Title: GraphFLEx: Structure Learning $\underline{\text{F}}$ramework for $\underline{\text{L}}$arge $\underline{\text{Ex}}$panding $\underline{\text{Graph}}$s
Venue: NeurIPS 2025
Accepted: no
Keywords: Graph Neural Networks

Unsupervised Graph Structure Learning

Graph Coarsening

Graph Clustering
Abstract: Graph structure learning is a core problem in graph-based machine learning, essential for uncovering latent relationships and ensuring model interpretability. However, most existing approaches are ill-suited for large-scale and dynamically evolving graphs, as they often require complete re-learning of the structure upon the arrival of new nodes and incur substantial computational and memory costs. In this work, we propose GraphFLEx—a unified and scalable framework for Graph Structure Learning in Large and Expanding Graphs. GraphFLEx mitigates the scalability bottlenecks by restricting edge formation to structurally relevant subsets of nodes identified through a combination of clustering and coarsening techniques. This dramatically reduces the search space and enables efficient, incremental graph updates. The framework supports 48 flexible configurations by integrating diverse choices of learning paradigms, coarsening strategies, and clustering methods, making it adaptable to a wide range of graph settings and learning objectives. Extensive experiments across 26 diverse datasets and Graph Neural Network architectures demonstrate that GraphFLEx achieves state-of-the-art performance with significantly improved scalability. Our implementation is publicly available on this link https://anonymous.4open.science/r/Scaling_Graph_Learning-CB67/README.md.

Abstract

Graph structure learning is a core problem in graph-based machine learning, essential for uncovering latent relationships and ensuring model interpretability. However, most existing approaches are ill-suited for large-scale and dynamically evolving graphs, as they often require complete re-learning of the structure upon the arrival of new nodes and incur substantial computational and memory costs. In this work, we propose GraphFLEx-a unified and scalable framework for Graph Structure Learning in Large and Expanding Graphs. GraphFLEx mitigates the scalability bottlenecks by restricting edge formation to structurally relevant subsets of nodes identified through a combination of clustering and coarsening techniques. This dramatically reduces the search space and enables efficient, incremental graph updates. The framework supports 48 flexible configurations by integrating diverse choices of learning paradigms, coarsening strategies, and clustering methods, making it adaptable to a wide range of graph settings and learning objectives. Extensive experiments across 26 diverse datasets and Graph Neural Network architectures demonstrate that GraphFLEx achieves state-of-the-art performance with significantly improved scalability. Our implementation is publicly available here.

1 Introduction

Graph representations capture relationships between entities, vital across diverse fields like biology, 18 finance, sociology, engineering, and operations research [1-4]. While some relationships, such as 19 social connections or sensor networks, are directly observable, many, including gene regulatory 20 networks, scene graph generation [5], brain networks, [6] and drug interactions, require inference [7]. 21 Even when available, graph data often contains noise, requiring denoising and recalibration. In such 22 cases, inferring the correct graph structure becomes more crucial than the specific graph model or 23 downstream algorithm. 24 Graph Structure Learning (GSL) offers a solution, enabling the construction and refinement of graph 25 topologies. GSL has been widely studied in both supervised and unsupervised contexts [8, 9]. In 26 supervised GSL (s-SGL), the adjacency matrix and Graph Neural Networks (GNNs) are jointly 27 optimized for a downstream task, such as node classification. Notable examples of s-GSL include 28 NodeFormer [10], Pro -GNN [11], WSGNN [12], and SLAPS [13]. Unsupervised GSL 29 (u-SGL), on the other hand, focuses solely on learning the underlying graph structure, typically 30 through adjacency or Laplacian matrices. Methods in this category include approximate nearest 31 neighbours ( A -NN ) [14, 15], k-nearest neighbours ( k -NN ) [16, 17], covariance estimation 32 ( emp.Cov. ) [18], graphical lasso ( GLasso ) [19], SUBLIME [8], and signal processing techniques 33 like l 2 -model, log -model and large -model [20, 21]. 34 Supervised structure learning (s-SGL) methods have demonstrated effectiveness in specific tasks; 35 however, their reliance on labeled data and optimization for downstream objectives-particularly node 36 classification-significantly constrains their generalizability to settings where annotations are scarce 37 or unavailable [8]. Unsupervised structure learning (u-SGL) methods, which constitute the focus 38

Input Data:a)Xt,At or b)Xo

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of this work, offer broader applicability. Nevertheless, both s-SGL and u-SGL approaches exhibit 39 critical limitations in their ability to scale to large graphs or adapt efficiently to expanding datasets. 40

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To address these challenges, we introduce unified and scalable framework for

GraphFLEx

, a

Graph Structure Learn- ing in Large and Expanding Graphs

. GraphFLEx is built upon the coordinated integration of three foundational

paradigms in graph processing:

coarsening

, and graph clustering

,

graph structure learning

. While each of these methodologies has been studied extensively in isolation,

their joint

application within

a

single framework has remained largely unexplored. The novelty of GraphFLEx

lies not merely in combining these components, but in the principled manner in which they are algorithmically

aligned to reinforce one another-clustering serves to localize the search space, coarsening reduces structural

redundancy while

preserving global

properties, and

structure learning operates efficiently within this refined context.

This integration enables GraphFLEx to scale effectively to large datasets and accommodate dynamic

graphs through

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incremental updates the framework supports

, eliminating the need for expensive re-training. Additionally,

, enabling broad adaptability across datasets,

48 modular configurations learning objectives, and deployment constraints.

Crucially, we establish theoretical guarantees

on edge recovery fidelity and computational complexity, offering rigorous foundations for the framework's efficiency and reliability. As illustrated in Figure 2, GraphFLEx significantly reduces

the candidate edge space by operating on structurally relevant node subsets. Empirical evaluations, summarized in Figure 1, demonstrate that GraphFLEx substantially outperforms existing baselines in

both runtime and scalability.

Key contributions of this work include: 66

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GraphFLEx unifies multiple structure learning strategies within a single flexible framework.

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GraphFLEx demonstrates effectiveness in handling growing graphs .

