each new coarsening ratio, resulting in unnecessary overhead. Moreover, most prior approaches are tailored to *homogeneous* graphs and fail to accommodate the semantic constraints of *heterogeneous* graphs, which comprise multiple node and edge types. To overcome these limitations, we introduce a novel framework that combines Locality-Sensitive Hashing (LSH) with Consistent Hashing to enable \textit{adaptive graph coarsening}. Leveraging hashing techniques, our method is inherently fast and scalable. For heterogeneous graphs, we propose a *type-isolated coarsening* strategy that ensures semantic consistency by restricting merges to nodes of the same type. Our approach is the first unified framework to support both adaptive and heterogeneous coarsening. Extensive evaluations on 23 real-world datasets—including homophilic, heterophilic, homogeneous, and heterogeneous graphs demonstrate that our method achieves superior scalability while preserving the structural and semantic integrity of the original graph. Our code is available here https://anonymous.4open.science/r/AdaptiveUGC-1912/README.md .

Abstract

Graph Coarsening (GC) is a prominent graph reduction technique that compresses large graphs to enable efficient learning and inference. However, existing GC methods generate only one coarsened graph per run and must recompute from scratch for each new coarsening ratio, resulting in unnecessary overhead. Moreover, most prior approaches are tailored to homogeneous graphs and fail to accommodate the semantic constraints of heterogeneous graphs, which comprise multiple node and edge types. To overcome these limitations, we introduce a novel framework that combines Locality-Sensitive Hashing (LSH) with Consistent Hashing to enable adaptive graph coarsening . Leveraging hashing techniques, our method is inherently fast and scalable. For heterogeneous graphs, we propose a type-isolated coarsening strategy that ensures semantic consistency by restricting merges to nodes of the same type. Our approach is the first unified framework to support both adaptive and heterogeneous coarsening. Extensive evaluations on 23 real-world datasets-including homophilic, heterophilic, homogeneous, and heterogeneous graphs demonstrate that our method achieves superior scalability while preserving the structural and semantic integrity of the original graph. Our code is available here.

1 Introduction

Graphs are ubiquitous and have emerged as a fundamental data structure in numerous real-world 19 applications [1-3]. Broadly, graphs can be categorized into two types: (a) Homogeneous graphs 20 [4-6], which consist of a single type of nodes and edges. For instance, in a homogeneous citation 21 graph, all nodes represent papers, and all edges represent the 'cite' relation between them; (b) 22 Heterogeneous graphs [7-9], which involve multiple types of nodes and/or edges, enabling the 23 modeling of richer and more realistic interactions. For example, in a recommendation system, a 24 heterogeneous graph may contain nodes of different types, such as users, items, and categories, and 25 edge types such as '(user, buys, item)', '(user, views, item)', and '(item, belongs-to, category)'. 26 Although many real-world datasets are inherently heterogeneous, early research in graph machine 27 learning predominantly focused on homogeneous graphs due to their modeling simplicity, availabil28 ity of standardized benchmarks, and theoretical tractability [10, 11]. However, the limitations of 29 homogeneous representations in capturing rich semantic information have shifted attention toward 30 heterogeneous graph modeling [8, 12]. 31 As real-world networks continue to grow rapidly in size and complexity, large-scale graphs have 32 become increasingly common across various domains [1, 13-15]. This surge in scale poses signifi33 cant computational and memory challenges for learning and inference tasks on such graphs. This 34 underscores the growing importance of developing efficient and effective methodologies for process35 ing large-scale graph data. To address the issue, an expanding line of research investigates graph 36 reduction methods that compress structures without compromising essential properties. Most existing 37

graph reduction techniques, including pooling [16], sampling-based [17], condensation [18], and 38

coarsening-based methods [4, 19, 20]. Coarsening methods have demonstrated effectiveness in 39 preserving structural and semantic information [4, 19, 20], this study focuses on graph coarsening 40 (GC) as the primary reduction strategy. Despite advancements in existing GC frameworks, two key 41 challenges remain: 42

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Lack of 'Adaptive Reduction'. Many applications, such as interactive visualization and real-time recommendations, benefit from multi-resolution graph representations. These scenarios often

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require dynamically adjusting the coarsening ratio based on user interaction or task demands. However, most existing methods generate a single fixed-size coarsened graph and must recompute from scratch for each new ratio, incurring high overhead. This highlights the need for adaptive coarsening frameworks that enable efficient, progressive refinement without redundant computation.

Lack of 'Heterogeneous Graph Coarsening' Framework. Existing methods typically assume homogeneous node types, making them unsuitable for heterogeneous graphs with semantically distinct nodes. This can result in invalid supernodes for example, merging an author with a paper node in a citation graph thus violating type semantics. Moreover, node types often have different feature dimensions, which standard coarsening techniques are not designed to handle.

Key Contribution. To address the dual challenges of adaptive reduction and heterogeneous GC, we propose AH-UGC , a unified framework for Adaptive and Heterogeneous Universal Graph Coarsening. We integrate locality-sensitive hashing (LSH) [4, 21, 22] with consistent hashing (CH) [23, 24]. While LSH ensures that similar nodes are coarsened together based on their features and connectivity, CH-a technique originally developed for load balancing-enables us to design a coarsening process that supports multi-level adaptive coarsening without reprocessing the full graph. To handle heterogeneous graphs, AH-UGC enforces type-isolated coarsening , wherein nodes are first grouped by their types, and coarsening is applied independently within each type group. This ensures that nodes and edges of incompatible types are never merged, preserving the semantic structure of the original heterogeneous graph. Additionally, AH-UGC is naturally suited for streaming or evolving graph settings, where new nodes and edges arrive over time. Our LSH- and CH-based method allows new nodes to be integrated into the existing coarsened structure with minimal recomputation. To summarize, AH-UGC is a general-purpose graph coarsening framework that supports adaptive, streaming, expanding, heterophilic, and heterogeneous graphs .

