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Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization
Robert Jankowski, Pedro Almagro-Blanco, Marián Boguñá, Melanie Weber, M. Ángeles Serrano
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Summary
This paper introduces a zero-shot transfer protocol for Graph Neural Networks (GNNs) called 'Train Small, Deploy Large'. It utilizes Geometric Renormalization (GR) to create scaled-down replicas of large graphs while preserving their geometric and topological structure. GNNs are trained on these coarse-grained graphs, and the learned weights are directly transferred to the original full-resolution graph for inference without retraining. The study demonstrates that this approach preserves predictive performance and aligns learned representations across scales, suggesting that structural similarity is more critical than network size for GNN transferability.
Entities (13)
Relation Signals (11)
Geometric Renormalization → enables → GNN Transfer Protocol
confidence 95% · We introduce a zero-shot transfer protocol in which a GNN is trained on a graph coarse-grained by geometric renormalization (GR)
cuMercator → isimplementationof → Mercator
confidence 92% · We release cuMercator, a GPU implementation of Mercator
HypBench → generates → synthetic networks
confidence 90% · synthetic networks of the HypBench framework
Geometric Renormalization → preserves → Structural Similarity
confidence 90% · These findings suggest that structural similarity may be more important than network size in determining GNN transferability
cuMercator → providesspeedupfor → hyperbolic embeddings
confidence 88% · achieves a speedup of more than 400×in computing hyperbolic embeddings
S1 Model → usedby → HypBench
confidence 85% · generated using the S1 /H 2 model
Geometric Renormalization → usedwith → GCN
confidence 85% · For GCN and GraphSAGE, performance remains high even at a compression factor
Geometric Renormalization → usedwith → GraphSAGE
confidence 85% · For GCN and GraphSAGE, performance remains high even at a compression factor
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Abstract
Abstract:Graph neural networks (GNNs) can operate on large graphs but become infrastructure-sensitive at the scale of millions of nodes and typically require scalable training techniques for even larger graphs. This raises a central question: when can a model trained on a smaller, scaled-down replica of a graph be deployed on the full-resolution graph without retraining? We introduce a zero-shot transfer protocol in which a GNN is trained on a graph coarse-grained by geometric renormalization (GR), and the resulting weights are transferred directly to the original network. Across synthetic and real-world networks, training on GR scaled-down replicas preserves much of the original-scale predictive performance while significantly reducing training cost. We further find that learned representations and predictive trajectories remain aligned across scales. These findings suggest that structural similarity may be more important than network size in determining GNN transferability, opening a path toward scale-equivariant graph architectures.
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- Source: https://arxiv.org/abs/2607.27767v1
- Canonical: https://arxiv.org/abs/2607.27767v1
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Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization Robert Jankowski TU Delft R.Jankowski@tudelft.nl Pedro Almagro-Blanco University of Seville palmagro@us.es Marián Boguñá University of Barcelona marian.boguna@ub.edu Melanie Weber Harvard University mweber@seas.harvard.edu M. Ángeles Serrano University of Barcelona and ICREA marian.serrano@ub.edu Abstract Graph neural networks (GNNs) can operate on large graphs but become infrastructure-sensitive at the scale of millions of nodes and typically require scalable training techniques for even larger graphs. This raises a central question: when can a model trained on a smaller, scaled-down replica of a graph be de- ployed on the full-resolution graph without retraining? We introduce a zero-shot transfer protocol in which a GNN is trained on a graph coarse-grained by geo- metric renormalization (GR), and the resulting weights are transferred directly to the original network. Across synthetic and real-world networks, training on GR scaled-down replicas preserves much of the original-scale predictive perfor- mance while significantly reducing training cost. We further find that learned representations and predictive trajectories remain aligned across scales. These findings suggest that structural similarity may be more important than network size in determining GNN transferability, opening a path toward scale-equivariant graph architectures. k layers 2. Embedding 1. Input network 3. Renormalization4. GNN training5. Zero-shot inference Predictions trained weights Figure 1: Overview of the train-small, deploy-large protocol. The input graphG 0 is embedded and renormalized to obtain a scaled-down graphG ℓ . A GNN is trained onG ℓ , and the learned weights are transferred without retraining to G 0 for zero-shot inference. 1 Introduction Message-passing GNNs learn node, edge, and graph representations by iteratively propagating and aggregating information among adjacent nodes of the input graph [1,2]. Although a given GNN architecture can, in principle, be applied to graphs of any size, successful transfer across graph scales is not guaranteed. A GNN trained at one scale may perform poorly at another when the structural properties encountered during training differ from those present at deployment [3]. The central challenge is therefore not merely computational scalability, but understanding how changes in Preprint. Preliminary work. arXiv:2607.27767v1 [cs.LG] 30 Jul 2026 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization graph resolution induce distribution shifts that can alter the statistics governing message passing and compromise transfer across scales. Addressing this challenge requires identifying the structural conditions under which GNN parameters learned on a scaled-down replica of a graph can be transferred to the full resolution graph without retraining. Existing results establish GNN transferability across graphs under restrictive assumptions, such as dense graphon limits [4]. However, real networks are often sparse, heterogeneous, and clustered [5], and standard message-passing GNNs can yield markedly different representations of the same network at different resolutions or levels of coarse-graining [6]. These observations suggest that cross-scale GNN transfer requires more than size-independent weights. It requires a scale transformation that preserves the statistics used by message passing. Graph coarsening provides one route to smaller training graphs, but most existing methods are designed to preserve global properties rather than the local neighborhood statistics that govern message passing in GNNs [7,8]. As a result, a coarsened graph may remain a good approximation of the original in the spectral sense while substantially altering degree distributions, clustering spectra, or community structure. Recent work has derived message-passing guarantees for graph coarsening, requiring a purpose-built propagation operator and approximate original-graph training [9]. Our approach builds on a different principle: many real networks are shaped by an underlying latent space [10,11] with hyperbolic geometry [12] in which pairwise distances between nodes determine the likelihood of connection. The geometric framework explains characteristic network properties such as the small-world property, heterogeneous degree distributions, high clustering, and self-similar organization [10,13]. Geometric renormalization (GR) [14] is a technique that merges nearby nodes along the similarity subspace, producing a hierarchy of graph representations at progressively lower resolutions that preserve the multiscale organization of the original network. Therefore, GR is not merely a graph compression procedure but a geometry-preserving scale transformation. In this work, we study zero-shot GNN weight transfer across network resolutions defined by a geometric renormalization flow. Given an original graphG 0 and a renormalized graphG ℓ , we train a GNN onG ℓ and deploy the same weights directly onG 0 . This train-small, deploy-large protocol tests whether the learned parameters