GraphFLEx enhances the scalability of graph structure learning on large-scale graphs.

GraphFLEx serves as a comprehensive framework applicable individually for clustering, coarsening, and learning tasks.

Time in Seconds vs Number of Nodes (Each Line Represents an Experiment)

Time in Seconds vs Number of Nodes (Each Line Represents an Experiment)

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We provide both empirical and theoretical results , demonstrating the effectiveness of GraphFLEx across a range of datasets.

2 Problem Formulation and Background

A graph G is represented using G ( V, A, X ) where V = { v 1 , v 2 ...v N } is the set of N nodes, each node v i has a d -dimensional feature vector x i in X ∈ R N × d and A ∈ R N × N is adjacency matrix representing connection between i th and j th nodes when entry A ij > 0 . An expanding graph E G can be considered a variant of graph G where nodes v now have an associated timestamp τ v . We can represent a expanding graph as a sequence of graphs, i.e., E G = {G 0 , G 1 , ... G T } where {G 0 ⊆ G 1 .... ⊆ G T } at τ ∈ { 0 , ...T } timestamps. New nodes arriving at different timestamps are seamlessly integrating into initial graph G 0 .

Problem statement. Given a partially known or missing graph structure, our goal is to incrementally learn the whole graph, i.e., learn adjacency or laplacian matrix. Specifically, we consider two unsupervised GSL tasks:

Goal 1. Large Datasets with Missing Graph Structure: In this setting, the graph structure is entirely unavailable, and existing methods are computationally infeasible for learning the whole graph in a single step. To address this issue, we first randomly partition the dataset into exclusive subsets. We then learn the initial graph G 0 ( V 0 , X 0 ) over a small subset of nodes and incrementally expand it by integrating additional partitions, ultimately reconstructing the full graph G T .

Goal 2. Partially Available Graph: In this case, we only have access to the graph G t at timestamp t , with new nodes arriving over time. The goal is to update the graph incrementally to obtain G T , without re-learning it from scratch at each timestamp.

GraphFlex addresses these challenges with a unified framework, outlined in Section 3. Before delving into the framework, we review some key concepts.

2.1 Graph Reduction

Graph reduction encompasses sparsification, clustering, coarsening, and condensation [22]. GraphFlex employs clustering and coarsening to refine the set of relevant nodes for potential connections. Graph Clustering. Graphs often exhibit global heterogeneity with localized homogeneity, making them well-suited for clustering [23]. Clusters capture higher-order structures, aiding graph learning. Methods like DMoN [24] use GNNs for soft cluster assignments, while Spectral Clustering (SC) [25] and K-means [16, 26] efficiently detect communities. DiffPool [27, 28] applies SC for pooling in GNNs.

Graph Coarsening. Graph Coarsening (GC) reduces a graph G ( V, E, X ) with N nodes and features X ∈ R N × d into a smaller graph G c ( ˜ V , ˜ E, ˜ X ) with n ≪ N nodes and ˜ X ∈ R n × d . This is achieved via learning a coarsening matrix P ∈ R n × N , mapping similar nodes in G to super-nodes in G c , ensuring ˜ X = P X while preserving key properties [29-32].

2.2 Unsupervised Graph Structure Learning

Unsupervised graph learning spans from simple k-NN weighting [17, 33] to advanced statistical and graph signal processing (GSP) techniques. Statistical methods, also known as probabilistic graphical models, assume an underlying graph G governs the joint distribution of data X ∈ R N × d Some approaches [36] prune elements in the inverse sample covariance matrix ̂ Σ = 1 d -1 XX T and sparse inverse covariance estimators, such as Graphical Lasso (GLasso) [19]: maximize log det Θ -

[19, 34, 35]. Θ tr( ̂ ΣΘ ) -ρ ∥ Θ ∥ 1 , where Θ is the inverse covariance matrix. However, these methods struggle with small sample sizes. Graph Signal Processing (GSP) techniques analyze signals on known graphs, ensuring properties like smoothness and sparsity. Signal smoothness on a graph G is quantified by the Laplacian quadratic form: Q ( L ) = x T Lx = 1 2 ∑ i,j w ij ( x ( i ) -x ( j )) 2 . For a set of vectors X , smoothness is measured using the Dirichlet energy [37]: tr( X T LX ) . State-of-the-art methods [20, 21, 38] optimize Dirichlet energy while enforcing sparsity or specific structural constraints. Table 7 in Appendix D compares various graph learning methods based on their formulations and time complexities. More recently, SUBLIME [8] learns graph structure in an unsupervised manner by leveraging self-supervised contrastive learning to align a learnable graph with a dynamically refined anchor graph derived from the data itself.

Remark 1 . Graph Structure Learning (GSL) differs significantly from Continual Learning (CL) [39-41] and Dynamic Graph Learning (DGL) [42-44], as discussed in Appendix C.

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3 GraphFLEx

In this section, we introduce GraphFLEx, which has three main modules:

Graph Clustering. Identifies communities and extracts higher-order structural information,

Graph Coarsening. Is used to coarsen down the desired community, if the community itself is large,

Graph Learning. Learns the graph's structure using a limited subset of nodes from the clustering and coarsening modules, enabling scalability .

For pseudocode, see Algorithm 1 in Appendix G.

3.1 Incremental Graph Learning for Large Datasets

Real-world graph data is continuously expanding. For instance, e-commerce networks accumulate new clicks and purchases daily [45], while academic networks grow with new researchers and publications [46]. To manage such growth, we propose incrementally learning the graph structure over smaller segments.

Given a large dataset L ( V L , X L ) , where V L is the node set and X L represents node features, we define an expanding dataset setting L E = {E T τ =0 } . Initially, L is split into: (i) a static dataset E 0 ( V 0 , X 0 ) and (ii) an expanding dataset E = {E τ ( V τ , X τ ) } T τ =1 . Both Goal 1 (large datasets with missing graph structure) and Goal 2 (partially available graphs with incremental updates), discussed in Section 2, share the common objective of incrementally learning and updating the graph structure as new data arrives. GraphFLEx handles these by decomposing the problem into two key components:

Initial Graph G 0 ( V 0 , A 0 , X 0 ) : For Goal 1 , where the graph structure is entirely missing, E 0 ( V 0 , X 0 ) is used to construct G 0 from scratch using structure learning methods (see Section 2.2). For Goal 2 , the initial graph G 0 ( V 0 , A 0 , X 0 ) is already available and serves as the starting point for incremental updates.