2 Background

Definition 2.1 (Graph) A graph is represented as G ( V, A, X ) , where V = { v 1 , . . . , v N } is 69 the set of N nodes, A ∈ R N × N is the adjacency matrix, and X ∈ R N × ˜ d is the node fea70

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ture matrix with each row X i ∈ R ˜ d denoting the feature vector of node v i . An edge be71 tween nodes v i and v j is indicated by A ij > 0 . Let D ∈ R N × N be the degree matrix with 72 D ii = ∑ j A ij , and let L = D -A denote the unnormalized Laplacian matrix. L ∈ S L , where 73 S L = { L ∈ R N × N ∣ ∣ ∣ L ij = L ji ≤ 0 for i = j ; L ii = -∑ j = i L ij } . For i = j , the matrices are 74 related by A ij = -L ij , and A ii = 0 . Hence, the graph G ( V, A, X ) may equivalently be denoted 75 G ( L, X ) , and we use either form as contextually appropriate. 76

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Definition 2.2 A heterogeneous graph can be represented in two equivalent forms, with either representation utilized as required within the paper.

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Entity-based: A heterogeneous graph extends the standard graph structure by incorporating multiple types of nodes and/or edges. Formally, a heterogeneous graph is defined as G ( V, E, Φ , Ψ) , where Φ : V →T V and Ψ : E →T E are node-type and edge-type mapping functions, respectively [9]. Here, T V and T E denote the sets of possible node types and edge types. When the total number of node types |T V | and edge types |T E | is equal to 1, the graph degenerates into a standard homogeneous graph (Definition 2.1).

Type-based: Alternatively, a heterogeneous graph can be described as G ( { X (node_type) } , { A (edge_type) } , { y (target_type) } ) , where feature matrices X , adjacency matrices A , and target labels y are grouped and indexed by their corresponding node, edge, and target types [25].

Definition 2.3 Following [4, 19, 20], The G raph C oarsening (GC) problem involves learning a coarsening matrix C ∈ R N × n , which linearly maps nodes from the original graph G to a reduced graph G c , i.e., V → ˜ V . This linear mapping should ensure that similar nodes in G are grouped into the same super-node in G c , such that the coarsened feature matrix is given by ˜ X = C T X . Each non-zero entry C ij denotes the assignment of node v i to super-node ˜ v j . The matrix C must satisfy the following structural constraints:

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where d ˜ V l means the number of nodes in the l th -supernode. The condition ⟨C T i , C T j ⟩ = 0 ensures that each node of G is mapped to a unique super-node. The constraint ∥C T i ∥ 0 ≥ 1 requires that each super-node contains at least one node.

2.1 Problem formulation and Related Work

We formalize the problem through two key objectives: Goal 1. Adaptive Coarsening and Goal 2. Graph Coarsening for Heterogeneous Graphs.

Goal 1. The objective is to compute multiple coarsened graphs {G ( r ) c } R r =1 from input graph G ( V, A, X ) , where each G ( r ) c corresponds to a target coarsening ratio r ∈ (0 , 1] , without recomputing from scratch for each resolution. Formally, the goal is to construct a family of coarsening matrices {C ( r ) ∈ R N × n ( r ) } such that

with the constraint that all C ( r ) are derived from a single, shared projection s = HASH ( X ) , thereby ensuring consistency across coarsening levels and enabling adaptive GC.

Goal 2. The objective is to learn a coarsening matrix C ∈ R N × n , such that the resulting coarsened graph G c ( ˜ V , ˜ E, ˜ Φ , ˜ Ψ) satisfies the following constraints:

where π : V → ˜ V is the node-to-supernode mapping induced by C . These constraints guarantee 112 that: a) nodes of different types are not merged into the same supernode, and b) edge types between 113 supernodes are consistent with the original heterogeneous schema. 114

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Related Work. Graph reduction methods have been extensively studied and can be broadly categorized into optimization-based and GNN-based approaches. Among optimization-driven heuristics, Loukas's spectral coarsening methods [20] including edge-based (LVE) and neighborhood-based (LVN) variants aim to preserve the spectral properties of the original graph. Other techniques, such as Heavy Edge Matching (HEM)[17, 26], Algebraic Distance[27], Affinity [28], and Kron reduction [29], rely on topological heuristics or structural similarity principles. FGC [19] incorporates node features to learn a feature-aware reduction matrix. Despite their diverse designs, a common drawback of these methods is that they are computationally demanding, often with time complexities ranging from O ( n 2 ) to O ( n 3 ) , and are not well suited for large-scale or adaptive graph reduction settings. UGC [4], a recent LSH-based framework, addresses these challenges by operating in linear time and supporting heterophilic graphs. However, it produces only a single coarsened graph and must recompute reductions for different coarsening levels, limiting its adaptability. GNN-based condensation methods like GCond [30] and SFGC [31] learn synthetic graphs through gradient matching but require full supervision, are model-specific, and lack scalability. HGCond [25] is the only approach designed for heterogeneous graphs, yet it inherits the inefficiencies of condensation-based techniques.

