From Euclidean to Graph-Structured Data: A Survey of Collaborative Learning

1 Introduction

• Data fragmentation, privacy legislation, ownership, integration, and coordination barriers make access to massive datasets a persistent obstacle for machine learning.Source: §1 · p. 2
• Graph collaboration includes independent graph collections and partitioned shared graphs, where topology-spanning relationships require information exchange during both learning and inference.Source: §1 · p. 3

1.1 The Objectives of the Survey and Related Work

• Collaborative graph learning is an emerging intersection of diffusion algorithms, graph representation learning, and collaborative learning, extending federated and decentralized approaches to private distributed graphs.Source: §1.1 · pp. 3–4
• The survey focuses on supervised learning and connects established results across the three domains to clarify relationships, accelerate research, and identify unanswered challenges.Source: §1.1 · pp. 3–4

1.2 The Organization of the Paper

• The paper first reviews Euclidean collaborative learning through effectiveness, efficiency, and privacy, then develops graph-data partition taxonomies and surveys corresponding solutions and open questions.Source: §1.2 · pp. 4–5

2 Collaborative Learning on Euclidean Data

• Collaborative learning differs from parallel distributed learning because private local data remains with its owner, encompassing federated and decentralized learning on Euclidean feature vectors.Parallel and model parallelism assume data can be centralized, whereas collaborative learning preserves distributed data ownership.Source: Figure 4 · p. 6, §2 · pp. 6–7

2.1 Collaborative Learning: Problem Formulation and the Federated Learning Framework

• Because local distributions are unknown and datasets are often too small, agents use empirical losses and collaborate under an assumption of shared distributional similarities to reduce estimation error.Minimizing empirical rather than population loss is computationally feasible, but its validity depends on a negligible generalization gap.Source: §2.1 · pp. 7–8
• Federated learning coordinates round-based training through a server that broadcasts a global model, receives local updates, aggregates them, and returns the jointly trained model for local inference.FedAvg samples agents, performs E local SGD updates, averages their models, and can converge to a solution under specific conditions.Source: Figure 5 · p. 7, §2.1 · p. 8

2.2 Data Partition, Heterogeneity and Privacy Challenges

• Realistic collaborative learning must address data partitioning, statistical imbalance, system heterogeneity, and privacy vulnerabilities that violate FedAvg’s idealized assumptions.These challenges affect how data is distributed, how local datasets differ, how agents train and communicate, and what information can be inferred.Source: §2.2 · p. 9

2.2.1 Data Partition

• Data may be horizontally partitioned across samples, vertically partitioned across feature and label domains, or transfer-partitioned across heterogeneous feature spaces with overlapping anchor samples.Horizontal partitioning is the default and most studied FL setting; transfer partitioning often involves substantial source-target data imbalance.Source: §2.2.1 · pp. 9–10, Figure 6 · p. 10

2.2.2 Statistical Imbalance

• Statistical imbalance includes quantity imbalance, covariate shift, and concept shift, with the latter changing the conditional feature-label relationship across agents.Quantity differences distinguish cross-device from cross-silo regimes, while covariate shift changes observed feature or label marginals and concept shift changes the underlying mapping.Source: §2.2.2 · p. 11

2.2.3 System Heterogeneity

• Computational, communication, and task heterogeneity create variable resource demands, straggler bottlenecks, dynamic-population complications, and potentially distinct local objectives or model architectures.Agents range from IoT devices to HPC clusters and use networks with different topologies and bitrates; agents may also join or leave and perform different tasks.Source: §2.2.3 · pp. 11–12

2.2.4 Privacy Vulnerabilities

• Although FedAvg limits data collection by communicating and aggregating model updates, collaborative learning provides no inherent privacy guarantee: models, gradients, and training trajectories can leak sensitive information.Attacks include model inversion, gradient inversion and reconstruction, membership inference, temporal observation, and active manipulation of contributions.Source: §2.2.4 · pp. 12–13

• Effective learning demands information exchange and complex computation, creating tensions with efficiency.Source: The Collaborative Learning Trilemma · p. 13
• Cryptographic techniques for privacy reduce efficiency, while noise injection decreases learning effectiveness.Source: The Collaborative Learning Trilemma · p. 13
• The collaborative learning trilemma balances effectiveness, efficiency, and privacy preservation.Effectiveness concerns accurate predictions, efficiency constrains communication and computation, and privacy preservation anonymizes agents’ data.Source: Figure 7 · p. 13, The Collaborative Learning Trilemma · p. 13