Expanding Dataset E = {E τ ( V τ , X τ ) } T τ =1 : In both cases, E consists of incoming nodes and features arriving over T timestamps. These nodes are progressively integrated into the existing graph, enabling continuous adaptation and growth.

The partition is controlled by a parameter r , which determines the proportion of static nodes: r = ∥ V 0 ∥ ∥ V L ∥ . For example, r = 0 . 2 implies that 20% of V L is treated as static, while the remaining 80%

arrives incrementally over T timestamps. In our experiments, we set r = 0 . 5 and T = 25 . Remark 2 . We can learn G τ ( V τ , A τ , X τ ) by aggregating E τ nodes in G τ -1 graph. Our goal is to learn G T ( V T , A T , X T ) after T th -timestamp.

3.2 Detecting Communities

From the static graph G 0 , our goal is to learn higher-order structural information, identifying potential communities to which incoming nodes ( V ∈ V τ ) may belong. We train the community detection/clustering model M clust once using G 0 , allowing subsequent inference of clusters for all incoming nodes. While our framework supports spectral and k-means clustering, our primary focus has been on Graph Neural Network (GNN)-based clustering methods. Specifically, we use DMoN [24, 47, 48], which maximizes spectral modularity. Modularity [49] measures the divergence between intra-cluster edges and the expected number. These methods use a GNN layer to compute the partition matrix C = softmax ( MLP ( ˜ X,θ MLP )) ∈ R N × K , where K is the number of clusters and ˜ X is the updated feature embedding generated by one or more message-passing layers. To optimize the C matrix, we minimize the loss function ∆( C ; A ) = -1 2 m Tr ( C T BC ) + √ k n | Σ i C T i | F -1 , which combines spectral modularity maximization with regularization to prevent trivial solutions, where B is the modularity matrix [24]. Our static graph G 0 and incoming nodes E follow Assumption 1.

Assumption 1. Based on the well-established homophily principle, which forms the basis of most graph coarsening and learning methods. We assume that the generated graphs adhere to the DegreeCorrected Stochastic Block Model (DC-SBM) [50], where intra-class (or intra-community) links are more likely than inter-class links. DC-SBM, an extension of SBM that accounts for degree heterogeneity, making it a more flexible and realistic choice for real-world networks.

For more details on DC-SBM, see Appendix A.

Lemma 1. M clust Consistency. We adopt the theoretical framework of [50] for a DC-SBM with N nodes and k classes. The edge probability matrix is parameterized as P N = ρ N P , where P ∈ R k × k is a symmetric matrix containing the between/within community edge probabilities and it is independent of N , ρ N = λ N /N , and λ N is the average degree of the network. Let

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ˆ y N = [ˆ y 1 , ˆ y 2 , . . . , ˆ y N ] denote the predicted class labels, and let ˆ C N be the corresponding N × k one-hot matrix. Let the true class label matrix is C N , and µ is any k × k permutation matrix. Under the adjacency matrix A ( N ) , the global maximum of the objective ∆( · ; A ( N ) ) is denoted as ˆ C ∗ N . The consistency of class predictions is defined as:

Strong Consistency.

Weak Consistency.

where ∥ · ∥ F is the Frobenius norm. Under the conditions of Theorem 3.1 from [50]:

The M clust objective is strongly consistent if λ N / log( N ) →∞ , and

It is weakly consistent when λ N →∞ .

Remark 3 . Structure Learning within Communities. In GraphFLEx , we focus on learning the structure within each community rather than the structure of the entire dataset at once. Strong consistency ensures perfect community recovery, meaning no inter-community edges exist representing the ideal case. Weak consistency, however, allows for a small fraction ( ϵ ) of inter-community edges, where ϵ is controlled by ρ n in P n = ρ n P , influencing graph sparsity.

By Lemma 1 and Assumption 1, stronger consistency leads to more precise structure learning, whereas weaker consistency permits a limited number of inter-community edges.

3.3 Learning Graph Structure on a Coarse Graph

After training M clust, we identify communities for incoming nodes, starting with τ = 1 . Once assigned, we determine significant communities those with at least one incoming node and learn their connections to the respective community subgraphs. For large datasets, substantial community sizes may again introduce scalability issues. To mitigate this, we first coarsen the large community graph into a smaller graph and use it to identify potential connections for incoming nodes. This process constitutes the second module of GraphFLEx, denoted as M coar, which employs LSH-based hashing for graph coarsening. The supernode index for i th node is given as:

where r (bin width) controls the coarsened graph size, W represents random projection matrix, X is the feature matrix, and b is the bias term. For further details, refer to UGC [32]. After coarsening the i th community ( C i ), M coar ( C i ) = {P i , S i } yields a partition matrix P i ∈ R ∥ S i ∥×∥ C i ∥ and a set of coarsened supernodes ( S i ), as discussed in Section 2.

To identify potential connections for incoming nodes, we define their neighborhood as follows:

Definition 1. The neighborhood of a set of nodes E i is defined as the union of the top k most similar nodes in C i for each node v ∈ E i , where similarity is measured by the distance function d ( v, u ) . A node u ∈ C i is considered part of the neighborhood if its distance d ( v, u ) is among the k smallest distances for all u ′ ∈ C i .

Goal 3. The neighborhood of incoming nodes N k ( E i ) represents the ideal set of nodes where the incoming nodes E i are likely to establish connections when the entire community is provided to a structure learning framework.. A robust coarsening framework must reduce the number of nodes within each community C i while ensuring that the neighborhood of the incoming nodes is preserved.