are approximately invariant across network scales. We evaluate transfer using node classification accuracy, similarity of learned representations [15], and alignment of predictive trajectories during training. On real networks and synthetic networks of the HypBench framework [16], generated using theS 1 /H 2 model, we find that predictive performance can be largely preserved under zero-shot transfer as GR preserves the relevant geometric and topological structure. Our contributions are five-fold: • We introduce a train-small, deploy-large protocol for studying zero-shot GNN weight transfer across graph resolutions. •We provide empirical evidence that GR preserves GNN transfer performance under substantial graph compression on synthetic and real networks. •We show that models trained at different GR scales learn compatible node representations, as quantified by centered kernel alignment (CKA) and orthogonal Procrustes (OP) similarity measures. •We demonstrate that GR preserves functional training trajectories across graph scales, as quantified by the Jensen-Shannon divergence between output-probability distributions. •We release cuMercator, a GPU implementation of Mercator that achieves a speedup of more than 400×in computing hyperbolic embeddings for networks with 10,000 nodes, with even greater speedups for larger networks; this implementation may be of independent interest. 2 Related work In machine learning, graph coarsening is widely used to reduce graph size or construct hierarchical representations. Learned pooling methods, such as DiffPool [17] and Graph U-Nets [18], build task-dependent coarse graphs inside the model. This line of work is closely related to graph pooling, where coarse representations are learned through node assignments, edge contractions, or geometric reductions [19–21]. Preprocessing approaches instead coarsen the input graph before training; for example, Huang et al. [22] used graph coarsening to scale GNN training. More recently, Joly and 2 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization Keriven [9] argued that classical spectral coarsening does not automatically preserve message passing and proposed coarsening operators with message-passing guarantees. These works primarily focus on efficient training, hierarchical pooling, or signal preservation on the coarse graph. In network science, network renormalization provides a framework for relating graph structure across scales [23]. Early work by Song, Havlin, and Makse [24] introduced box-covering renormalization and showed that many real networks exhibit self-similar organization. Hidden-metric-space models later connected network self-similarity to latent geometry [10,12]. Geometric renormalization builds on this idea by embedding a graph in the hyperbolic plane and merging nearby nodes in the similarity subspace, producing a multiscale flow that preserves the probability of connection and therefore key topological observables [14]. This framework has been applied to empirical systems such as the human connectome [25] and extended to weighted networks [26]. Other renormalization schemes define scales without hidden geometry, including Laplacian renormalization for heterogeneous networks [27], higher-order Laplacian renormalization for simplicial and hypergraph structures [28], and geometry-free multiscale renormalization based on hidden variables [29]. Among these methods, GR is distinguished by its direct connection to analytically tractable hidden-metric-space models, which makes its coarse-graining and rescaling rules interpretable as a renormalization flow in the underlying similarity subspace rather than merely as an algorithmic graph contraction. These approaches define principled graph-scale transformations, but have not been used to study the scale dependence of learned GNN weights. GNNs can be applied to graphs of different sizes because their parameters are shared across neigh- borhoods. However, this architectural size agnosticism does not guarantee cross-size generalization. More broadly, GNN performance has been shown to depend sensitively on graph-specific data properties, including degree distributions, joint topological and feature structure, and spectral char- acteristics [30,31]. Yehudai et al. [3] showed that GNNs may fail to generalize from small to large graphs when local structures differ across sizes. Graphon-based analyses provide transferability guarantees for graph sequences sampled from a common limiting object [4,32], while recent work shows that standard message-passing GNNs can be discontinuous across graph resolutions [6]. These results demonstrate that GNN transfer across scales is nontrivial. 3 Network geometry and geometric renormalization Random geometric graphs describe complex networks by embedding nodes in a latent metric space, where geometric distances encode similarity and link formation emerges from a trade-off between node popularity and similarity [10], with emerging hyperbolic geometry [12]. The canonical model in this framework is theS 1 model [10], which places nodes on a circle of radiusR = N/2π, representing the similarity space, and connects every pair i, j with probability p ij = 1 1 + χ β = 1 1 + R∆θ ij μκ i κ j β ,(1) whereμcontrols the average degree of the network,βits clustering coefficient, and∆θ ij is the angular distance between nodesiandj. The hidden degreesκreflect node popularity or importance. TheS 1 model is isomorphic to the purely geometricH 2 model [12], in which nodes are placed in a two-dimensional hyperbolic disk of radiusR H 2 = 2 ln 2R μκ 2 0 , whereκ 0 = minκ i . By mapping each κ i to a radial coordinate r i , r i = R H 2 − 2 ln κ i κ 0 , the connection probability, Eq. 1, becomes p ij = 1 1 + e β 2 (x ij −R H 2 ) ,(2) wherex ij = r i +r j + 2 ln(∆θ ij /2)is a good approximation of the hyperbolic distance between two points with coordinates (r i ,θ i ) and (r j ,θ j ) in the native representation of hyperbolic space [13]. Geometric renormalization (GR) unveils the self-similarity of complex networks to construct scaled- down representations [14]. It operates by merging neighboring nodes in the similarity subspace into supernodes and rescaling connections, averaging over short-range interactions and progressively preserving longer-range connections. The procedure is shown in Figure 2a. Given an input network G 0 at layer 0, we first construct its hyperbolic map using the embedder Mercator [33,34], obtaining the hidden degree (or radial position) and angular position of each node,(κ i ,θ i ). Then, we partition 3 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization a l = 0 l = 1 l = 2 r = 2 r = 2 r = 4 10 0 10 1 10 2 k 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 P c (k) l= 0 l= 1 l= 2 l= 3 l= 4 10 1 10 2 k 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ̄ c(k) 10 0 10 1 10 2 k 0 5 10 15 20 25 30 ̄ k n (k) 0 20 ̄ k 0 1 H 01234 Renormalization level` 0 1 μ L 2 b Figure 2: (a) Schematic of geometric renormalization. Each layer is obtained by applying a renormalization step with resolutionr, starting from the original network atl = 0. Orange nodes are placed on the circle, with sizes proportional to the logarithm of their hidden degrees, and solid lines denote links within each layer. Blue shaded regions indicate coarse-graining blocks, and dashed lines connect nodes to their supernodes in layerℓ + 1. Two supernodes are linked in layerℓ + 1if at least one pair of nodes belonging to them is linked in layerℓ. Under composition, GR forms an Abelian semigroup: repeated transformations at a given resolution are equivalent to a single transformation at higher resolution. If the number of nodes is not divisible byr, the final supernode contains fewer than rnodes, as shown atℓ = 1. Reproduced from [14]. (b) Topological validation of GR for synthetic networks in the small-world regime. In the top-left panel, the complementary cumulative degree distributionP c (k)is shown across renormalization levelsl = 0tol = 4. The top-right panel shows the clustering spectrum ̄c(k), and the bottom-left panel shows the average nearest-neighbor degree ̄ k n (k). In both panels, solid lines denote the median and shaded regions denote the interquartile range (IQR). The three panels on the bottom right show, as a function of scalel, the mean degree ̄ k , the homophily ratioH, and the spectral gap μ L 2 . the nodes into nonoverlapping blocks ofrconsecutive nodes along the circle and coarse-grain each block into a supernode, independently of whether the nodes within the block are connected. Each supernode is placed within the angular region spanned by its corresponding block, thereby preserving the circular order of nodes in the original embedding throughout the renormalization process. Finally, all links between nodes belonging to two different supernodes, if any, are rescaled into a single link between those supernodes. The GR transformation generates a version of the original network with the same statistical properties while typically increasing the average degree. The additional links are then selectively pruned to preserve the network’s structure, yielding a scaled-down replica of the original network. In Appendix A.1, we provide a detailed description of the algorithm. 