FlatLand: Personalized Graph Federated Learning via Tailored Lorentz Space

Jiahong Liu; Ram Samarth B B; Xinyu Fu; Menglin Yang; Weixi Zhang; Rex Ying; Irwin King

cs.LGAug 21, 2026

FlatLand: Personalized Graph Federated Learning via Tailored Lorentz Space

Authors: Jiahong Liu, Ram Samarth B B, Xinyu Fu, Menglin Yang, Weixi Zhang, Rex Ying, Irwin King

Organizations: Department of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong SAR, China

Abstract

Federated learning enables privacy-preserving collaborative training, but highly heterogeneous client data remain challenging, especially in graph federated learning where clients possess structurally diverse graphs. Existing personalized federated learning (PFL) methods ignore the intrinsic geometric properties of diverse graph structures. We propose FlatLand, a novel personalized federated learning method that embeds different clients' data in tailored Lorentz space of hyperbolic geometry. Our key insight is that hyperbolic geometry naturally accommodates the intrinsic negative curvature prevalent in real-world graphs, while the time-like dimension in Lorentz space provides a principled way to encode client-specific heterogeneity. We develop a parameter decoupling strategy that separates heterogeneous information (captured in time-like parameters) from common knowledge (preserved in space-like parameters), enabling direct aggregation without requiring client similarity estimation and extra calculation modules. Empirical results on diverse federated graph learning tasks demonstrate that FlatLand achieves superior performance, particularly in low-dimensional settings.

Figures & tables

Figure 1: Example: (a) KDE of degree distributions from three CiteSeer clients ( Davis et al., 2011 ) , and (b) their respective 2D Lorentz Spaces with different Lorentz scale parameters $K$ .

Figure 2: Averaged Forman-Ricci curvature across datasets (Cora, ogbn-arxiv, and Amazon-Photo). Higher bars indicate more pronounced non-Euclidean characteristics in these datasets.

Figure 3: The $\mathsf{FlatLand}$ framework. It comprises three stages: (1) Initialization of the client-specific Lorentz scale ( Section 5.1 ), personalized, and shared parameters; (2) Local updates within client-specific Lorentz spaces ( Appendix C.2.3 ); and (3) Server updates, aggregating only shared parameters while preserving personalization by keeping personalized parameters locally ( Section 5.2 ).

	Cora		CiteSeer		ogbn-arxiv		Photo
# clients	$\mathbf{10}$	$\mathbf{20}$	$\mathbf{10}$	$\mathbf{20}$	$\mathbf{10}$	$\mathbf{20}$	$\mathbf{10}$	$\mathbf{20}$
Local ( $E$ )	$79.94\pm 0.24$	$80.30\pm 0.25$	$67.82\pm 0.13$	$65.98\pm 0.17$	$64.92\pm 0.09$	$65.06\pm 0.05$	$91.80\pm 0.02$	$90.47\pm 0.15$
Local ( $L$ )	$78.35\pm 0.05$	$80.46\pm 0.18$	$72.30\pm 0.04$	$69.52\pm 0.25$	$65.85\pm 0.09$	$66.75\pm 0.05$	$91.76\pm 0.10$	$90.12\pm 0.20$
FedAvg	$69.19\pm 0.67$	$69.50\pm 3.58$	$63.61\pm 3.59$	$64.68\pm 1.83$	$64.44\pm 0.10$	$63.24\pm 0.13$	$83.15\pm 3.71$	$81.35\pm 1.04$
FedProx	$60.18\pm 7.04$	$48.22\pm 6.81$	$63.33\pm 3.25$	$64.85\pm 1.35$	$64.37\pm 0.18$	$63.03\pm 0.04$	$80.92\pm 4.64$	$82.32\pm 0.29$
FedPer	$79.35\pm 0.04$	$78.01\pm 0.32$	$70.53\pm 0.28$	$66.64\pm 0.27$	$64.99\pm 0.18$	$64.66\pm 0.11$	$91.76\pm 0.23$	$90.59\pm 0.06$

Table 1: Comparison of node classification performance across real-world datasets with varying numbers of clients. The results, presented as mean and standard deviation, are based on five separate trials. Performances that are statistically significant (

p<0.05

) are highlighted in bold.