3.4 Graph Learning only with Potential Nodes

As we now have a smaller representation of the community, we can employ any graph learning algorithms discussed in Section 2.2 to learn a graph between coarsened supernodes S i and incoming nodes ( V i τ ∈ V τ ). This is the third module of GraphFLEx, i.e., graph learning; we denote it as M gl . The number of supernodes in S i is much smaller compared to the original size of the community, i.e., ∥ S i ∥ ≪ ∥ C i ∥ ; scalability is not an issue now. We learn a small graph first using M gl ( S i , X i τ ) = ˜ G i τ ( V c τ , A c τ ) where X i τ represents features of new nodes belonging to i th

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community at time τ , ˜ G i τ ( V c τ , A c τ ) representing the graph between supernodes and incoming nodes. Utilizing the partition matrix P i obtained from M coar, we can precisely determine the set of nodes associated with each supernode. For every new node V ∈ V i τ , we identify the connected supernodes and subsequently select nodes within those supernodes. This subset of nodes is denoted by ω V i τ , the sub-graph associated with ω V i τ represented by G i τ -1 ( ω V i τ ) then undergoes an additional round of graph learning M gl ( G i τ -1 ( ω V i τ ) , X i τ ) , ultimately providing a clear and accurate connection of new nodes V i τ with nodes of G τ -1 , ultimately updating it to G τ . This multi-step approach, characterized by coarsening, learning on coarsened graphs, and translation to the original graph, ensures scalability.

Theorem 1. Neighborhood Preservation. Let N k ( E i ) denote the neighborhood of incoming nodes E i for the i th community. With partition matrix P i and M gl ( S i , X i τ ) = G c τ ( V c τ , A c τ ) we identify the supernodes connected to incoming nodes E i and subsequently select nodes within those supernodes; this subset of nodes is denoted by ω V i τ . Formally,

̸

Then, with probability Π { ∈ } p ( c ) , it holds that N k ( E i ) ⊆ ω i where

and ϕ is a set containing all pairwise distance values ( c = ∥ v -u ∥ ) between every node v ∈ E i and the nodes u ∈ ω V i τ . Here, π -1 ( s ) denotes the set of nodes mapped to supernode s, r is the bin-width hyperparameter of M coar .

Proof. The proof is deferred in Appendix B.

Remark 4 . Theorem 1 establishes that, with a constant probability of success, the neighborhood of incoming nodes N k ( E i ) can be effectively recovered using the GraphFLEx multistep approach, which involves coarsening and learning on the coarsened graph, i.e., N k ( E i ) ⊆ ω V i τ . The set ω V i τ , estimated by GraphFLEx, identifies potential candidates where incoming nodes are likely to connect. The probability of failure can be reduced by regulating the average degree of connectivity in M gl ( S i , X i τ ) = G c τ ( V c τ , A c τ ) . While a fully connected network G c τ ensures all nodes in the community are candidates, it significantly increases computational costs for large communities.

3.5 GraphFLEx: Multiple SGL Frameworks Each module in Figure 3 controls a distinct aspect of the graph learning process: clustering influences community detection, coarsening reduces graph complexity via supernodes, and the learning module governs structural inference. Altering any of these modules results in a new graph learning method. Currently, we support 48 different graph learning configurations, and this number scales exponentially with the addition of new methods to any module. The number of possible frameworks is given by α × β × γ , where α , β , and γ represent the number of clustering, coarsening, and learning methods, respectively.

3.6 Run Time Analysis

UGC

Log model

FGC

L2 model

Kmeans

LVE

KNN

Spectral

Clustering

LVN

Empirical

Covariance

DMoN

HEM

GLasso

DiffPool

Kron

Large model

GCond

ANN

Graph Clustering

Graph Coarsening

GraphFLEx computational time is always bounded by existing approaches, as it operates on a significantly reduced set of nodes. We evaluate the run-time complexity of GraphFLEx in two

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scenarios: (a) the worst-case scenario, where computationally intensive clustering and coarsening modules are selected, providing an upper bound on time complexity, and (b) the best-case scenario, where the most efficient modules are chosen. Table 1 presents a summary of this analysis for both cases. Due to space limitations, a more comprehensive analysis is provided in Appendix E.

4 Experiments

Tasks and Datasets. To validate GraphFLEx's utility, we evaluate it across four key dimensions: (i) computational efficiency, (ii) scalability to large graphs, (iii) quality of learned structures, and (iv) adaptability to dynamically growing graphs. To validate the characteristics of GraphFLEx, we conduct extensive experiments on 26 different datasets, including (a) datasets that already have a complete graph structure (allowing comparison between the learned and the original structure), (b) datasets with missing graph structures, (c) synthetic datasets, (d) small datasets for visualizing the graph structure, and (e) large datasets, including datasets with even 2 . 4 M nodes. More details about datasets and system specifications are presented in Table 8 in Appendix F.

4.1 Computational Efficiency.

Existing methods like k -NN and log -model struggle to learn graph structures even for 20k nodes due to out-of-memory (OOM) or out-of-time (OOT) issues, while l 2 -model and large -model struggle beyond 50k nodes. Although A -NN and emp -Covar. are faster, GraphFLEx outperforms them on sufficiently large graphs (Table 2). While traditional methods may be efficient for small graphs, GraphFLEx scales significantly better, excelling on large datasets like Pubmed and Syn 5 , where most methods fail. It accelerates structure learning, making A -NN 3× faster and emp -Covar. 2× faster.

4.2 Node Classification Accuracy

Experimental Setup. We now evaluate the prediction performance of GNN models when trained on graph structures learned from three distinct scenarios: 1) Original Structure: GNN models trained

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on the original graph structure, which we refer to as the Base Structure, 2) GraphFLEx Structure: GNN models trained on the graph structure learned from GraphFLEx, and 3)Vanilla Structure: GNN models trained on the graph structure learned from other existing methods.

For each scenario, a unique graph structure is obtained. We trained GNN models on each of these three structure. For more details on GNN model parameters, see Appendix H.