While some methods are model-agnostic, others offer partial support for heterophilic or streaming graphs. Yet, no existing approach simultaneously addresses all these challenges-model-agnosticism, adaptability, and support for heterophilic, heterogeneous, and streaming graphs. As illustrated in Figure 2, HA-UGC is the first framework to meet all six criteria comprehensively. For details on LSH and consistent hashing, see Appendix B.

3 The Proposed Framework: Adaptive and Heterogeneous Universal Graph Coarsening

In this section we propose our framework AH-UGC to address the issues of adaptive and heterogeneous graph coarsening. Figure 1 shows the outline of AH-UGC.

3.1 Adaptive Graph Coarsening(Goal 1)

The AH-UGC pipeline closely follows the recently proposed structure of UGC but incorporates consistent hashing principles to enable adaptive i.e., multi-level coarsening. Our framework introduces an innovative and flexible approach to graph coarsening that removes the UGC's dependency on fixed bin widths and enables the generation of multiple coarsened graphs. Similar to UGC [4], AH-UGC employs an augmented representation to jointly encode both node attributes and graph topology. For a given graph G ( V, A, X ) , we compute a heterophily factor α ∈ [0 , 1] , which quantifies the relative emphasis on structural information based on label agreement between connected nodes i.e., α = |{ ( v,u ) ∈ E : y v = y u }| | E | . This factor is then used to blend node features X i and adjacency vectors A i . For each node v i we calculate F i = (1 -α ) · X i ⊕ α · A i where ⊕ denotes concatenation. This hybrid representation ensures that both local attribute similarity and topological proximity are captured before the coarsening process. Importantly, this design enables our framework to handle heterophilic graphs robustly by incorporating structural properties beyond mere feature similarity.

Adaptive Coarsening via Consistent and LSH Hashing. Let F i ∈ R d denote the augmented feature vector for node v i . AH-UGC applies l random projection functions using a projection matrix W ∈ R d × l and bias vector b ∈ R l , both sampled from a p -stable distribution [32]. The scalar hash score for each projection for i th node is given by:

UGC relies on a bin-width parameter ( r ) to control the coarsening ratio ( R ), but determining appropriate bin-widths for different target ratios can be computationally expensive. In contrast, AH-UGC eliminates the need for bin width by leveraging consistent hashing. Once the hash scores ( s i ) across projections are computed, AH-UGC enables efficient construction of coarsened graphs at multiple coarsening ratios without requiring reprocessing, making it well-suited for adaptive settings. We define an AGGREGATE function to combine projection scores across multiple random projectors. For each node i , the final score s i is computed as:

Alternative aggregation functions such as max, median, or weighted averaging can also be used, 163 depending on the design objectives. After computing the scalar hash scores { s i } for all nodes v i ∈ V , 164

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we sort the nodes in increasing order of s i to form an ordered list L , represented as a list of super-node and mapped nodes: L = [ { u 1 : { v 1 }} , { u 2 : { v 2 }} , . . . , { u n : { v n }} ] , where each key u j denotes a super-node index, and the associated value is the set of nodes currently assigned to that super-node. Initially, each node is its own super-node, and the number of super-nodes is | V (0) c | = | V | . At each iteration t , a super-node u j is randomly selected from the current list L ( t ) and merged with its immediate clockwise neighbor u j +1 . The updated super-node entry is given by:

followed by the removal of u j +1 from the list. This reduces the number of super-nodes by one: | V ( t +1) c | = | V ( t ) c | -1 . The process is repeated until the desired coarsening ratio is reached: R = | V c | | V | . Furthermore, this coarsening strategy is inherently adaptive, enabling transitions between any two coarsening ratios R → T directly from the sorted list without reprocessing.

Since the list L is constructed using locality-sensitive hashing (LSH) principles [32], similar nodes are positioned adjacently. Through Theorem 3.1 and Lemma 1, we show that the clockwise merging operations in Consistent Hashing (CH) are locality-aware and effectively preserve feature similarity.

Theorem 3.1 Let x, y ∈ R d , and let the projection function be defined as: h ( x ) = ∑ ℓ j =1 r ⊤ j x, r j ∼ N (0 , I d ) i.i.d. Then the difference h ( x ) -h ( y ) ∼ N (0 , ℓ ∥ x -y ∥ 2 ) , and for any ε > 0 :

Proof: The proof is deferred in Appendix D.

This gives the probability that two nodes, initially close in the feature space, are projected within an ϵ -range in the projection space.

Lemma 1 Let x, y, z ∈ R d , with ∥ x -y ∥ ≪ ∥ x -z ∥ . Then the probability that a distant point z lies between x and y after projection is:

where Φ is the cumulative distribution function (CDF) of the standard normal distribution. This result ensures that distant nodes rarely interrupt merge candidates that are close in feature space, preserving the structural consistency of coarsened regions.

Remark 1 Our framework also supports de-coarsening i.e., given the final sorted list and merge history, the graph can be reconstructed to finer resolutions by reversing the merging process. However, in this work, we restrict our focus to the coarsening direction only.