	CHEM (1)	BIO-CHEM-SN (3)
# datasets	7	13
Local ( $E$ )	$75.54\pm 1.73$	$67.17\pm 1.76$
Local ( $L$ )	$75.72\pm 2.41$	$65.31\pm 2.13$
FedAvg	$75.88\pm 2.17$	$66.91\pm 1.94$
FedProx	$76.05\pm 1.92$	$66.34\pm 2.26$
FedPer	$75.81\pm 2.17$	$66.27\pm 2.09$

Table 3: Performance on graph classification tasks. The results are reported as mean

\pm

standard deviation over five runs. Bold indicates statistical significance (

p<0.05

Figure 6: Performance comparison between FedAvg, FedPer, FedHGCN, and

\mathsf{FlatLand}

under different client participation rates on Cora with 50 clients.

	Cora (10)	Cora (20)	CiteSeer (10)	CiteSeer (20)
# datasets	10	20	10	20
FedAvg	$69.19\pm 0.67$	$69.50\pm 3.58$	$63.61\pm 3.59$	$64.68\pm 1.83$
FedPer	$\underline{79.35}\pm 0.04$	$\underline{78.01}\pm 0.32$	$70.53\pm 0.28$	$\underline{66.64}\pm 0.27$
$\mathsf{FlatLand}$ ( $E$ )	$78.53\pm 0.73$	$76.23\pm 0.43$	$\underline{70.68}\pm 0.52$	$66.29\pm 0.35$
$\mathsf{FlatLand}$ (ours)	$\mathbf{80.46}\pm 0.28$	$\mathbf{82.49}\pm 0.25$	$\mathbf{73.90}\pm 0.23$	$\mathbf{72.24}\pm 0.24$

Table 4: Ablation study results about the necessity of using Lorentz space to do parameter decoupling.

Figure 7: Client-level performance comparison on Cora (left) and CiteSeer (right) with 10 clients. Each group corresponds to one client and reports accuracy of

\mathsf{FlatLand}

w/o DS,

\mathsf{FlatLand}

, and Local (

L

\mathsf{FlatLand}

improves most clients, while removing DS causes client-wise degradation, supporting the role of time-like parameter decoupling.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8: Averaged Forman-Ricci curvature across datasets (Cora, ogbn-arxiv, and Amazon-Photo). Higher bars indicate more pronounced non-Euclidean characteristics in these datasets.

Figure 9: Averaged Forman-Ricci curvature across heterophilic datasets (Questions and Tolokers). Higher bars indicate more pronounced non-Euclidean characteristics in these datasets.

Dataset	CHEM							BIO			SN
Dataset	MUTAG	BZR	COX2	DHFR	PTC_MR	AIDS	NCI1	ENZYMES	DD	PROTEINS	COLLAB	IMDB-BINARY	IMDB-MULTI
# Graphs	188	405	467	467	344	2000	4110	600	1178	1113	5000	1000	1500
Avg. # Nodes	17.93	35.75	41.22	42.43	14.29	15.69	29.87	32.63	284.32	39.06	74.49	19.77	13.00
Avg. # Edges	19.79	38.36	43.45	44.54	14.69	16.20	32.30	62.14	715.66	72.82	2457.78	96.53	65.94
# Classes	2	2	2	2	2	2	2	6	2	2	3	2	3
Node Features	original	original	original	original	original	original	original	original	original	original	degree	degree	degree

Appendix

Table 5: Statistics of graph classification datasets. We report the (average) number of graphs, nodes, edges, classes, and node features of each dataset.