GNNModels. Graph neural networks (GNNs) such as GCN [51], GraphSage [52], GIN [53], and GAT [54] rely on accurate message passing, dictated by the graph structure, for effective embedding. We use these models to evaluate the above-mentioned learned structures. Table 3 reports node classification performance across all methods. Notably, GraphFLEx outperforms vanilla structures by a significant margin across all datasets, achieving accuracies close to those obtained with the original structure. Figure 8 in Appendix H illustrates GraphSage classification results, highlighting GraphFLEx's superior performance. For the CS dataset, GraphFLEx ( large -model) and GraphFLEx ( empCovar. -model) even surpass the original structure, demonstrating its ability to preserve key structural properties while denoising edges, leading to improved accuracy.

4.3 Scalability of GraphFLEx on Large-Scale Graphs.

To comprehensively evaluate GraphFLEx's scalability to large-scale graphs, we consider four datasets with a high number of nodes: (a) Flickr(89k nodes) [55], (b) Reddit (233k nodes) [55], (c) Ogbn-arxiv (169k nodes) [46], and (d) Ogbn-products (2.4M nodes) [56]. As shown in Table 4, GraphFLEx consistently demonstrates superior scalability across all datasets, outperforming all baseline methods in runtime. In particular, methods such as log-model , l2-model , and large-model fail to run even on Flickr , while GraphFLEx successfully scales them on Flickr , Ogbn-arxiv , and Reddit , enabling structure learning where others cannot. For the most computationally demanding dataset, Ogbn-products , these methods remain prohibitively expensive even for GraphFLEx. Nonetheless, GraphFLEx efficiently supports scalable structure learning on Ogbn-products using the Covar , ANN , and KNN modules. Table 4 also reports node classification accuracy, demonstrating that GraphFLEx maintains performance comparable to the original (base) structure across all datasets. These results confirm that GraphFLEx not only scales effectively, but also preserves the quality of learned structures.

4.4 GraphFLEx for Link Prediction and Graph Classification.

To further validate the generalization of our framework, we evaluate GraphFLEx on the link prediction task. The results are presented in Table 5, following the same setting as Table 3. The structure learned by GraphFLEx demonstrates strong predictive performance, in some cases even outperforming the base structure. This highlights the effectiveness of GraphFLEx in preserving and even enhancing relational information relevant for link prediction. While our primary focus is on structure learning

for node-level tasks, we briefly discuss the applicability of GraphFLEx to graph classification. In such tasks, especially in domains like molecule or drug discovery, each data point often corresponds to a small individual subgraph. For these cases, applying clustering and coarsening is typically redundant and may introduce unnecessary computational overhead. Nevertheless, GraphFLEx

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(a) 10 incoming

nodes

(b) 20 incoming

nodes

(c) 30 incoming

nodes

(d) ANN as

M

gl

(e) Emp. Covr. as

M

gl

(f) kNN as

M

gl

remains flexible-its learning module can be directly used without the clustering or coarsening steps, making it suitable for graph classification as well. This adaptability reinforces GraphFLEx's utility across a broad range of graph learning tasks.

4.5 Clustering Quality

We measure three metrics to evaluate the resulting clusters or community assignments: a) Normalized Mutual Information (NMI) [24] between the cluster assignments and original labels; b) Conductance ( C ) [57] which measures the fraction of total edge volume that points outside the cluster; and c) Modularity ( Q ) [49] which measures the divergence between the intra-community edges and the expected one. Table 6 illustrates these metrics for single-cell

RNA and the MNIST dataset (where the whole structure is missing), and Figure 12 in Appendix K shows the PHATE [58] visualization of clusters learned using GraphFLEx's clustering module M clust . We also train the aforementioned GNN models for the node classification task in order to illustrate the efficacy of the learned structures; the accuracy values presented in Table 6, clearly highlight the significance of the learned structures, as reflected by the high accuracy values.

4.6 Structure Visualization

We evaluate the structures generated by GraphFLEx through visualizations on four small datasets: (i) MNIST [59], consisting of handwritten digit images, where Figure 5(a) shows that images of the same digit are mostly connected; (ii) Pre-trained GloVe embeddings [60] of English words, with Figure 5(b) revealing that frequently used words are closely connected; (iii) A synthetic H.E dataset (see Appendix F), demonstrating GraphFLEx's ability to handle

expanding networks without requiring full relearning. Figure 4(a-c) shows the graph structure evolving as 30 new nodes are added over three timestamps; and (iv) Zachary's karate club network [61], which highlights GraphFLEx's multi-framework capability. Figure 4(d-f) shows three distinct graph structures after altering the learning module. For a comprehensive ablation study, refer to Appendix L.

5 Conclusion

Large or expanding graphs challenge the best of graph learning approaches. GraphFLEx, introduced in this paper, seamlessly adds new nodes into an existing graph structure. It offers diverse methods for acquiring the graph's structure. GraphFLEx consists of three key modules: Clustering, Coarsening, and Learning which empowers GraphFLEx to serves as a comprehensive framework applicable individually for clustering, coarsening, and learning tasks. Empirically, GraphFLEx outperforms state-of-the-art baselines, achieving up to 3× speedup while preserving structural quality. It achieves accuracies close to training on the original graph, in most instances. The performance across multiple real and synthetic datasets affirms the utility and efficacy of GraphFLEx for graph structure learning. Limitations and Future Work. GraphFLEx is designed assuming minimal inter-community connectivity, which aligns well with many real-world scenarios. However, its applicability to heterophilic graphs may require further adaptation. Future work will focus on extending the framework to supervised GSL methods and heterophilic graphs, broadening its scalability and versatility.

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Expanding S 2 ( c ) : 592

Appendix

A Degree-Corrected Stochastic Block Model(DC-SBM)

The DC-SBM is one of the most commonly used models for networks with communities and postulates that, given node labels c = c 1 , ...c n , the edge variables A ′ ij s are generated via the formula

, where θ i is a "degree parameter" associated with node i , reflecting its individual propernsity to form ties, and P is a K × K symmetric matrix containing the between/withincommunity edge probabilities and P c P c denotes the edge probabilities between community c i and c j .

i j For DC-SBM model [50] assumed P n on n nodes with k classes, each node v i is given a label/degree pair( c i , θ i ), drawn from a discrete joint distribution Π K × m which is fixed and does not depend on n. This implies that each θ i is one of a fixed set of values 0 ≤ x 1 ≤ .... ≤ x m . To facilitate analysis of asymptotic graph sparsity, we parameterize the edge probability matrix P as P n = ρ n P where P is independent of n , and ρ n = λ n /n where λ n is the average degree of the network.