Construction of Coarsening Matrix C . Given the score-based node assignments π : V → ˜ V , where π [ v i ] is the super-node index of v i , the binary coarsening matrix C ∈ { 0 , 1 } N × n is defined such that C ij = 1 if π [ v i ] = ˜ v j , and C ij = 0 otherwise. Each entry C ij of the coarsening matrix is set to 1 if node v i is assigned to super-node ˜ v j . Since each node receives a unique hash value h i , it is exclusively mapped to a single super-node. This one-to-one assignment guarantees that every super-node has at least one associated node. As a result, each row of C contains exactly one non-zero entry, ensuring that its columns are mutually orthogonal. The matrix C therefore adheres to the structural properties defined in Equation 2.3. The adaptiveness of C stems from its sensitivity to local projection scores rather than fixed bin constraints.

Construction of the Coarsened Graph G c . The final coarsened graph G c = ( ˜ V , ˜ A, ˜ F ) is constructed from the coarsening matrix C . Two super-nodes ˜ v i and ˜ v j are connected if there exists at least one edge ( u, v ) ∈ E with u ∈ π -1 ( ˜ v i ) and v ∈ π -1 ( ˜ v j ) . The weighted adjacency matrix is obtained via matrix multiplication: ˜ A = C T A C . The super-node features are computed as the average of the features of the original nodes merged into the super-node: ˜ F i = 1 | π -1 ( ˜ v i ) | ∑ u ∈ π -1 ( ˜ v i ) F u . This ensures that the coarsened representation preserves the aggregate semantic and structural content of its constituent nodes. Since each super-edge aggregates multiple edges from the original graph, ˜ A is significantly sparser than A , leading to lower memory and computation requirements downstream. Algorithm 1 in Appendix G outlines the sequence of steps in our AH-UGC framework.

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3.2 Heterogeneous Graph Coarsening

In this section, we present AH-UGC's capability to handle heterogeneous graphs. Given a heterogeneous graph,

AH-UGC proceeds by first partitioning G by node type and independently applying the coarsening framework to each subgraph. This ensures that only semantically similar nodes are grouped into supernodes and that type-specific structure and features are preserved. Our approach naturally supports varying feature dimensions and allows different coarsening ratios η type across node types. Figure 7 in Appendix H illustrates this process, highlighting how AH-UGC preserves semantic meaning compared to other GC methods that merge heterogeneous nodes indiscriminately.

Construction of the Coarsened Heterogeneous Graph G c . The output of AH-UGC consists of a set of coarsening matrices

each of which maps original nodes of type t i.e., V ( t ) to their corresponding super-nodes ˜ V ( t ) . Using these mappings, we construct the coarsened graph

For each node type t , the coarsened feature matrix is computed as: ˜ X ( t ) = C ( t ) · X ( t ) , where rows of C ( t ) are row-normalized so that super-node features represent the average of their constituent nodes. The label matrix ˜ y ( paper ) is computed by majority voting over the labels of nodes merged into each super-node. To compute the coarsened edge matrices, for each edge type T e ∈ T E , we consider the interaction between supernodes of types node-type 1 and node-type 2 , corresponding to the edge relation e = ( node-type 1 , T e , node-type 2 ) ∈ ˜ E . The coarsened adjacency matrix ˜ A ( e ) is then computed as:

This formulation accumulates the edge weights between the original nodes to define the intersupernode connections, thereby preserving the structural connectivity patterns between different node-types of the original graph. Since each edge type is coarsened independently based on the mappings from its corresponding node types, G c preserves the heterogeneous semantics and topological relationships of the original graph G . Algorithm 2 in Appendix G outlines the sequence of steps in our AH-UGC framework. By leveraging consistent hashing, our method ensures balanced supernode formation. Theorem 3.2 provides a probabilistic upper bound on the number of nodes mapped to any supernode, thereby guaranteeing load balance across supernodes with high probability.

Theorem 3.2 (Explicit Load Balance via Random Rightward Merges) Let n nodes be sorted according to the consistent hashing scores defined earlier. Let k supernodes be formed by performing n -k random rightward merges in the sorted list. Then, for any constant c > 0 , the maximum number of nodes in any supernode S i satisfies:

Proof: The proof is deferred in Appendix C.

4 Experiments

We conduct comprehensive experiments to evaluate the effectiveness of AH-UGC. First, we validate its ability to perform adaptive graph coarsening . Second, we assess the quality of coarsened graphs using node classification accuracy and spectral similarity. Finally, we demonstrate AH-UGC's generalizability by evaluating its performance on heterogeneous graphs .

Datasets: We experiment on 23 widely-used benchmark datasets grouped into four categories:

Homophilic : Cora ,Citeseer, Pubmed [33], CS, Physics [34], DBLP [35];

Heterophilic : Squirrel, Chameleon, Texas, Cornell, Film, Wisconsin [36-39], Penn49, deezereurope, Amherst41, John Hopkins55, Reed98 [11];

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Heterogeneous : IMDB, DBLP, ACM [7, 25];

Large-scale : Flickr, Yelp , [14] ogbn-arxiv [6] , Reddit [40].

These datasets enable us to evaluate all six key components outlined in Section 2.1. For detailed dataset statistics and characteristics, refer to Table 5 in Appendix A.

System Specifications: All experiments are conducted on a server equipped with two NVIDIA RTX A6000 GPUs (48 GB memory each) and an Intel Xeon Platinum 8360Y CPU with 1 TB RAM .