4 Methods 4.1 Experimental setup In this work, we use synthetic networks generated with HypBench [16], together with a set of real networks. The schematic pipeline is shown in Figure 1. Starting from the input graphG 0 , we first embed the network into the hyperbolic disk using Mercator. For synthetic networks, this embedding step is not required because the ground-truth coordinates are known. We then apply GR to construct a coarse-grained networkG ℓ , which is used for GNN training. After training the GNN onG ℓ for node classification, we evaluate the model directly onG 0 without any retraining: node representations are computed onG 0 using the weights learned onG ℓ , and predictions are obtained for every node in the original test set. We refer to the resulting accuracy, computed on the original test nodes ofG 0 , as the transfer test accuracy. 4 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization Results are averaged over 10 random train/test splits per network (60/40 ratio), and, for synthetic networks, over 10 independently generated network realizations per configuration. To generate scale-down replicas of a given input network, we use the geometric renormalization procedure based on the family of hyperbolic random graph modelsS 1 /H 2 as described in Section 3. Because nodes are divided into train and test sets, the merging procedure must avoid label leakage. Each supernode label is assigned by majority vote over the training labels within the block; test labels are ignored. In the event of a tie, one of the tied labels is selected at random. This yields a fixed compression factor ofrper level. Unless stated otherwise, we user = 2in all experiments. The feature vector of each supernode is obtained by averaging the feature vectors of its constituent nodes. Random Baselines. As a baseline, we also consider random renormalization, in which both the network topology and the node coordinates are randomized before renormalization (Random). To disentangle the contribution of each of these two factors, we additionally consider two partial baselines in which only the topology (Random-T ) or only the geometry (Random-G) is randomized, while the other is kept unaltered, following the same topology/geometry disentanglement strategy used by Zheng et al. [25]. In the main text, we report the results for GR against the fully randomized baseline (Random). Results for Random-T and Random-G are reported in Appendix A.8.3. Parameters of HypBench.The HypBench framework allows us to generate synthetic networks with nodes’ features by varying parameters of theS 1 /H 2 and bipartite-S 1 /H 2 models [16]. The unipartite networks are generated withN = 2 17 = 131,072nodes,⟨k⟩ = 10andβ = 1.5,N c = 6classes, and homophily parameterα = 6. We consider two degree-distribution exponents,γ = 2.5andγ = 3.5, corresponding to the ultra-small-world (USW) and small-world (SW) regimes, respectively [35]. These regimes differ in the scaling of the average shortest-path length with network size: it grows asO(log logN )in USW networks and asO(logN )in SW networks. The bipartite networks used to encode node features are generated withN f = 1000features andγ n = γ f = 2.5, which are the degree-distribution exponents for node and feature vertices, respectively. We set⟨k n ⟩ = 10, the average node degree in the bipartite network. Real networks.For real networks, we consider eight benchmark datasets: Cora, PubMed [36,37], Computers, Photo, CS, Physics [38], WikiCS [39], and Flickr [40]. Table 2 summarizes the main properties of each dataset. For each network, we inferβusing Mercator [33] and estimate the exponentγof the degree distribution following [41]. Based on the classification in [35], networks withβ < 2and2 < γ < 3are categorized as ultra-small-world (USW), whereas networks with β < 2 and γ > 3 are categorized as small-world (SW). We used the original dataset features. GNN training details.We compare three GNN models: GCN [1], GraphSAGE [2], and GAT [42], each implemented with two layers and 32 hidden dimensions using the PyTorch Geometric library [43]. We train the models using the Adam optimizer with an initial learning rate of 0.01 and a cosine learning-rate scheduler. We also evaluated SGD, but found that it led to weaker performance on some real-world networks. Further details are provided in Appendix A.7. Hyperbolic embeddings of real networks. Mercator infers node coordinates by maximizing the likelihood that the observed network was generated by theS 1 /H 2 model. It requires no input parameters beyond the graph itself and returns, for each node, an angular coordinateθ i and hidden degreeκ i , as well as the model parametersβandμ. A limitation of Mercator [33,34] is the computational cost of embedding large networks, because its time complexity scales asO(N 2 ). In this work, we introduce cuMercator, an accelerated implementation of Mercator that uses CUDA kernels for GPU execution and negative sampling to approximate the contribution of non-neighbors to the maximum-likelihood calculation. Detailed computational comparisons and estimates of the time required to embed large networks are provided in Appendix A.6. 4.2 Representational similarity measures To investigate why transfer is possible, we compare the similarity between representations learned by GNNs trained on the original network and on its GR counterpart. Although many representation- similarity measures are available (see [15]), we restrict our analysis to two measures that capture complementary aspects of representational similarity: Centered Kernel Alignment (CKA) [44,45] and Orthogonal Procrustes (OP) analysis [46, 47]. 5 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization For each renormalization levelℓ, we train a GNNf ℓ on the coarse-grained graphG ℓ = (V ℓ ,E ℓ ) using labels induced from the original graphG 0 . To compare this model with a GNN trained directly onG 0 , we must first put their representations on the same set of nodes. We do this using the hierarchical coarse-graining mapφ ℓ : V 0 → V ℓ , which records which supernode inG ℓ each original node belongs to. Thus, if two original nodesiandjare merged into the same supernode, they are assigned the same coarse-scale representation when we lift back toV 0 . We then compute CKA and OP between the original-scale and lifted coarse-scale representations. CKA compares the node–node geometry induced by the two representations, whereas OP measures how closely they can be aligned by a rotation or reflection. Both measures range from 0 to 1, with larger values indicating greater representational similarity. Formal definitions and further details are provided in Appendix A.2. 4.3 Functional trajectory preservation during training To test whether a renormalized graph preserves the predictive dynamics of training, we compare the class-probability distributions produced during training on the original graphG 0 and a coarse graphG ℓ . LetP j→i (t)denote the collection of node-wise categorical distributions over class labels obtained at training step t by a model trained on G j and evaluated on G i . First, we compute the training-graph divergence D TG JS = D JS (P 0→0 (t),P ℓ→0 (t)),(3) whereD JS is the Jensen-Shannon divergence. Small values ofD TG JS indicate that the renormalized graph acts as a good training surrogate for the original graph in output space. Second, we examine the evaluation-graph divergence: it quantifies how much a model trained onG ℓ changes when evaluated on G 0 rather than G ℓ , D EG JS = D JS (U ℓ P ℓ→ℓ (t),P ℓ→0 (t)).(4) Here,U ℓ denotes the coarse-to-fine lifting operator that maps coarse-node predictions fromG ℓ back to the node set of G 0 . Further definitions and implementation details are provided in Appendix A.3. 