B Neighbourhood Preservation

Theorem 2. Neighborhood Preservation. Let N k ( E i ) denote the neighborhood of incoming nodes E i for the i th community. With partition matrix P i and M gl ( S i , X i τ ) = G c τ ( V c τ , A c τ ) we identify the supernodes connected to incoming nodes E i and subsequently select nodes within those supernodes; this subset of nodes is denoted by ω V i τ . Formally,

̸

Then, with probability Π { c ∈ ϕ } p ( c ) , it holds that N k ( E i ) ⊆ ω V i τ where

and ϕ is a set containing all pairwise distance values ( c = ∥ v -u ∥ ) between every node v ∈ E i and the nodes u ∈ ω V i τ . Here, π -1 ( s ) denotes the set of nodes mapped to supernode s, r is the bin-width hyperparameter of M coar .

Proof: The probability that LSH random projection [32, 62] preserves the distance between two nodes v and u i.e., d ( u, v ) = c , is given by:

where f 2 ( x ) = 2 √ 2 π e -x 2 / 2 represents the Gaussian kernel when the projection matrix is randomly sampled from p -stable( p = 2 ) distribution [62].

The probability p ( c ) can be decomposed into two terms:

S 1 ( c ) and S 2 ( c ) are defined as follows:

593

Thus, the probability p ( c ) can be bounded as: 594

Now, let ϕ be the set of all pairwise distances d ( u, v ) , where v ∈ E i and node ω V i τ . The probability 595 that all nodes in N k ( E i ) are preserved within ω V i τ , requires that all distances c ∈ ϕ are also preserved. 596 The probability is then given by: 597

C Continual Learning and Dynamic Graph Learning

In this subsection, we highlight the key distinctions between Graph Structure Learning (GSL) and related fields to justify our specific selection of related works in Section 2.2. GSL is often confused with topics such as Continual Learning (CL) and Dynamic Graph Learning (DGL).

CL [39-41] addresses the issue of catastrophic forgetting, where a model's performance on previously learned tasks degrades significantly after training on new tasks. In CL, the model has access only to the current task's data and cannot utilize data from prior tasks. Conversely, DGL [42-44] focuses on capturing the evolving structure of graphs and maintaining updated graph representations, with access to all prior information.

While both CL and DGL aim to enhance model adaptability to dynamic data, GSL is primarily concerned with generating high-quality graph structures that can be leveraged for downstream tasks such as node classification [51], link prediction [63], and graph classification [64]. Moreover, in CL and DGL, different tasks typically involve distinct data distributions, whereas GSL assumes a consistent data distribution throughout.

D Related Work

Table 7 presents the formulations and associated time complexities of various unsupervised Graph Structure Learning methods.

E Run Time Analysis

In the context of clustering module, k -NN is the fastest algorithm, while Spectral Clustering is the slowest. Suppose we aim to learn the structure of a graph with N nodes. The clustering module, however, is only applied to a randomly sampled, smaller, static subgraph with k nodes, where k ≪ N . In the worst-case scenario, spectral clustering requires O ( k 3 ) time, whereas in the best case, k -NN requires O ( k 2 ) time. For coarsening module, LSH-based coarsening framework [30], has the best time complexity of O ( k τ c ) while FGC denotes the worst case with a time-complexity of O (( k τ c ) 2 ∥ S i τ ∥ ) where c is the number of communities detected by clustering module M clust , ∥ S i τ ∥ is the number of coarsened node in the relevant community at τ timestamp and k τ denotes number of nodes at τ timestamp. For learning module, A -NN is the most efficient algorithm with time

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complexity as O ( NlogN ) , while GLasso has the worst computational cost of O ( N 3 ) . So, the effective time complexity of GraphFLEx is upper bounded by O ( k 3 +( k τ c ) 2 ∥ S i τ ∥ + α 3 ) and lower bounded by O ( k 2 + k τ c + αlogα ) where α = ∥ S i τ ∥ + ∥E i τ ∥ . GraphFLEx's efficiency in term of computational time is evident in Figure 1 and further quantified in Table 2. Out of the three modules of GraphFLEx first module( M clust) is trained once, and hence its run time is always bounded; computational time for second module( M coar) can also be controlled because some of the methods either needs training once [65] or have linear time complexity [30]. Consequently, both the clustering and coarsening modules contribute linearly to the overall time complexity, denoted as O ( N ) . Thus, the effective time complexity of GraphFLEx is given by O ( N + O ( M gl ( ∥ S i , X i τ ∥ )) . The overall complexity scales either linearly or sub-linearly, depending on α and the M gl module. For instance, when M gl is A -NN the complexity remains linear, if α log( α ) ≈ N , whereas for GLasso , a linear behavior is observed when α 3 ≈ N .

F Datasets

Datasets used in our experiments vary in size, with nodes ranging from 1k to 60k. Table 8 lists all the datasets we used in our work. We evaluate our proposed framework GraphFlex on realworld datasets Cora ,Citeseer, Pubmed [66], CS, Physics [67], DBLP [68], all of which include graph structures. These datasets allow us to compare the learned structures with the originals. Additionally, we utilize single-cell RNA pancreas datasets [69], including Baron, Muraro, Segerstolpe, and Xin, where the graph structure is missing. The Baron dataset was downloaded from the Gene Expression Omnibus (GEO) (accession no. GSE84133). The Muraro dataset was downloaded from GEO (accession no. GSE85241). The Segerstolpe dataset was accessed from ArrayExpress (accession no. E-MTAB-5061). The Xin dataset was downloaded from GEO (accession no. GSE81608). We simulate the expanding graph scenario by splitting the original dataset across different T timestamps. We assumed 50% of the nodes were static, with the remaining nodes arriving as incoming nodes at different timestamps.