2.3 Improving Learning Effectiveness

• FedAvg can become suboptimal under non-i.i.d. data and unbalanced participation, motivating data-based, model-based, and vertical-partition strategies to improve local-objective minimization.The taxonomy includes data-based and model-based techniques for horizontal partitions, plus vertical federated learning with collaborative inference.Source: §2.3 · pp. 13–14, Figure 8 · p. 14

2.3.1 Data-based Techniques

• Data-based techniques mitigate client drift while retaining a shared global model through local regularization, data augmentation, and client selection.Regularization reduces update variance, augmentation modifies local distributions, and selection changes which clients contribute to aggregation, trading effectiveness against efficiency and sometimes privacy.Source: §2.3.1 · pp. 14–16

2.3.2 Model-based Techniques

• Model-based techniques personalize parameters or architectures because heterogeneous agents may have different optimal solutions, using adaptation, hypernetworks, clustering, similarity weighting, decoupling, multiple models, or distillation.Personalization can improve local performance but increases coordination and communication costs; explicit agent comparison or data sharing can expose agent-specific information.Source: §2.3.2 · pp. 16–17

2.3.3 Training and Inference under Vertical Partitioning

• Under vertical partitioning, agents combine complementary features of shared samples by exchanging intermediate representations and gradients for collaborative inference and optimization.Training requires a collaborative inference loss and privacy-preserving sample alignment, commonly using Private Set Intersection; exchanged representations and gradients may leak sensitive information.Source: §2.3.3 · pp. 19–20
• Collaborative inference exchanges local representations, aggregates them at an active agent, computes loss and gradients, and updates each agent’s local parameters across rounds.The protocol performs collaborative inference followed by a collaborative model-update phase.Source: §2.3.3 · p. 20
• Pre-training can reduce communication and scale vertical federated learning but may introduce bias from local data distributions, while self-supervised objectives support learning without labels.Self-supervised extensions include contrastive losses and autoencoder reconstruction losses.Source: §2.3.3 · p. 21
• Isolated inference reduces inference-time communication by transferring collaborative knowledge through distillation for overlapping feature spaces or transfer learning when source agents have richer data.Knowledge distillation aligns representations using shared unlabeled samples, while transfer learning assumes semantically similar representations from unaligned samples and may use feature, pseudo-label, or adversarial matching.Source: §2.3.3 · pp. 21–22
• Vertical partitioning improves learning effectiveness when agents hold complementary features of shared samples, using collaborative or isolated inference according to deployment requirements.Collaborative inference jointly optimizes over the full feature space, whereas isolated inference supports autonomous deployment.Source: §2.3.3 · p. 22
• The vertical-partitioning trilemma is that collaborative inference improves accuracy but incurs persistent coordination overhead, whereas isolated inference avoids inference communication but requires complex alignment and exposes privacy risks.Privacy exposure arises from embedding exchange, sample alignment through PSI protocols, and representation sharing during knowledge transfer.Source: §2.3.3 · p. 22

2.4.1 Model Aggregation

• Model aggregation balances convergence, communication cost, scalability, robustness, and privacy across centralized, decentralized, and hierarchical topologies.Centralized schemes simplify coordination but create bottlenecks and single points of failure; decentralized methods remove central coordination, while hierarchical architectures localize communication and support larger systems. Aggregation timing and topology create tradeoffs among convergence, accuracy, communication efficiency, and privacy.Source: §2.4.1 · pp. 24–25

2.4.2 Asynchronous Aggregation

• Under bounded staleness, asynchronous distributed learning preserves synchronous convergence rates and accuracy, although delays can reduce actual convergence speed.Immediate aggregation improves responsiveness but increases communication and privacy exposure, whereas batched aggregation reduces communication costs while potentially slowing convergence or losing useful straggler information.Source: §2.4.2 · p. 26

2.4.3 Communication Efficiency

• Communication-efficient learning exploits aggregation’s tolerance to noise through transmission reduction, wireless over-the-air aggregation, and client scheduling.Quantization and sparsification reduce transmitted information; wireless channel superposition performs aggregation during transmission with costs independent of agent count; scheduling selects clients using channel quality, update importance, or total learning time.Source: §2.4.3 · p. 27