4.1 Adaptive Coarsening Run-Time.

Given a graph G , we evaluate AH-UGC's ability to adaptively coarsen it to multiple resolutions, targeting a set of coarsening ratios R = { 55 , 50 , 45 , 40 , 35 , 30 , 25 , 20 , 15 , 10 } . As described in Section 3, AH-UGC leverages LSH and consistent hashing to group similar nodes into supernodes, enabling the construction of multiple coarsened graphs in a single pass. This adaptivity significantly reduces computational overhead compared to existing methods, which typically require reprocessing the entire graph for each target resolution. The computational advantages of our approach are evident in Table 1, where AH-UGC outperforms all

baseline methods by a significant margin, achieving the lowest coarsening time across all datasets and coarsening ratios, while maintaining scalability even on large-scale graphs where other methods fail.

4.2 Spectral Properties Preservation.

Following the experimental setup of [4, 19, 20] we use Hyperbolic Error (HE), Reconstruction Error (RcE) and Relative Eigen Error (REE) to indicate the structural similarity between G and G c . A more detailed discussion about these properties is included in Appendix F. Across three spectral evaluation metrics AH-UGC delivers performance that is comparable to, and in several cases surpasses, state-of-the-art methods, see Table 2. While there are minor dips in performance on a few datasets, this trade-off can be justified given the significant computational efficiency and scalability gains offered by our framework. These results underscore that AH-UGC achieves strong structural fidelity without compromising on runtime, making it especially suitable for large-scale or adaptive coarsening scenarios.

LSH and consistent hashing results. We empirically validates Theorem 3.1, see Figure 3. As ϵ increases, Pr[ | h ( x ) -h ( y ) | ≤ ε ] approaches 1, consistent with the theoretical erf-based bound. These results justify the use of consistent hashing, where each node is merged with its nearest clockwise neighbor. Theorem 3.1 and Figure 3 together guarantee that similar nodes are projected to nearby locations and are thus highly likely to be merged into a supernode.

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4.3 Node Classification Accuracy

Graph Neural Networks (GNNs) are widely used for node classification tasks [5, 40-42], where the goal is to predict labels for nodes based on both node features and the underlying graph structure. In this context, we evaluate the effectiveness of AH-UGC by examining how well it preserves predictive performance when downstream models are trained on coarsened graphs [43]. Specifically, we train several GNN models on the coarsened version of the original graph while evaluating their performance on the original graph's test nodes. As discussed earlier, our experimental setup spans a diverse collection of datasets, each with distinct structural characteristics. Following established practice in the literature, we employ different GNN backbones tailored to each graph type. For 'homophilic' datasets, we use GCN [5], Sage [40], GAT [41], GIN [42] and APPNP [43], which are well-suited to leverage dense neighborhood similarity. For 'heterophilic' datasets, we adopt GPRGNN [44], MixHop [45], H2GNN [46], GCN-II [47], GatJK [48] and SGC [49], which are designed to handle weak or inverse homophily. For 'heterogeneous' graphs, we use HeteroSGC, HeteroGCN, HeteroGCN2 [25] models that respect node and edge types during message passing. Complete architectural and hyperparameter details are provided in Appendix E. Due to space constraints, Table 3 reports node classification accuracy for homophilic and heterophilic graphs on a representative subset of datasets and GNN models. Please refer to Table 8 in Appendix E for comprehensive results across additional datasets and architectures. The AH-UGC framework consistently delivers results that are either on par with or exceed the performance of existing coarsening methods. As shown in Table 3, the framework is independent of any particular GNN architecture, highlighting its robustness and model-agnostic characteristics.

Performance on Heterogeneous Graphs: As outlined in Section 3, conventional graph coarsening techniques struggle with preserving the semantic integrity of heterogeneous graphs. In contrast,

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AH-UGC explicitly enforces type-aware coarsening, ensuring that supernodes are composed of nodes from a single type, thus maintaining the heterogeneity semantics. Table 4 presents node classification accuracies across various heterogeneous GNN models. AH-UGC consistently outperforms other methods due to its ability to preserve type purity within supernodes. This structural consistency enables all tested GNN architectures to achieve significantly higher classification performance. Figure 4 illustrates the degree of supernode impurity for each method. Each bar corresponds to a supernode and depicts the percentage distribution of node types within it. While supernodes generated by AH-UGC are entirely type-pure, those produced by baseline methods exhibit substantial cross-type mixing, leading to semantic drift and reduced model performance. Figure 5 analyzes the effect of increasing coarsening ratios on node classification accuracy. As expected, all methods experience performance degradation with aggressive coarsening. However, the drop is exponential for existing approaches due to rising impurity levels. In contrast, AH-UGC maintains structural purity across coarsening levels, resulting in a gradual, near-linear decline in accuracy. This robustness demonstrates AH-UGC's superior capacity to coarsen heterogeneous graphs while preserving their semantic and structural fidelity.

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Coarsening Ratio

5 Conclusion

In this paper, we propose AH-UGC, a unified framework for adaptive and heterogeneous graph coarsening. By integrating Locality-Sensitive Hashing (LSH) with Consistent Hashing, AH-UGC efficiently produces multiple coarsened graphs with minimal overhead. Additionally, its typeaware design ensures semantic preservation in heterogeneous graphs by avoiding cross-type node merges. The framework is model-agnostic, scalable, and capable of handling both heterophilic and heterogeneous graphs. We demonstrate that AH-UGC preserves key spectral properties, making it applicable across diverse graph types. Extensive experiments on 23 real-world datasets with various GNN architectures show that AH-UGC consistently outperforms existing methods in scalability, classification accuracy, and structural fidelity.