Synthetic datasets: Different data generation techniques validate that our results are generalized to different settings. Please refer to Table 8 for more details about the number of nodes, edges, features, and classes, Syn denotes the type of synthetic datasets. Figure 6 shows graphs generated using different methods. We have employed three different ways to generate synthetic datasets which are mentioned below:

PyGSP(PyGsp): We used synthetic graphs created by PyGSP [70] library. PyG-G and PyG-S denotes grid and sensor graphs from PyGSP.

Watts-Strogatz's small world(SW): [71] proposed a generation model that produces graphs with small-world properties, including short average path lengths and high clustering.

Heterophily(HE): We propose a method for creating synthetic datasets to explore graph behavior across a heterophily spectrum by manipulating heterophilic factor α , and classes. α is determined by dividing the number of edges connecting nodes from different classes by the total number of edges in the graph.

Visulization Datasets: To evaluate, the learned graph structure, we have also included three datasets: (i) MNIST [59], consisting of handwritten digit images; (ii) Pre-trained GloVe embeddings [60] of English words; and (iii) Zachary's karate club network [61].

Large Datasets: To comprehensively evaluate GraphFLEx's scalability to large-scale graphs, we consider four datasets with a high number of nodes: (a) Flickr(89k nodes) [55], (b) Reddit (233k nodes) [55], (c) Ogbn-arxiv (169k nodes) [46], and (d) Ogbn-products (2.4M nodes) [56].

System Specifications: All the experiments conducted for this work were performed on an Intel Xeon W-295 CPU with 64GB of RAM desktop using the Python environment.

(a) PyGSP-Sensor, N = 50,

α

=3

(b) PyGSP-Grid, N = 80,

α

=3

(c) SW, N = 50,

α

=3

(d) HE, N = 50,

α

=3

G Algorithm 672

Algorithm 1 GraphFlex: A Unified Structure Learning framework for expanding and Large Scale Graphs

H Other GNN models 673

We used four GNN models, namely GCN, GraphSage, GIN, and GAT. Table 9 contains parameter 674

details we used to train GraphFlex. We have used these parameters across all methods. 675

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Figure 7 illustrates the pipeline for training our GNN models. Graph structures were learned using 676 both existing methods and GraphFlex, and GNN models were subsequently trained on both structures. 677

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We randomly split data in 60%, 20%, 20% for training-validation-test. The results for these models 679 on synthetic datasets are presented in Table 10. 680

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Figure 7 illustrates the pipeline for training our GNN models. Graph structures were learned using both existing methods and GraphFlex, and GNN models were subsequently trained on both structures.

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I Computational Efficiency

Table 11 illustrates the remaining computational time for learning graph structures using GraphFLEx with existing Vanilla methods on Synthetic datasets. While traditional methods may be efficient for small graphs, GraphFLEx scales significantly better, excelling on large datasets like Pubmed and Syn 5 , where most methods fail.

J Visualization of Growing graphs

This section helps us visualize the phases of our growing graphs. We have generated a synthetic graph of 60 nodes using PyGSP-Sensor and HE methods mentioned in Appendix F. We then added 40 new nodes denoted using black color in these existing graphs at four different timestamps. Figure 9 and Figure 10 shows the learned graph structure after each timestamp for two different Synthetic graphs.

K Clustering Quality

Figure 12 shows the PHATE [58] visualization of clusters learned using GraphFLEx's clustering module M clust for 6 single-cell RNA datasets, namely Xin , MNIST , Baron -Human , Muraro , BaronMouse , and Segerstolpe datasets.

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In this section, we present an ablation study to analyze the role of individual modules within 698 GraphFLEx and their influence on the final graph structure. Specifically, we focus on two aspects: (i) 699 the significance of the clustering module, and (ii) the effect of varying module configurations on the 700 learned graph topology. 701

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To evaluate the effectiveness of the clustering module, we compute standard metrics such as Normal703

ized Mutual Information (NMI), Conductance (C), and Modularity (Q) across various datasets (see 704

Table 6 in Section 4.5). These metrics collectively validate the quality of the discovered clusters, 705

thereby justifying the use of a clustering module as a foundational step in GraphFLEx. Since cluster706

ing in GraphFLEx is applied only once on a randomly sampled small set of nodes, selecting the right 707

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method can be considered as part of hyperparameter tuning, where these clustering measures can guide the optimal choice based on dataset characteristics.

L.2 Impact of Module Choices on Learned Graph Structure

This section involves a comparison of the graph structure learned from GraphFlex with existing 711 methods. Six nodes were randomly selected and considered as new nodes. Figure 13 visually depicts 712 the structures learned using GraphFlex compared to other methods. It is evident from the figure that 713 the structure known with GraphFlex closely resembles the original graph structure. Figure 11 shows 714 the original structure of Zachary's karate club network [61]. We assumed six random nodes to be 715 dynamic nodes, and the structure learned using GraphFlex compared to existing methods is shown in 716 Figure 13. 717