2.4 Improving Learning Efficiency

• Efficient learning must manage tradeoffs between resource savings and optimization quality, including accuracy loss, coordination complexity, fairness, privacy exposure, and delayed convergence.These tradeoffs arise from quantization and sparsity, synchronization and channel-state requirements, selection bias toward well-connected clients, frequent individual updates, and staleness.Source: §2.4.1 · p. 25, §2.4.2 · p. 26, §2.4.3 · p. 27

Blind computations

• Blind computations use TEEs, HE, and SMPC to cryptographically prevent access to private intermediate computations, which are especially exposed in VFL.Intermediate values such as local models, gradients, and representations can leak private information; VFL exchanges representations, whereas horizontal FL mainly aggregates models or gradients.Source: §2.5 · p. 28

Differential privacy

• Differential privacy limits inference about private data by randomizing collaborative learning, with smaller ε providing stronger guarantees and δ allowing rare bound violations.DP-SGD and related methods use subsampling and shuffling, per-sample gradient clipping or quantization, and calibrated noise before model updates.Source: §2.5 · p. 28
• Collaborative DP typically protects each agent’s entire local dataset through agent-level DP, implemented with local, distributed, or server-integrated mechanisms.Local DP protects data from an honest-but-curious aggregator but has high communication and computation demands and can degrade model estimates; distributed DP amplifies mild local protections through server-side shuffling and subsampling.Source: §2.5 · p. 29
• DP variants of FedAvg balance privacy, learning, and communication by combining mild local noise or quantization with server-side clipping, scaling, shuffling, or subsampling; DP-FTRL instead uses tree aggregation and correlated noise.Naively distributing DP-SGD is infeasible because the server lacks control over local data sampling and updates.Source: §2.5 · p. 29

Trilemma

• Privacy protection creates a trilemma: stronger guarantees can reduce learning effectiveness through noise, clipping, and subsampling while increasing computation, communication, and protocol complexity.Distributed DP has been demonstrated in large-scale deployments at Google, Apple, and Microsoft, while newer work extends privacy to gossip learning, transparent aggregation and inference, and verifiable mechanisms.Source: §2.5 · pp. 29–30

• The section first reviews state-of-the-art solutions for learning on graph-structured data before examining collaborative-learning-specific challenges.This overview establishes the foundation for the subsequent graph-domain analysis.Source: §3 · p. 30
• It formulates collaborative learning for graph-structured data and characterizes the different types of heterogeneity that arise.Source: §3 · p. 30
• The analysis organizes graph-domain design choices around learning effectiveness, efficiency, and privacy preservation, with the surveyed papers summarized in Table 2.This structure follows the same trilemma used for Euclidean data and is formalized in Figure 7.Source: §3 · p. 30

Graph data and learning problem

• Graphs model relationships through nodes, edges, features, and task-dependent labels, supporting node classification, link prediction, and graph classification.A graph is represented as G = (V, E), with adjacency matrix A and feature matrix X; datasets contain graph samples (G, y) drawn from an unknown distribution.Source: §3.1 · p. 30

Graph representation learning

• Graph representation learning embeds nodes, edges, or whole graphs into fixed-dimensional Euclidean spaces while preserving topology and node attributes.A node encoder ϕnode produces node embeddings, which can be combined by ϕlink or ϕgraph and passed to a predictor fψ for downstream tasks.Source: §3.1 · p. 32, Figure 11 · p. 32

Graph Neural Networks

• Modern GNNs predominantly use Message Passing Neural Networks, which iteratively aggregate permutation-invariant information from each node’s L-hop neighborhood to compute representations.Different message, aggregation, and update functions yield architectures such as GCN (Kipf & Welling, 2017), GraphSAGE (Hamilton et al., 2017), and GATs (Veličković et al., 2018).Source: §3.1 · pp. 32–33

Spectral and spatial GNNs

• Spectral GNNs filter graph signals in the Laplacian eigenbasis, but eigendecomposition is expensive and must be recomputed after graph modifications; polynomial approximations reduce these operations to iterative neighborhood propagation.Polynomial bases include Cheb-Nets (Defferrard et al., 2016; He et al., 2022), BernNet (He et al., 2021), APPNP (Gasteiger et al., 2019), JacobiNet (Wang & Zhang, 2022), and GPRGNN (Chien et al., 2021).Source: §3.1 · pp. 32–33