5 Results We report results for synthetic and real networks. In both cases, we first assess whether networks preserve key structural properties along the GR flow. We then evaluate the transfer test accuracy of GNNs trained on renormalized networks. Next, we show that learned representations and the training dynamics remain aligned across graph scales. 5.1 Zero-shot GNN transfer on synthetic networks Figure 2b presents the topological validation of geometric renormalization for synthetic networks in the small-world regime. The degree distribution, clustering spectrum, and average nearest-neighbor- degree spectrum are preserved during coarse-graining. Global properties, such as the average degree and homophily level, are also maintained. The spectral gap is slightly larger for smaller networks, as expected from its asymptotic vanishing in the large-network limit [48]. Similar results are obtained in the ultra-small-world regime, see Figure 8 in Appendix A.8.1. In Figure 3, we summarize our results for synthetic networks in the small-world regime. Panel a shows the transfer test accuracy across different renormalization levels. In most cases, it closely matches the test accuracy obtained at the original graph scale. For GCN and GraphSAGE, performance remains high even at a compression factor of2 5 = 32, whereas GAT exhibits a slight decrease. In contrast, the random baseline declines in performance at higher renormalization levels. However, performance does not fall to the level of random chance because the models continue to leverage the original node features, which are not randomized. Reducing the network size also substantially improves training time, as shown in Fig. 3b. For example, for GraphSAGE, training on a coarse-grained network with 4,096 nodes instead of the original network with 131,072 nodes yields a 20-fold speedup, see also Figure 3c for the network size at each renormalization level. Figures 3d and 3e report the representational-similarity results. GNNs trained independently at different graph scales converge to similar hidden representations, indicating that geometric renormalization preserves the learned representation space across scales. Both linear CKA and OP similarity remain high even under strong 6 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization 012345 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 Transfer accuracy a GR Random GCN GAT GraphSAGE 012345 0 20 40 60 Training time (s) b 012345 Renormalization level 0 50k 100k Nodes c G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.870.830.810.790.78 0.040.860.820.800.79 0.020.140.900.860.84 0.010.030.140.920.89 0.000.010.020.140.94 0.000.000.010.020.12 d CKA G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.910.880.870.870.86 0.190.910.880.870.86 0.140.370.930.900.88 0.100.160.380.950.92 0.060.070.150.350.96 0.030.030.060.140.34 e OP similarity 060120180240300360420480 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 D TG JS f 060120180240300360420480 Epoch 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 D EG JS g GR Random 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 train GNN G 0 G 0 evaluate train GNN G l evaluate G 0 G l evaluate train GNN G l evaluate Figure 3: Zero-shot GNN transfer on synthetic networks. (a) Transfer test accuracy as a function of renormalization levelℓfor different GNN architectures. Solid lines denote GR and dashed lines denote the random baseline. (b) Training time for GNNs across renormalization levels. (c) Number of nodes at each renormalization levelℓ. (d, e) Representation similarity between levels, measured by linear CKA and orthogonal Procrustes similarity, respectively (higher is better). The upper-triangular entries correspond to GR and the lower-triangular entries to the random baseline. (f, g) Training alignment of output prediction distributions quantified by Jensen-Shannon divergenceD JS (lower is better). Panel f showsD TG JS (Eq. 3), whereas panel g showsD EG JS (Eq. 4). Solid and dashed lines indicate GR and random renormalization, respectively. In panels f and g, we use a GCN model and a GR resolution ofr = 32to produce the coarse-grained graphG ℓ . All results are shown for the synthetic network in the small-world regime. compression, whereas random merging produces almost no similarity across scales. Figures 3f,g show the alignment of prediction dynamics across graph scales. The training-graph divergenceD TG JS (Fig. 3f) remains close to zero throughout training for GR, indicating that a model trained on the renormalized graph produces prediction distributions onG 0 that closely match those of a model trained directly onG 0 . In contrast, the random baseline leads to a rapid and sustained increase in D TG JS , showing that random coarse graphs do not preserve the output-space training trajectory. The evaluation-graph divergenceD EG JS (Fig. 3g) remains small for both GR and the random baseline, indicating that models trained on coarse graphs are relatively stable when evaluated across graph resolutions. However, this stability alone is not sufficient to preserve predictive performance: despite similarly lowD EG JS , only GR maintains high transfer test accuracy across renormalization levels, whereas the random baseline leads to a pronounced accuracy degradation (Fig. 3a). Together, these results suggest that GR preserves the functional training trajectory. 5.2 Zero-shot GNN transfer on real networks We apply the same methodology to a set of real-world networks. Figure 4 summarizes the results for the Photo dataset, while results for the remaining datasets are reported in Appendix A.8.2. We also report in Appendix A.8.4 the topological validation of the GR transformation. Overall, we observe qualitatively similar behavior to that seen in the synthetic networks. Under geometric renormalization, transfer test accuracy remains relatively high across renormalization levels (Fig. 4a). Among the architectures considered, the simpler GCN model appears to be the most suitable for weight transfer, outperforming both GAT and GraphSAGE. Although the transfer accuracy of GraphSAGE decreases by approximately10%, this reduction is accompanied by a roughly fourfold speedup in training time. We note that transfer accuracy on real-world networks is not preserved as strongly as in the synthetic 7 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization Table 1: Transfer test accuracy for eight real-world datasets. For each dataset and levelℓ, the GCN is trained on the coarse-grained graphG ℓ obtained via GR, and its weights are directly applied, without retraining, to the original graphG 0 . We report the mean±standard deviation of transfer test accuracy over 10 random train/test splits. The bottom row shows the compression ratioN 0 /N ℓ at each renormalization level, obtained with a fixed resolution r = 2 per level. OriginalCoarse-grained ℓ = 0ℓ = 1ℓ = 2ℓ = 3ℓ = 4ℓ = 5 Computers0.92± 0.010.90± 0.000.88± 0.010.87± 0.000.84± 0.010.80± 0.03 Cora0.87± 0.010.80± 0.010.77± 0.010.77± 0.010.76± 0.010.71± 0.02 CS 0.94± 0.000.90± 0.000.87± 0.000.87± 0.000.88± 0.010.88± 0.01 Flickr0.52± 0.000.49± 0.020.49± 0.010.50± 0.000.49± 0.000.50± 0.00 Photo0.95± 0.000.94± 0.000.93± 0.010.92± 0.010.90± 0.010.85± 0.02 Physics0.96± 0.000.94± 0.000.92± 0.000.92± 0.010.92± 0.010.91± 0.01 PubMed 0.88± 0.000.85± 0.000.85± 0.000.85± 0.010.84± 0.010.83± 0.00 WikiCS0.83± 0.000.81± 0.000.79± 0.010.75± 0.010.68± 0.010.64± 0.01 Compression ratio1×2×4×8×16×32× examples, likely because topology, node features, and labels are correlated in a more complex and nontrivial manner [49]. 