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A Datasets

We experiment on 24 widely-used benchmark datasets grouped into four categories: (a) Homophilic : Cora ,Citeseer, Pubmed [33], CS, Physics [34], DBLP [35]; (b) Heterophilic : Squirrel, Chameleon, Texas, Cornell, Film, Wisconsin [36-39], Penn49, deezer-europe, Amherst41, John Hopkins55, Reed98 [11]; (c) Heterogeneous : IMDB, DBLP, ACM [7, 25]; and (d) Large-scale : Flickr, Yelp , [14] ogbn-arxiv [6] , Reddit [40]. These datasets enable us to evaluate all six key components outlined in Section 2.1. Please refer to Table 5 and 6 for detailed dataset statistics and characteristics.

B Locality-Sensitive Hashing and Consistent Hashing

)

Locality-Sensitive Hashing (LSH) is a technique for hashing high-dimensional data points so that similar items are more likely to collide (i.e., hash to the same bucket) [32, 50, 51]. It is commonly used in approximate nearest neighbor search, dimensionality reduction, and randomized algorithms [52]. For example, a hash function h ( · ) is locality-sensitive with respect to a similarity measure s ( · , · if Pr[ h ( x ) = h ( y )] increases with s ( x, y ) . Gaussian LSH schemes, such as those using random projections, are particularly effective for preserving Euclidean distances [4, 22].

In the consistent hashing (CH) [23, 24] scheme, objects/requests are hashed to random bins/servers on the unit circle, as shown in Figure 6. Objects are then assigned to the closest bin in the clockwise direction. CH was originally proposed for load balancing in distributed systems; it maps data points

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and its tail probability is:

Choose t = log k + c k . Then:

to buckets such that small changes in input (e.g., adding or removing an object) do not drastically affect the overall assignment. We aim to employ CH for adaptive graph coarsening, as it enables stable and scalable grouping of similar objects/nodes. When combined with LSH, consistent hashing offers a powerful mechanism for adaptive graph reduction.

C Proof of Theorem 3.2

Theorem C.1 (Explicit Load Balance via Random Rightward Merges) Let n nodes be sorted according to the consistent hashing scores defined earlier. Let k supernodes be formed by performing n -k random rightward merges in the sorted list. Then, for any constant c > 0 , the maximum number of nodes in any supernode S i satisfies:

Proof Let U 1 , . . . , U k -1 ∼ Uniform (0 , 1) and let U (1) < · · · < U ( k -1) be their order statistics. Define the spacings:

Then ( I 1 , . . . , I k ) form a random partition of the unit interval [0 , 1] . It is a classical result (e.g., [53]) that:

The vector ( I 1 , . . . , I k ) ∼ Dirichlet (1 , . . . , 1) ,

Each individual spacing I i ∼ Beta (1 , k -1) .

Tail bound on I i . The PDF of I i is:

Union bound. Over all k intervals: 531

Scaling to n nodes. We model the sorted list of n nodes as uniformly spaced over [0 , 1] . Each 532 spacing I i then corresponds to a fraction of the list, and multiplying by n yields the expected number 533

of nodes in that supernode: 534

This completes the proof.

D Proof of Theorem 3.1

Theorem D.1 (Projection Proximity for Similar Points) Let x, y ∈ R d , and define the projection function:

Then the difference h ( x ) -h ( y ) ∼ N (0 , ℓ ∥ x -y ∥ 2 ) , and for any ε > 0 : 539

Proof Let z = x -y ∈ R d . Then: 540

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Each term r ⊤ j z is a linear projection of a standard Gaussian vector, hence:

Since the r j are independent, the sum of ℓ such independent variables is:

This is the cumulative probability within ε of a zero-mean Gaussian with variance ℓ ∥ x -y ∥ 2 . Let σ 2 = ℓ ∥ x -y ∥ 2 . Then:

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Now consider the probability:

as required.

E Node Classification Accuracy

Graph Neural Networks (GNNs), designed to operate on graph data [4, 54], have demonstrated strong performance across a range of applications [55-58]. Nevertheless, their scalability to large graphs remains a significant bottleneck. Motivated by recent efforts in scalable learning [43], we explore how our graph coarsening framework can improve the efficiency and scalability of GNN training, enabling more effective processing of large-scale graph data. Specifically, we train several GNN models on the coarsened version of the original graph while evaluating their performance on the original graph's test nodes. As discussed earlier in 4.3, our experimental setup spans a diverse collection of datasets, each with distinct structural characteristics. For homophilic graph settings, we follow the architectural configurations proposed in UGC [4], see Table 7. For heterophilic graphs, the GNN model designs are based on the implementations introduced in [11]. The heterogeneous GNN architectures are adopted directly from [25].

Table 8 reports node classification accuracy for homophilic and Table 9 reports node classification accuracy for heterophilic graphs. The AH-UGC framework consistently delivers results that are either on par with or exceed the performance of existing coarsening methods. As shown in Table 3, the framework is independent of any particular GNN architecture, highlighting its robustness and model-agnostic characteristics.

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F Spectral Properties

Relative Eigen Error (REE): REE used in [4, 19, 20] gives the means to quantify the measure of the eigen properties of the original graph G that are preserved in coarsened graph G c .

Definition F.1 REE is defined as follows:

where λ i and ˜ λ i are top k eigenvalues of original graph Laplacian ( L ) and coarsened graph Laplacian ( L c ) matrix, respectively.

Hyperbolic error (HE): HE [59] indicates the structural similarity between G and G c with the help of a lifted matrix along with the feature matrix X of the original graph.