Dataset	Cora			CiteSeer			ogbn-arxiv			Amazon-Photo
# Clients	1	10	20	1	10	20	1	10	20	1	10	20
# Classes	7			6			40			8
Avg. # Nodes	2,485	249	124	2,120	212	106	169,343	16,934	8,467	7,487	749	374
Avg. # Edges	10,138	891	422	7,358	675	326	2,315,598	182,226	86,755	238,086	19,322	8,547
Avg. Clustering Coefficient	0.238	0.259	0.263	0.170	0.178	0.180	0.226	0.259	0.269	0.410	0.457	0.477
Heterogeneity	N/A	0.606	0.665	N/A	0.541	0.568	N/A	0.615	0.637	N/A	0.681	0.751

Appendix

Table 6: Statistics of homophilic node classification datasets. We report the (average) number of nodes, edges, classes, clustering coefficient, and heterogeneity for different numbers of clients.

Dataset	Roman-empire			Minesweeper			Tolokers			Questions
# Clients	1	10	20	1	10	20	1	10	20	1	10	20
# Classes	18			2			2			2
Avg. # Nodes	22,662	2,326	1,124	10,000	1,008	513	11,758	1,138	605	48,921	5,037	2,484
Avg. # Edges	32,927	6,682	3,268	39,402	7,696	3,820	519,000	40,000	15,000	153,540	20,000	7,702

Appendix

Table 7: Statistics of heterophilic node classification datasets. We report the (average) number of nodes, edges, and classes.

Group	Variant	Cora	CiteSeer	Source
Full model	$\mathsf{FlatLand}$	82.49	72.24	Sec. 7.2 , Table 2
Representation	Local ( $E$ )	80.30	65.98	Sec. 7.2 , Table 2
Representation	Local ( $L$ )	80.46	69.52	Sec. 7.2 , Table 2
Decoupling	$\mathsf{FlatLand}$ ( $E$ )	76.23	66.29	Sec. 7.5 , Table 4
Decoupling	w/o DS	71.76	63.08	Sec. 7.5 , Figure 5
Tailored curvature	w/o TS	78.83	67.93	Sec. 7.5 , Figure 5

Appendix

Table 8: Unified ablation summary on Cora and CiteSeer with 20 clients. Results are reported as mean accuracy.

Clients	Ricci	Ollivier	Const.	MLP
10	$73.90\pm 0.23$	$73.51\pm 0.18$	$72.91\pm 0.20$	$72.96\pm 0.46$
20	$72.24\pm 0.24$	$72.21\pm 0.26$	$71.89\pm 0.31$	$72.59\pm 0.50$

Appendix

Table 9: Performance with different curvature initialization methods on CiteSeer. Results are reported as mean

\pm

standard deviation over five runs.

Figure 10: Convergence curves of $\mathsf{FlatLand}$ and strong baselines.

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Graph Federated Learning (GFL) enables collaborative representation learning across distributed subgraphs while preserving privacy. However, heterogeneity remains a critical challenge, as subgraphs across clients typically differ significantly in both semantics and structures. Existing methods address heterogeneity by enforcing the rigid alignment of model parameters or prototypes between clients and the server. However, these alignments implicitly rely on a restrictive global linearity assumption that summarizes local data distributions using a single and globally consistent representation space. This severely compresses the personalized representation space of clients and fails to preserve diverse local graph distributions. To overcome these limitations, we propose Federated Graph Manifold Calibration (FedGMC), a novel paradigm that tackles semantic heterogeneity and structural heterogeneity from a unified manifold perspective. Instead of enforcing rigid alignment, FedGMC introduces a dual manifold calibration mechanism that preserves global commonalities while maximizing the personalized representation space of local clients. Specifically, for semantic heterogeneity, the server constructs a geometrically optimal semantic manifold via equidistant semantic anchors, so as to guide the calibration of local semantic manifolds. For structural heterogeneity, the server constructs a global structural manifold by building global structural templates, so as to guide the calibration of local structural manifolds. Finally, the server dynamically refines both global semantic manifolds and structural manifolds by aggregating local manifolds. Extensive experiments on eleven homophilic and heterophilic graphs demonstrate that FedGMC effectively balances global commonality and local personalization, thereby significantly outperforming state-of-the-art baseline methods.

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