012345 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Transfer accuracy a GCN GAT GraphSAGE GR Random 012345 2 3 4 Training time (s) b 012345 Renormalization level 0 2k 4k 6k Nodes c G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.560.380.300.260.22 0.160.640.480.420.36 0.070.420.740.640.56 0.030.190.410.840.73 0.010.090.180.430.84 0.010.030.080.190.41 d CKA G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.750.630.570.530.50 0.350.820.720.680.64 0.230.600.860.810.76 0.160.400.600.900.85 0.110.270.390.640.91 0.070.170.250.420.63 e OP similarity 060120180240300360420480 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 D TG JS f 060120180240300360420480 Epoch 0.00 0.02 0.04 0.06 0.08 0.10 0.12 D EG JS g GR Random 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 train GNN G 0 G 0 evaluate train GNN G l evaluate G 0 G l evaluate train GNN G l evaluate Figure 4: Zero-shot GNN transfer on the Photo dataset. See the caption of Fig. 3 for more detail. The representation-similarity measures remain higher under GR than under the random baseline (Fig. 4d,e), indicating that the GR graphs better preserve the learned representation structure. As in the synthetic case, representational similarity does not fall to negligible levels under the random baseline because the models continue to leverage the original node features, which are not randomized. The functional trajectories show a qualitatively similar pattern to the synthetic networks (Fig. 4f,g): GR preserves the training-graph trajectory more effectively than the random baseline, while the evaluation-graph divergence remains relatively small. In Table 1, we show the transfer test accuracy in the 8 considered datasets. Several other coarse-graining methods have been proposed in the literature. We compare GR with the Laplacian renormalization group (LRG) [27] and with two edge-contraction pooling methods, MagEdgePool and SpreadEdgePool [20]. 8 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization 0.00.51.01.52.02.5 Combined topology preservation error 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Acc = Acc original Acc transfer Lower is better Lower is better GR LRG MagEdgePool SpreadEdgePool Reduction rate ~6x ~31x ~76x Figure 5: Combined topology preservation er- ror as a function of the difference between the transfer and original test accuracies for differ- ent coarse-graining methods for the PubMed dataset.Marker color denotes the coarse- graining method, while marker size represents the compression rate. Larger markers corre- spond to smaller coarse-grained networks. Re- sults are averaged over 10 train-test splits. We repeat the zero-transfer weight evaluation for each method and network reduction rate. Specifi- cally, we measure the difference between the trans- fer and original test accuracies and the combined topology preservation error. The latter quantifies how well the coarse-graining procedure preserves key topological properties, including the degree distribution, clustering spectrum, average nearest- neighbor degree, average clustering coefficient, and spectral gap. See Appendix A.5 for more details. Figure 5 shows that GR produces the smallest ac- curacy gap, indicating that it maintains high trans- fer test accuracy even at high compression rates. It also achieves the lowest combined topological error, demonstrating superior preservation of the original network structure. The other methods gen- erally perform worse, except at low compression rates. Together, these results suggest that GR pro- vides a faithful scaled-down replica of the original graph. 6 Discussion We introduced a train-small, deploy-large proto- col for zero-shot GNN transfer through geometric renormalization. Our results show that GNN pa- rameters learned on substantially compressed GR graphs can be transferred directly to the original network while retaining much of the full-scale predictive performance and reducing training cost. This behavior is not explained by graph-size reduction alone: random aggregation substantially degrades transfer, demonstrating that preserving network structure is essential. Models trained across GR scales also learn compatible representation spaces and follow similar predictive trajectories during optimization. These observations indicate that, within a finite range of scales, GR approximately preserves both the representations learned by a GNN and the effective objective it optimizes. 7 Limitations While the results are promising, we highlight several limitations of our study. First, we renormalize graph structure but not node features. We used simple mean feature aggregation rather than deriving a principled scale transformation for node features, which we leave for future work. Second, we keep the GNN architecture and hyperparameters fixed across scales; whether hyperparameters optimized at a lower scale remain optimal at the original scale is therefore unknown. Third, transfer may depend on the dimensionality of the underlying hyperbolic representation [34,50,51]. However, geometric renormalization is currently limited to a one-dimensional similarity space. Fourth, we focus on homophilic graphs, whether a similar scale transfer approach is possible in heterophilic settings remains open. Finally, we study node-level learning tasks; extending the framework to graph classification [6], link prediction, and other graph-learning problems is an important direction for future work. Acknowledgments The authors acknowledge support from Grants PID2022-137505NB-C22, PID2024-156576OB-C33, and PID2023-147198NB-I00, funded by MICIU/AEI/10.13039/501100011033 and by ERDF/EU. M. B. acknowledges support from the Agency for Management of University and Research Grants of the Generalitat de Catalunya through the Acadèmia d’Excel·lència grant. M. W. acknowledges partial support from NSF award DMS-2406905, an Alfred P. Sloan Research Fellowship in Mathematics, the AI2050 program at Schmidt Sciences (Grant G-25-69786), and an Aramont Fellowship for Emerging Science Research. 9 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization References [1]Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. 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The renormalized networks remain maximally congruent with the hidden metric space model by assigning a new hidden degreeκ (l+1) i to supernodeiin layer l + 1 as a function of the hidden degrees of nodes it contains in layer l according to κ (l+1) i = r X j=1 κ (l) j β 1/β (5) as well as an angular coordinate θ (l+1) i given by θ (l+1) i = r P j=1 θ (l) j κ (l) j β r P j=1 κ (l) j β 1/β .(6) The global parameters also need to be rescaled asμ (l+1) = μ (l) /r,β (l+1) = β (l) , andR (l+1) = R (l) /r. This implies that the probabilityp (l+1) ij for two supernodesiandjto be connected in layer l + 1maintains its original form (Eq. 1). This applies both to the model and to real networks as long as they admit a good embedding. During the renormalization flow, the average degree increases according to⟨k⟩ (l+1) = r ν ⟨k⟩ (l) , whereνis a scaling exponent determined by the connectivity structure of the original network. We therefore adjust the average degree of the downscaled network replicas. Specifically, to match the average degree of a renormalized network to that of the original network, we prune links according to the underlying metric model with which networks at all layers are congruent. A.2 Representational similarity measures Letn = |V 0 |and letddenote the common representation dimension of all models, which use the same GNN architecture. We write Z 0 ∈ R n×d , Z ℓ ∈ R |V ℓ |×d (7) for the node representations learned on the original and coarse-grained graphs. The coarse-scale representations are lifted to the original node set so thatZ ↑ ℓ ∈ R n×d . We center each feature across nodes, obtaining X 0 and X ↑ ℓ . We compute linear CKA as CKA(X 0 ,X ↑ ℓ ) = ∥X ⊤ 0 X ↑ ℓ ∥ 2 F ∥X ⊤ 0 X 0 | F ∥(X ↑ ℓ ) ⊤ X ↑ ℓ ∥ F ,(8) and normalized Orthogonal Procrustes similarity as OP(X 0 ,X ↑ ℓ ) = ∥X ⊤ 0 X ↑ ℓ ∥ ∗ ∥X 0 ∥ F ∥X ↑ ℓ ∥ F ,(9) Here,∥·∥ F is the Frobenius norm, i.e., the square root of the sum of squared matrix entries, and ∥·∥ ∗ is the nuclear norm, i.e., the sum of the singular values. For nonzero representation matrices, both measures are in[0, 1]. A value of 1 indicates maximal similarity under the corresponding criterion, whereas 0 indicates no shared linear structure. CKA compares the geometry induced among nodes and is invariant to isotropic scaling and orthogonal transformations. OP measures the maximal similarity after rotating or reflecting one representation. 13 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization A.3 Functional trajectory diagnostics Representation-similarity measures characterize learned embeddings but do not directly determine whether training on a graph-renormalization replica follows the same predictive trajectory as training on the original graph. We therefore compare model outputs throughout training rather than gradients in parameter space, which may be sensitive to parameterization and training noise. LetY v ∈1,...,Cdenote the class-label random variable for nodev, whereCis the number of classes. Letω (j) t denote the parameters of a GNN after training fortsteps on graphG j . For a nodev in the evaluation graph G i , define P (v) j→i (t) = softmax f ω (j) t (G i ) v .