Definition F.2 HE is defined as follows:

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where L is the Laplacian matrix and X ∈ R N × d is the feature matrix of the original input graph, L lift is the lifted Laplacian matrix defined in [20] as L lift = C L c C T where C ∈ R N × n is the coarsening matrix and L c is the Laplacian of G c .

Definition F.3 Let L be the original Laplacian matrix and L lift be the lifted Laplacian matrix, then the reconstruction error (RE) [19, 60] is defined as:

G Algorithms 579

H Heterogenous graph coarsening 580

Algorithm 2 Heterogeneous Graph Coarsening

Require: Graph G ( { X ( node_type ) } , { A ( edge_type ) } , { y ( target_type ) } ) , compression ratio η Ensure: Condensed graph G c ( { ˜ X ( node_type ) } , { ˜ A ( edge_type ) } , { ˜ Y ( target_type ) } ) 1: for each node type t do 2: r t ← η · | V t | 3: G coarse t , C t ← AH-UGC ( X t , A t , r t ) 4: ˜ X t ← node features from G coarse t 5: if t is target type then 6: ˜ y t [ i ] ← majority vote of y j for v j ∈ C t [ i ] 7: for each edge type e = ( t 1 , t 2 ) do 8: Initialize ˜ A e ∈ R | ˜ V t 1 |×| ˜ V t 2 | 9: for each ( v i , v j ) ∈ A e do 10: u ← super-node index of v i via C t 1 11: v ← super-node index of v j via C t 2 12: ˜ A e [ u, v ] ← ˜ A e [ u, v ] + 1 13: return G c ( { ˜ X ( node_type ) } , { ˜ A ( edge_type ) } , { ˜ Y ( target_type ) } )