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes GraphFLEx, a scalable and modular framework for unsupervised graph structure learning in large and dynamically expanding graphs. By combining graph clustering, coarsening, and structure learning, GraphFLEx enables efficient incremental updates and supports multiple method configurations. Experiments on 26 datasets demonstrate its advantages in scalability, runtime, and predictive performance compared to existing methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The paper addresses a clear and important problem: scalable structure learning for expanding graphs.\n\n2. The integration of clustering, coarsening, and learning in one flexible framework is well-motivated.\n\n3. The framework is modular and extensible, supporting 48 combinations of clustering/coarsening/learning strategies.\n\n4. The experiments are comprehensive, covering a wide range of datasets (small to large-scale, synthetic and real-world).\n\n5. The framework consistently outperforms or matches the performance of existing baselines in classification and link prediction tasks, especially under scalability constraints.\n\nWeaknesses:\n\n1. Several technical components (e.g., parameter settings, coarsening via LSH, community detection) are described at a high level and need more detail.\n\n2. The assumption of strong community structure (homophily/DC-SBM) limits generalizability. Performance on heterophilic graphs is not explored.\n\n3. GraphFLEx introduces significant overhead on small graphs compared to vanilla methods. The cost-benefit tradeoff is unclear in such cases.\n\n4. The incremental graph learning setting is a major claim of the paper, but lacks a direct comparison to full re-training baselines to highlight its practical value.\n\n5. The guidance on selecting among the 48 configuration variants is limited.\n\nQUESTIONS:\n1. In Table 2, GraphFLEx performs slower than vanilla methods on small datasets. Could the authors explain the source of this overhead? Which modules (clustering, coarsening, etc.) contribute most to the added cost?\n\n2. How should users choose the partitioning ratio r and number of timestamps T? Are there any guidelines or ablation results that indicate their impact on final accuracy and efficiency?\n\n3. In some cases, GraphFLEx achieves performance even better than the original graph structure. Could the authors explain whether this is due to denoising effects or improved structure inference? What conditions enable such gains?\n\n4. The method relies heavily on the assumption of community homophily. How would the method perform under heterophilic settings, where inter-community links are common? Is there a way to relax this assumption?\n\n5. Given the framework’s flexibility, is there a practical recommendation on how to select clustering, coarsening, and learning components for new datasets? Some empirical guidelines would be very useful.\n\nLIMITATIONS:\nPartially addressed. The authors briefly acknowledge that the method may not generalize well to heterophilic graphs due to the DC-SBM assumption. However, this point should be discussed more explicitly, possibly with experimental illustration. Furthermore, potential overhead for small datasets and the dependency on hyperparameter tuning are also notable limitations worth mentioning.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes GraphFLEX, a unified structure learning framework for large-scale and expanding graphs. The method incorporates three main components: a clustering module, a coarsening module, and a graph learner. Experiments are conducted on a wide range of tasks, including node classification, link prediction, clustering quality, and scalability evaluation.\n\nSTRENGTHS AND WEAKNESSES:\nStregnths:\n\n1. The paper addresses a significant and underexplored challenge, i.e., scalable and flexible structure learning on expanding graphs.\n\n2. The authors conduct comprehensive experiments covering node classification, link prediction, scalability (on large graphs like ogbn-products), and include metrics like NMI, conductance, and modularity to assess clustering quality.\n\n3. The proposed GraphFLEX offers a clean abstraction that decouples clustering, coarsening, and structure learning, which makes it adaptable to different tasks and datasets.\n\n\nWeaknesses\n\n1. The paper does not compare GraphFLEX with existing state-of-the-art graph structure learning methods (e.g., NodeFormer, ProGNN, etc.). It primarily compares to vanilla structures (e.g., KNN, ANN), which are relatively basic baselines and do not fully validate the performance gains from a structural learning perspective.\n\n2. The implementation of the graph learner is not clearly described, lacking architectural or algorithmic details. That is, it’s unclear whether it uses a similarity-based function, attention mechanism, or neural architecture for edge prediction. This limits reproducibility and understanding of its novelty.\n\n3. It's unclear whether the clustering, coarsening, and learning modules are jointly optimized using supervision signals, or if they are decoupled. The training objective and whether gradients flow through all modules need clarification.\n\n4. Several key parts are underexplained. For example, 1) in line 164-165, authors claim \"these methods use a GNN layer to compute the partition matrix C=softmax(MLP())\", but it is unclear how the output of MLP integrates with the GNN layer. 2) The purpose of Equation (1), which lacks sufficient context or follow-up.\n\nQUESTIONS:\n1. What is the architecture and training objective of the graph learner module? Please clarify its inputs, architecture, and how it is supervised.\n\n2. Is the entire framework trained end-to-end using downstream supervision, or are clustering and coarsening frozen after initial pretraining?\n\n3. There are no comparisons to more advanced structure learning baselines, such as ProGNN. How does GraphFLEX compare when those methods are applicable (e.g., static datasets)?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nGraph Structure Learning (GSL) is always an important problem for machine learning, since it can adaptively extract meaningful structure or topological information between nodes. The objective of this work is to address the drawbacks of existing methods when the confront ill-suited issues for large-scale and dynamically evolving graphs. Spcifically, this work proposes a new GraphFLEx model for GSL problem. The GraphFLEx mainly has the  following key computational procedure, i.e., the graph clustering, the graph coarsening, and the structure learning. Since the GraphFLEx restricts the edge to relevant node subsets based on the clustering and coarsening process, it can mitigates the computational buerdensome problem. Moreover, the coarsening process can guarantee that the GraphFLEx has less searching space, and can be employed for large-scale graph datasets. Experiments demonstrate the performance of the new GraphFLEx.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nThis paper is overall well-written and the idea is also interesting. The solved problem of this work is also very important, and is a very significant research for the community. The authors also utilized various experiments to demonstrate the performance of the GraphFLEx, e.g., the Computational Efficiency, the Node Classification, the Scalability for Large-Scale Graphs, the Link Prediction, the Graph Classification, etc.\n\nWeaknesses:\nThis work has some obvious weaknesses.\n\nW1: The motivation need to be enhanced. Although the idea of this paper is interesting, the novelty is still a little limited.The author stated that “the novelty of GraphFLEx lies not merely in combining these components”, but the expected principled manner is not well motivated. Thus, it seems that the GraphFLEx combines some existing techniques together, with less theoretical explanations. \n\nW2: Although this paper provides various experiments, the results are still not convincing. Because the authors only compare the GraphFLEx on very old methods. For example, the alternative methods for node classification are the GCN, GraphSage, GIN, and GAT, where many of them were published nearly 10 years ago. For link predictions, the alternative methods are even older. \n\nW3: Some contents seems a little verbose, some basic concepts take some space of the manuscript.\n\nQUESTIONS:\nQ1：First,  can you give some theoretical analysis and explanation to further enhance the motivation?\n\nQ2：Second, why not compare the new method to the alternative methods in recent 3-5 years? \n\nQ3：Third, the process of the clustering and coarsening seems similar to the classical DiffPool, can you explain the differences?\n\nFor more problems, please refer to the weaknesses.\n\nLIMITATIONS:\nPlease see the weaknesses and questions for details.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-05"}