(10) We useP j→i (t)to denote the collection of these node-level distributions over the evaluated node set. In particular,P 0→0 (t)corresponds to training and evaluation onG 0 ,P ℓ→ℓ (t)to training and evaluation on G ℓ , and P ℓ→0 (t) to training on G ℓ followed by evaluation on G 0 . D TG JS (t) = D JS (P 0→0 (t),P ℓ→0 (t)).(11) This quantity isolates the effect of replacingG 0 withG ℓ during training while keeping the evaluation graph fixed. Small values indicate thatG ℓ acts as an effective training surrogate forG 0 in output space. D EG JS (t) = D JS (U ℓ P ℓ→ℓ (t),P ℓ→0 (t)).(12) This quantity keeps the trained model fixed and compares its predictions at the coarse and original graph resolutions. The operatorU ℓ maps each coarse-node prediction to the corresponding nodes ofG 0 . Small values therefore indicate that the coarse-trained model transfers stably to the original graph. Both divergences are evaluated throughout training. Together, they distinguish the quality of G ℓ as a training surrogate from the stability of evaluating a coarse-trained model at the original graph resolution. A.4 Properties of real networks Table 2: Properties of the real datasets.N: number of nodes;E: number of edges;N f : number of node features;N c : number of classes; ̄ k: average degree;H: homophily ratio;γ: degree-distribution exponent;β: clustering-coefficient parameter of theS 1 /H 2 model, inferred with Mercator; ̄c: average clustering coefficient; Regime: structural classification of the network as ultra-small-world (USW) or small-world (SW), following [35], based on the inferred values of β and γ. DatasetNEN f N c ̄ k H γ β ̄cRegime Computers13,381245,7787671036.70.782.961.460.36USW Cora2,4855,069143374.10.813.081.540.29SW CS18,33381,8946805158.90.816.111.720.37SW Flickr89,250449,878500710.10.322.641.010.08USW Photo7,487119,043745831.80.832.991.650.41USW Physics34,493247,9628415514.40.935.01.710.38SW PubMed19,71744,32450034.50.84.181.120.11SW WikiCS11,311215,5543001038.10.653.721.610.5SW A.5 Comparison with other coarse-graining methods We compare geometric renormalization (GR) with the Laplacian renormalization group (LRG), which identifies appropriate spatiotemporal scales in heterogeneous networks [27], and with two edge-contraction pooling methods, MagEdgePool and SpreadEdgePool. The latter methods iteratively contract the least important edges and are computed before GNN training [20]. For LRG, we first fix the network’s temporal resolution parameterτ. For GR, we choose the resolution parameterrso that the final size of the coarse-grained network matches the target compression level. We use the corresponding pooling ratio for MagEdgePool and SpreadEdgePool. 14 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization 10 0 10 1 10 2 10 3 k 10 4 10 3 10 2 10 1 10 0 P c ( k ) a Original Laplacian RG Geometric RG MagEdgePool SpreadEdgePool 10 1 10 2 10 3 k 10 5 10 4 10 3 10 2 10 1 10 0 c ( k ) b 10 0 10 1 10 2 10 3 k 10 0 10 1 10 2 10 3 k n ( k ) c 0 5 4.5 2.9 4.6 2 2.5 d k 0 1 0.11 0.56 0.042 0.6 0.98 e c 0.0 0.1 0.014 0.079 0.018 0.0057 0.14 f 2 0.0 0.5 1.0 0.79 0.54 0.69 0.49 0.18 g Transfer test accuracy Figure 6: Comparison of coarse-graining methods on the PubMed dataset. We report: (a) the complementary cumulative degree distribution, (b) the clustering spectrum, (c) the average-neighbor- degree spectrum, (d) the average degree, (e) the average clustering coefficient, (f) the spectral gap, and (g) the transfer test accuracy. For LRG, we useτ = 1.2, which yields approximately 1100 nodes, compressing the original network by a factor of approximately 17. For GR, MagEdgePool, and SpreadEdgePool, we match this number of nodes by varying the resolution parameterrfor GR and the pooling ratio for MagEdgePool and SpreadEdgePool. For the transfer task, we train a two-layer GCN with 32 hidden dimensions. We first compare the topological properties of the coarse-grained networks produced by each method. In Figure 6a-f, we show the degree distribution, clustering spectrum, average nearest-neighbor degree spectrum, and global network properties, including the average degree, average clustering coefficient, and spectral gap. Overall, GR preserves both local and global network properties more accurately than the other methods. We then evaluate transfer accuracy (Fig. 6g). GR yields the smallest drop in transfer accuracy compared with a GNN trained on the original PubMed network. We extend the comparison by varying the parameterτ, which controls the compression rate. In Figure 5, we considerτ ∈ 0.5, 1.6, 2, corresponding to approximate reductions in network size of6×,31×, and76×, respectively. We compute the combined topology-preservation error as the sum of the normalized discrepancies across all topology metrics except average degree, because GR explicitly preserves the average degree through network pruning [14]. This combined topology preservation error is defined as the sum of these normalized topology errors as Err topo (G ℓ ,G 0 ) = g Err P (k) + g Err c(k) + g Err k n (k) + g Err ̄c + g Err λ 2 ,(13) 15 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization where g Err i is the normalized error for topology statistici, obtained by dividing the corresponding raw error by the maximum raw error of the same statistic across all coarse-grained graphs included in the comparison. Lower values indicate better preservation of the original topology. The degree- distribution term Err P (k) is the Anderson–Darling statistic between the node-degree samples of G 0 andG ℓ . The spectrum termsErr c(k) andErr k n (k) are root-mean-square errors (RMSE) between the original and coarse-grained degree-dependent clustering spectrumc(k)and average-neighbor-degree spectrumk n (k), respectively. The mean-clustering term is the relative errorErr ̄c = | ̄c ℓ − ̄c 0 | ̄c 0 , and the spectral-gap term is the absolute log-ratio errorErr λ 2 = log λ 2,ℓ +ε λ 2,0 +ε , whereλ 2 is the second-smallest eigenvalue of the normalized graph Laplacian and ε = 10 −6 . Finally, we compare the computational complexity of the methods. Assuming that the input graph is sparse, i.e.,E = O(N ), the GR coarse-graining procedure scales linearly with the network size,O(N ). However, GR first requires a hyperbolic embedding of the network using Mercator, whose standard time complexity isO(N 2 ). As shown in Appendix A.6, GPU acceleration can substantially reduce the practical runtime of this embedding step. LRG requiresO(N 3 )time due to the eigendecomposition and matrix exponential operations. MagEdgePool and SpreadEdgePool have time complexityO(N 3 ) on sparse graphs when approximate diffusion distances are used. A.6 Computational analysis of Mercator on GPU Mercator [33,34] is a reliable method for embedding complex networks into their underlying hyperbolic latent geometry. The algorithm combines machine-learning techniques with maximum- likelihood (ML) optimization to infer node coordinates in the hyperbolic disk by maximizing the agreement between the observed network topology and the geometric model. The original implemen- tation has a time complexity that scales quadratically with the network size. 10 3 10 4 Network size 10 1 10 2 10 3 Time [s] a CPU GPU 5 × 10 4 1 × 10 5 5 × 10 5 Network size 10 2 10 3 10 4 Time [s] b Exact S=64 5 × 10 4 1 × 10 5 5 × 10 5 Network size 0.995 0.996 0.997 C-score c Exact S=64 Figure 7: Computational performance of Mercator. (a) Embedding time as a function of network size for the CPU and GPU implementations. (b) Embedding time on the GPU using either exact maximum-likelihood (ML) calculations or negative sampling withS = 64. (c)c-score between the original and inferred angular coordinates for embeddings obtained with exact ML and negative sampling. Model parameters areβ = 2,γ = 10, and⟨k⟩ = 10. All experiments were performed on a single H200 GPU. In this work, we accelerate the ML stage by moving the most computationally expensive operations to a GPU. In Figure 7a, we report the time required to embed networks of different sizes. The GPU implementation, called cuMercator, provides substantial speedup compared with the CPU version. However, the asymptotic time complexity remainsO(N 2 ). To reduce this cost, we introduce negative sampling into the ML calculation. Instead of computing the log-likelihood contribution from all non-neighbors at each newly proposed position, we sample onlySnon-neighbors. This significantly reduces the cost of evaluating the non-neighbor contribution. In Figure 7b, we compare the embedding time obtained with exact ML calculations against the negative-sampling approximation withS = 64. We observe a substantial reduction in runtime while retaining high-quality embeddings. Negative sampling may prevent the final log-likelihood from reaching the exact optimum. Nevertheless, Figure 7c shows that the resultingc-score [52] remains very high, indicating strong agreement between the ordering of nodes in the original and inferred angular coordinates. These changes allow us to embed very large networks efficiently and use the inferred node coordinates for geometric renormalization. 