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a novel framework for graph coarsening, Adaptive and Heterogeneous-Universal Graph Coarsening (AH-UGC), that addresses limitations in existing approaches. Traditional graph coarsening techniques often produce a single fixed coarsened graph, are inefficient for multiple target resolutions, and struggle with heterogeneous graphs involving different node and edge types. AH-UGC combines Locality-Sensitive Hashing (LSH) with Consistent Hashing (CH) to efficiently produce multiple coarsened graphs adaptively without redundant recomputation. Furthermore, it incorporates type-isolated coarsening to preserve semantic integrity in heterogeneous graphs by ensuring nodes of different types are not merged.\n\nContributions:\nThe authors propose AH-UGC, a unified framework for Adaptive and Heterogeneous Universal Graph Coarsening which integrate locality-sensitive hashing (LSH) with consistent hashing (CH). This framework supports adaptive, streaming, expanding, heterophilic, and heterogeneous graphs.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The proposed framework, AH-UGC, implements type-isolated coarsening by grouping nodes based on their types and performing coarsening separately within each group. This approach prevents the merging of nodes and edges with incompatible types, thereby preserving the semantic integrity of the original heterogeneous graph.\n2. AH-UGC is well-suited for streaming or dynamic graph scenarios, where new nodes and edges are continuously introduced. The proposed method efficiently integrates new nodes into the existing coarsened structure with minimal recomputation.\n\nWeaknesses:\n1. The paper is not well-written. It is a little difficult to understand the paper clearly which including motivation and the key contributions of the paper. \n2. The overall structure of the paper appears to lack clarity. While the authors present the addressed problems as two distinct goals, only one goal seems to be clearly pursued throughout the paper. Additionally, the meaning and significance of the main theorems, as well as the logical flow of their proofs, are not clearly conveyed. Furthermore, it is unclear which specific conclusions the experimental results are intended to support.\n\nQUESTIONS:\n1. What is the motivation and contributions of this paper? I hope authors can explain a bit clear since I am a little confused about the background of the problems addressed in the paper. As I mentioned in the weakness, I think it is only a writing problem and I hope it can help me understand the paper correctly.\n2. In the problems formulation section, the goal 2 is to learn a coarsening matrix. But the learning details is not clear in section 3.2 (if I understand correctly that section 3.2 is for it).\n3. The proofs for the theorems are unclear. In Appendix D, it appears that the authors did not provide a proof for Theorem 3.1 and instead proved a theorem that seems unrelated.\n4. What is the significance of Lemma 1? Its purpose and role within the paper are unclear.\n5.Which conclusions do the experiments support? This is somewhat unclear and leaves me a bit confused.\n\nLIMITATIONS:\nAuthors don't address the limitations of their work. I suggest authors respond my questions above.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel graph coarsening method based on locality-sensitive hashing. The approach applies random projections on the designed node feature vectors, and the coarsening process is then performed based on the aggregated outcomes of these projections. A key advantage of the method is its adaptability. It allows different coarsening ratios without requiring recomputation of the full coarsening pipeline. The authors provide theoretical insights to analyze which node pairs are more likely to be merged. The proposed method is evaluated on several different datasets, demonstrating competitive or superior performance against several baselines.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- The paper is well written and clearly structured.\n- Empirical results show competitive or superior performance against baselines.\n\n**Weaknesses**\n- Several experimental details are either unclear or missing.\n- Certain assumptions made in the theoretical results are not clearly stated or justified, and some theoretical claims (e.g., Lemma 1) lack complete proofs.\n- The paper proposes a limited novelty.\n\nQUESTIONS:\n**Questions**\n- The heterophily factor in Line 147 requires node labels to compute the vectors $F_i$​, but it's unclear how this is handled in scenarios where node labels are unavailable or partially known. If label information is used during coarsening but then withheld during node classification, this may create an unfair advantage in the downstream tasks compared to other methods. Please clarify this point.\n- The current node merging strategy relies on sorting nodes by their final hashing scores. However, due to the randomness inherent in LSH, structurally dissimilar nodes can have close similarity scores (node attributes might also be impactful) so it is unclear why this approach should be preferred over more structure-aware alternatives.\n- Is the proposed coarsening approach specifically designed for node classification tasks, or is it also applicable to other downstream tasks such as link prediction? If the latter, could the authors clarify how well the method preserves structural properties relevant for link prediction performance? A discussion or empirical evaluation in this context would help demonstrate the generality of the proposed framework.\n- Lemma 1 is given without any proof or reference. Please consider including the proof or a reference.\n- Theoretical results very similar to Theorem 3.1 have been studied in the literature, so please consider citing relevant works such as [1].\n- For the runtime comparisons in Table 1, do the reported times represent averages across all coarsening ratios or cumulative totals?\n- In Tables 2 and 3, spectral preservation and classification accuracy are only reported for a 50% coarsening ratio. Why was this specific value chosen, and how does performance vary with different coarsening levels?\n- Are the reported results based on a single run or averaged over multiple trials? How does the performance distribution look across different runs? How is the variation for different runs?\n- What is the dimensionality of the projection space used in the hashing step? Have you observed any sensitivity to this hyperparameter in terms of performance or stability?\n- While the paper claims that the proposed method can be applied to various graph types (e.g., streaming or dynamic graphs), the experiments focus only on node-attributed and knowledge graphs. Authors might provide empirical or theoretical justification to show its applicability to dynamic/streaming scenarios.\n- The caption of Table 1 refers to ratios, and according to the definition given in Line 172, they should lie between 0 and 1, but this is not the case. Could you please clarify it?\n\n[1] Matoušek, Jiří. \"On variants of the Johnson–Lindenstrauss lemma.\" Random Structures & Algorithms 33.2 (2008): 142–156.\n\n**Additional Comments**\n- Line 38 contains an incomplete sentence: it lacks a verb.\n- In Definition 2.2, the meaning of \"target labels y\" is unclear, as it has not been defined prior to this point.\n- Definitions 2.1 and 2.2 use inconsistent notation: in one case, the second entry denotes an adjacency matrix, and in another, it seems to represent the edge set.\n- In the appendix, the proof of Theorem 3.2 is given before the proof of Theorem 3.1\n\nLIMITATIONS:\nThe paper does not provide a clear discussion of the limitations of the proposed method. For instance, it is not evident whether the approach can be extended to downstream tasks beyond node classification. Addressing this point, along with the other potential limitations outlined in the Questions section, would strengthen the overall contribution and help clarify the scope of the method's applicability.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes AH-UGC, a new method for graph coarsening that makes large graphs smaller while preserving their structure and meaning. Unlike previous methods, it can 1) adaptively coarsen a graph to different sizes without starting from scratch, and 2) handle heterogeneous graphs by merging only nodes of the same type. It does this by combining two techniques: LSH to group similar nodes, and Consistent Hashing to efficiently merge them. The method is fast, scalable, and works for different types of graphs, including homophilic, heterophilic, and heterogeneous ones. Extensive experiments show that AH-UGC is more efficient and often more accurate than existing methods.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n1. The paper is clearly written and easy to follow.\n2. The proposed method is supported by solid theoretical analysis.\n\n### Weaknesses\n\n1. **Similarity to UGC** The proposed method is highly similar to UGC. Like UGC, AH-UGC uses LSH, runs in linear time, and supports heterophilic graphs. The main differences lie in adaptivity and heterogeneous support, but the core framework remains largely the same.\n2. **Reliance on Random LSH** The method depends on LSH, which is inherently randomized. It's unclear how robust the results are across different random seeds. Do the authors average over multiple runs or use a fixed seed? Additionally, is there potential to improve performance by learning the hashing function instead of using random projections?\n3. **Questionable Effectiveness of Coarsening** The benefits of graph coarsening are unclear. As shown in Tables 3 and 4, applying a 30%-50% coarsening ratio often leads to a drop of over 10% in accuracy. Figure 5 shows that pushing the coarsening ratio beyond 40% causes a sharp decline in performance. If coarsening brings only modest efficiency gains but significantly harms model quality, it's worth questioning whether the tradeoff is worthwhile.\n\nQUESTIONS:\n1. Is it possible to replace random LSH with a learned hashing function? Would this improve performance?\n2. How does graph coarsening impact other downstream tasks, such as graph-level prediction?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-04"}

comments: SUMMARY:
The paper seeks to empower current Graph Coarsening methods in two ways:
1) do GC in a "dynamic" fashion, allowing the GC ratio to be dynamically adjusted rather than computing new GC from scratch everytime,
2) handle heterogeneous nodes, i.e. different nodes types in a graph. 

The main method to achieve these goals is to integrate LSH and CH. Here CH handles the first problem and LSH handles the second problem.

STRENGTHS AND WEAKNESSES:
Strengths
1) The exposition is quite clear and lucid. 
2) The proposed model attains several attributes simultaneously, as compared to existing GC models. 
3) The experimental results are quite extensive.

Weaknesses 
1) I am not sure about the degree of technical novelty in the paper, in particular regarding the use of consistent hashing. 
2) For GNN node classification, the baselines (such as GCN) are too outdated. 
3) An ablation analysis would be required to compare the effect of LSH and CH.

QUESTIONS:
-

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 2
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-06