16 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization A.7 Choice of optimizer In this work, we used Adam as the optimizer for GNN training. We also tested stochastic gradient de- scent (SGD) as a baseline. While SGD led to convergence on the synthetic networks, its performance was weaker on some real-world networks. Therefore, we used Adam in all experiments to ensure stable training and high accuracy across all considered networks. It is worth noting one limitation of using Adam when analyzing training trajectories. With SGD on synthetic networks, we were able to extract raw gradients and directly compare the gradients of GNNs trained on the original and coarse-grained graphs. With Adam, however, the raw gradients were noisier and less stable for comparison. For this reason, we instead compared the softmax prediction trajectories, which provided a more stable basis for analyzing training alignment. A.8 Additional results A.8.1 Synthetic networks Figure 8 shows the topological validation of geometric renormalization in the ultra-small-world (USW) regime. Similarly to the small-world regime, the topological properties are preserved. Figure 9 summarizes the zero-shot transfer results. 10 0 10 1 10 2 10 3 k 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 P c (k) l= 0 l= 1 l= 2 l= 3 l= 4 10 1 10 2 10 3 k 0.0 0.2 0.4 0.6 0.8 ̄ c(k) 10 0 10 1 10 2 10 3 k 0 50 100 150 ̄ k n (k) 0 20 ̄ k 0 1 H 01234 Renormalization level` 0 1 μ L 2 Figure 8: Topological validation of geometric renormalization in the ultra-small-world (USW). See the caption of Fig. 2 for panel details. Across all renormalization levels, these topological and mesoscale properties remain stable, confirming that geometric renormalization preserves the structural organization of the network in this regime as well. 012345 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 Transfer accuracy a GR Random GCN GAT GraphSAGE 012345 0 20 40 60 Training time (s) b 012345 Renormalization level 0 50k 100k Nodes c G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.820.730.650.640.63 0.060.820.710.690.68 0.030.150.800.740.72 0.010.030.150.830.77 0.000.010.030.140.87 0.000.000.010.020.12 d CKA G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.890.830.770.760.75 0.230.890.820.800.79 0.160.380.880.840.82 0.110.170.390.900.85 0.070.080.160.360.92 0.040.040.070.140.34 e OP similarity 060120180240300360420480 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 D TG JS f 060120180240300360420480 Epoch 0.000 0.005 0.010 0.015 0.020 D EG JS g GR Random 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 train GNN G 0 G 0 evaluate train GNN G l evaluate G 0 G l evaluate train GNN G l evaluate Figure 9: Zero-shot GNN transfer on synthetic networks in the USW regime. See the caption of Figure 3 for more details. 17 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization A.8.2 Real networks Figures 10–16 show transfer accuracy and training time across renormalization levels for the remain- ing real-world datasets. 012345 0.60 0.65 0.70 0.75 0.80 0.85 0.90 Transfer accuracy a GCN GAT GraphSAGE GR Random 012345 4 6 Training time (s) b 012345 Renormalization level 0 10k 20k Nodes c G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.610.440.340.280.24 0.110.680.540.450.40 0.050.200.740.620.56 0.020.050.160.780.70 0.010.010.030.120.81 0.000.000.010.020.08 d CKA G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.780.660.580.510.46 0.330.820.730.650.60 0.220.440.850.760.70 0.140.220.400.860.79 0.090.100.170.350.86 0.040.050.070.120.27 e OP similarity 060120180240300360420480 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 D TG JS f 060120180240300360420480 Epoch 0.00 0.01 0.02 0.03 0.04 0.05 D EG JS g GR Random 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 train GNN G 0 G 0 evaluate train GNN G l evaluate G 0 G l evaluate train GNN G l evaluate Figure 10: Zero-shot GNN transfer on PubMed dataset. See the caption of Fig. 3 for more detail. 012345 0.5 0.6 0.7 0.8 0.9 Transfer accuracy a GCN GAT GraphSAGE GR Random 012345 0 10 20 Training time (s) b 012345 Renormalization level 0 5k 10k 15k Nodes c G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.570.440.400.390.38 0.130.670.570.540.52 0.060.330.770.720.69 0.020.110.300.860.81 0.010.040.090.260.90 0.000.010.030.080.24 d CKA G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.690.590.540.530.51 0.280.740.650.620.60 0.180.450.780.730.70 0.120.230.420.840.79 0.080.130.210.400.89 0.040.080.120.200.39 e OP similarity 060120180240300360420480 0.0 0.1 0.2 0.3 0.4 D TG JS f 060120180240300360420480 Epoch 0.00 0.05 0.10 0.15 0.20 0.25 D EG JS g GR Random 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 11: Zero-shot GNN transfer on CS dataset. See the caption of Fig. 3 for more detail. 18 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization 012345 0.65 0.70 0.75 0.80 0.85 0.90 0.95 Transfer accuracy a GCN GAT GraphSAGE GR Random 012345 0 25 50 75 Training time (s) b 012345 Renormalization level 0 10k 20k 30k Nodes c G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.810.770.760.760.75 0.100.870.840.820.80 0.040.290.910.880.86 0.020.100.280.940.91 0.010.040.110.250.96 0.000.020.040.080.27 d CKA G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.830.790.780.770.76 0.280.870.830.820.80 0.180.460.910.880.86 0.120.240.450.940.91 0.070.150.240.430.95 0.040.090.130.220.45 e OP similarity 060120180240300360420480 0.00 0.05 0.10 0.15 0.20 0.25 0.30 D TG JS f 060120180240300360420480 Epoch 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 D EG JS g GR Random 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 12: Zero-shot GNN transfer on Physics dataset. See the caption of Fig. 3 for more detail. 012345 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Transfer accuracy a GCN GAT GraphSAGE GR Random 012345 2.5 5.0 7.5 10.0 Training time (s) b 012345 Renormalization level 0 5k 10k Nodes c G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.500.300.200.170.16 0.240.570.370.320.29 0.110.460.610.510.46 0.050.200.440.720.66 0.020.100.210.480.80 0.010.040.100.220.47 d CKA G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.730.590.490.460.44 0.420.780.650.600.57 0.290.640.800.730.68 0.190.430.650.840.79 0.130.290.450.690.88 0.090.200.300.460.68 e OP similarity 060120180240300360420480 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 D TG JS f 060120180240300360420480 Epoch 0.00 0.02 0.04 0.06 0.08 0.10 0.12 D EG JS g GR Random 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 13: Zero-shot GNN transfer on Computers dataset. See the caption of Fig. 3 for more detail. 19 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization 012345 0.3 0.4 0.5 0.6 0.7 0.8 Transfer accuracy a GCN GAT GraphSAGE GR Random 012345 4 6 8 Training time (s) b 012345 Renormalization level 0 5k 10k Nodes c G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.470.270.180.150.14 0.300.570.420.360.35 0.150.450.660.540.50 0.080.210.470.750.67 0.040.100.220.450.82 0.010.040.090.180.40 d CKA G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.710.560.470.410.37 0.410.780.660.580.54 0.290.590.820.720.66 0.200.400.660.850.77 0.140.270.440.660.88 0.090.170.280.420.63 e OP similarity 060120180240300360420480 0.00 0.05 0.10 0.15 0.20 0.25 0.30 D TG JS f 060120180240300360420480 Epoch 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 D EG JS g GR Random 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 14: Zero-shot GNN transfer on WikiCS dataset. See the caption of Fig. 3 for more detail. 012345 0.25 0.30 0.35 0.40 0.45 0.50 Transfer accuracy a GCN GAT GraphSAGE GR Random 012345 10 20 Training time (s) b 012345 Renormalization level 0 25k 50k 75k Nodes c G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.320.140.070.040.02 0.110.350.150.080.04 0.070.180.350.190.09 0.040.070.250.410.18 0.020.030.100.300.41 0.010.010.040.120.35 d CKA G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.510.330.220.170.12 0.250.550.360.260.18 0.200.360.570.420.29 0.140.220.430.620.43 0.100.140.260.470.64 0.070.090.170.300.54 e OP similarity 060120180240300360420480 0.00 0.05 0.10 0.15 0.20 D TG JS f 060120180240300360420480 Epoch 0.000 0.005 0.010 0.015 0.020 0.025 0.030 D EG JS g GR Random 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 15: Zero-shot GNN transfer on Flickr dataset. See the caption of Fig. 3 for more detail. 20 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization 012345 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Transfer accuracy a GCN GAT GraphSAGE GR Random 012345 1 2 3 4 Training time (s) b 012345 Renormalization level 0 1k 2k Nodes c G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.560.420.410.400.35 0.070.570.500.470.43 0.030.120.610.540.51 0.010.030.090.690.60 0.010.010.030.120.73 0.000.010.010.040.10 d CKA G 0 G 1 G 2 G 3 G 4 G 5 G 0 G 1 G 2 G 3 G 4 G 5 0.700.570.540.530.50 0.250.680.610.580.54 0.150.310.700.640.59 0.100.150.270.760.68 0.070.090.150.300.78 0.050.080.110.180.28 e OP similarity 060120180240300360420480 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 D TG JS f 060120180240300360420480 Epoch 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 D EG JS g GR Random 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 16: Zero-shot GNN transfer on Cora dataset. See the caption of Fig. 3 for more detail. A.8.3 Comparison with Random-T and Random-G baselines In the main text, we compared geometric renormalization (GR) against a fully randomized baseline (Random), in which both the network topology and the geometric coordinates used to merge nodes are randomized before coarse-graining. To disentangle the individual contribution of these two factors, we here report results for two partial baselines: Random-T, in which only the topology is randomized while node coordinates are kept as inferred by Mercator, and Random-G, in which only the geometric coordinates are randomized while the original topology is preserved. Comparing GR against these two variants isolates whether preserving topology, preserving geometry, or their combination is responsible for the transfer performance observed in the main text. Figure 17 shows the zero-shot transfer accuracy for the synthetic networks in both the ultra-small- world (USW) and small-world (SW) regimes, comparing GR against Random-T and Random-G. Figure 18 shows the zero-shot transfer accuracy for each of the eight real-world datasets, comparing GR against Random-T and Random-G. 012345 Renormalization level` 0.6 0.7 0.8 0.9 Accuracy Small world 012345 Renormalization level` 0.6 0.7 0.8 0.9 Accuracy Ultra-small world GCN GAT GraphSAGE GR Random-G Random-T Figure 17: Zero-shot GNN transfer accuracy for SW and USW regimes, against Random-T and Random-G renormalization methods. See the caption of Fig. 3 for details on models and renormalization methods shown. Interestingly, the three architectures respond differently to the Random-T baseline. We hypothesize that this reflects how each model aggregates neighborhood information. One possible explanation is that each architecture aggregates neighborhood information differently: GraphSAGE separates self and neighbor transformations, GCN applies a shared transformation across both, and GAT learns attention weights over neighbors, which may make them differently sensitive to randomized topology. 21 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization 012345 Renormalization level` 0.4 0.5 0.6 0.7 0.8 0.9 Accuracy computers 012345 Renormalization level` 0.4 0.5 0.6 0.7 0.8 0.9 Accuracy cora 012345 Renormalization level` 0.6 0.7 0.8 0.9 Accuracy cs 012345 Renormalization level` 0.25 0.30 0.35 0.40 0.45 0.50 Accuracy flickr 012345 Renormalization level` 0.4 0.6 0.8 Accuracy photo 012345 Renormalization level` 0.7 0.8 0.9 Accuracy physics 012345 Renormalization level` 0.6 0.7 0.8 0.9 Accuracy pubmed 012345 Renormalization level` 0.4 0.6 0.8 Accuracy wikics GCN GAT GraphSAGE GR Random-G Random-T Figure 18: Zero-shot GNN transfer accuracy across all real-world datasets, against Random-T and Random-G renormalization methods. Each panel corresponds to one dataset. See the caption of Fig. 3 for details on models and renormalization methods shown. A.8.4 Topological validation Figures 19 and 20 show the topological validation of GR across all real-world networks considered. Figures 21 and 22 further show the topological validation of the SW regime under the random renor- malization variants (Random, Random-T, and Random-G), the latter broken down by renormalization level l = 1, 2, 3, 4. 22 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization 10 1 10 3 k 10 −4 10 −3 10 −2 10 −1 10 0 P c (k) computers l= 0 l= 1 l= 2 l= 3 l= 4 10 1 10 2 10 3 k 0.00 0.25 0.50 0.75 1.00 ̄ c(k) 10 1 10 2 10 3 k 0 100 200 300 400 ̄ k n (k) 34 39 ̄ k 0 1 H 01234 Renormalization level` 0 1 μ L 2 10 0 10 1 10 2 k 10 −3 10 −2 10 −1 10 0 P c (k) cora 10 1 10 2 k 0.0 0.2 0.4 0.6 0.8 ̄ c(k) 10 1 10 2 k 0 10 20 30 ̄ k n (k) 1 7 ̄ k 0 1 H 01234 Renormalization level` 0 1 μ L 2 10 0 10 1 10 2 k 10 −4 10 −3 10 −2 10 −1 10 0 P c (k) cs 10 1 10 2 k 0.00 0.25 0.50 0.75 1.00 ̄ c(k) 10 1 10 2 k 0 10 20 30 ̄ k n (k) 6 11 ̄ k 0 1 H 01234 Renormalization level` 0 1 μ L 2 10 1 10 3 k 10 −4 10 −2 10 0 P c (k) flickr 10 1 10 2 10 3 k 0.0 0.2 0.4 0.6 0.8 ̄ c(k) 10 1 10 2 10 3 k 0 200 400 ̄ k n (k) 8 13 ̄ k 0 1 H 01234 Renormalization level` 0 1 μ L 2 Figure 19: Topological validation of GR for the real-world datasets considered (part 1 of 2). 23 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization 10 1 10 3 k 10 −4 10 −3 10 −2 10 −1 10 0 P c (k) photo l= 0 l= 1 l= 2 l= 3 l= 4 10 1 10 2 10 3 k 0.00 0.25 0.50 0.75 1.00 ̄ c(k) 10 1 10 2 10 3 k 0 100 200 ̄ k n (k) 29 35 ̄ k 0 1 H 01234 Renormalization level` 0 1 μ L 2 10 0 10 1 10 2 k 10 −4 10 −3 10 −2 10 −1 10 0 P c (k) physics 10 1 10 2 k 0.00 0.25 0.50 0.75 1.00 ̄ c(k) 10 1 10 2 k 0 20 40 60 ̄ k n (k) 12 17 ̄ k 0 1 H 01234 Renormalization level` 0 1 μ L 2 10 0 10 1 10 2 k 10 −4 10 −3 10 −2 10 −1 10 0 P c (k) pubmed 10 1 10 2 k 0.0 0.2 0.4 ̄ c(k) 10 1 10 2 k 0 10 20 30 40 ̄ k n (k) 2 7 ̄ k 0 1 H 01234 Renormalization level` 0 1 μ L 2 10 1 10 3 k 10 −4 10 −3 10 −2 10 −1 10 0 P c (k) wikics 10 1 10 2 10 3 k 0.00 0.25 0.50 0.75 1.00 ̄ c(k) 10 1 10 2 10 3 k 0 200 400 600 800 ̄ k n (k) 36 41 ̄ k 0 1 H 01234 Renormalization level` 0 1 μ L 2 Figure 20: Topological validation of GR for the real-world datasets considered (part 2 of 2). 24 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization 10 0 10 1 10 2 k 10 −4 10 −2 10 0 P c (k) Random-G l= 0 l= 1 l= 2 l= 3 l= 4 10 1 10 2 k 0.0 0.1 0.2 0.3 ̄ c(k) 10 0 10 1 10 2 k 0 10 20 30 ̄ k n (k) 0 20 ̄ k 0 1 H 01234 Renormalization level` 0 1 μ L 2 10 0 10 1 10 2 k 10 −4 10 −2 10 0 P c (k) Random-T 10 1 10 2 k 0.0 0.1 0.2 0.3 ̄ c(k) 10 0 10 1 10 2 k 0 10 20 30 ̄ k n (k) 0 20 ̄ k 0 1 H 01234 Renormalization level` 0 1 μ L 2 10 0 10 1 10 2 k 10 −4 10 −2 10 0 P c (k) Random 10 1 10 2 k 0.0 0.1 0.2 0.3 ̄ c(k) 10 0 10 1 10 2 k 0 10 20 30 ̄ k n (k) 0 20 ̄ k 0 1 H 01234 Renormalization level` 0 1 μ L 2 Figure 21: Topological validation of the SW regime under three random variants of the renormaliza- tion method. 25 Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization 10 0 10 1 10 2 k 10 −4 10 −2 10 0 P c (k) l = 1 GR Random Random-G Random-T Original (l= 0) 10 1 10 2 k 0.00 0.25 0.50 0.75 1.00 ̄ c(k) 10 0 10 1 10 2 k 0 10 20 30 ̄ k n (k) 0 20 ̄ k 0 1 H GR Random Random-G Random-T 0 1 μ L 2 10 0 10 1 10 2 k 10 −4 10 −2 10 0 P c (k) l = 2 10 1 10 2 k 0.00 0.25 0.50 0.75 1.00 ̄ c(k) 10 0 10 1 10 2 k 0 10 20 30 ̄ k n (k) 0 20 ̄ k 0 1 H GR Random Random-G Random-T 0 1 μ L 2 10 0 10 1 10 2 k 10 −4 10 −2 10 0 P c (k) l = 3 10 1 10 2 k 0.00 0.25 0.50 0.75 1.00 ̄ c(k) 10 0 10 1 10 2 k 0 10 20 30 ̄ k n (k) 0 20 ̄ k 0 1 H GR Random Random-G Random-T 0 1 μ L 2 10 0 10 1 10 2 k 10 −4 10 −2 10 0 P c (k) l = 4 10 1 k 0.00 0.25 0.50 0.75 1.00 ̄ c(k) 10 0 10 1 10 2 k 0 10 20 30 ̄ k n (k) 0 20 ̄ k 0 1 H GR Random Random-G Random-T 0 1 μ L 2 Figure 22: Topological validation of the SW regime under GR and three random variants-Random, Random-G, and Random-T of the renormalization method, shown for renormalization levelsl = 1, 2, 3, 4. Dashed black lines indicate the reference values of the original (non-renormalized) network at l = 0. 26