GNN training and distributed systems

• MPNNs are a natural fit for collaborative machine learning because their parameters and gradients propagate along graph edges like distributed message-passing algorithms.GNNs are commonly trained with SGD and backpropagation, sometimes augmented by link-prediction or contrastive self-supervision, while empirical risk serves as a proxy for true risk.Source: §3.1 · p. 34

Problem settings

• Collaborative graph learning is divided into multiple graph instances and subgraphs of a single global graph.The two settings are illustrated in Figure 13.Source: §3.2 · p. 34, Figure 13 · p. 35

Learning from multiple graph instances

• For multiple graph instances, each agent holds graphs from a local distribution and typically performs graph-level tasks such as molecule-property prediction.When local distributions are i.i.d., FedAvg can identify optimal GNN parameters.Source: §3.2 · pp. 34–35

Learning from subgraphs

• For subgraphs, agents observe node-induced portions of a global graph, supporting node-, link-, and subgraph-level tasks in applications including recommender systems and social networks.A GNN’s L-hop neighborhood may extend beyond an agent’s local subgraph, requiring information from other agents.Source: §3.2 · pp. 35–36
• Subgraph learning supports isolated inference on local data or collaborative inference that exchanges boundary-node embeddings across agents to reproduce distributed message passing.Algorithm 3 combines centralized model aggregation with server storage and exchange of node representations during collaborative inference.Source: §3.2 · pp. 36–37

3.3.1 Graph Partition Taxonomy

• Graph data partitions span full graph instances and subgraphs, organized by topology, features, and schema, with horizontal, vertical, edge-cut, vertex-cut, and ego-graph variants.These partition dimensions can be combined, while vertical topology partition remains unimplemented in distributed training.Source: §3.3.1 · pp. 37, 39, 40, Figure 16 · p. 40

3.3.2 Statistical Heterogeneity in Graphs

• Graph statistical heterogeneity includes quantity, feature, structure, and concept shifts, with structure shifts comprising size, density, and motif variation across agents.MPNNs experience these differences through local L-hop ego-graph distributions; evolving graphs couple topology and features, limiting static shift categories.Source: §3.3.2 · pp. 42–43
• Structure concept shifts create homophily conflict between agents, while feature concept shifts challenge GNNs that assume stable feature–label correlations.Differences in homophily levels across distributed graphs decrease learning accuracy.Source: §3.3.2 · p. 43

3.3.3 System Heterogeneity in Collaborative Learning over Graph Data

• Collaborative subgraph inference requires exchanging boundary-node embeddings, so graph topology determines dependencies and amplifies computational, memory, communication, and synchronization heterogeneity.Costs increase with local edges, message-passing layers, boundary nodes, and boundary centrality, particularly for dense or widely connected subgraphs.Source: §3.3.3 · p. 44
• Partitioning graph data must trade lower communication from fewer edge cuts against balanced computation, while stragglers delay parameter updates, message passing, and inference.The trade-off becomes harder as the number of agents and system constraints increase, and late embeddings degrade inference speed and quality.Source: §3.3.3 · p. 44
• Dynamic graphs compound system heterogeneity because new nodes change inference topologies and local neighborhood distributions.Source: §3.3.3 · p. 44
• Task heterogeneity spans node-, edge-, and graph-level prediction or different domains, while modular GNNs support shared encoders with task-specific prediction heads.This flexibility follows from separating encoding, message passing, and decoding components.Source: §3.3.3 · p. 44

3.3.4 Privacy Vulnerabilities in Collaborative Learning over Graphs

• Graph collaboration inherits aggregation and inference privacy vulnerabilities, and exchanged boundary or final embeddings can expose node features, edges, and training participation.Inference attacks can extract private node features, reconstruct edge existence, or identify nodes and edges involved in training (Duddu et al., 2020; Wu et al., 2022; He et al., 2021; Olatunji et al., 2021; Zhang et al., 2021).Source: §3.3.4 · p. 45
• The analysis assumes honest-but-curious agents; active subgraph manipulation, inference attacks, and training poisoning by malicious agents remain outside scope.Such attacks are associated with (Dai et al., 2018).Source: §3.3.4 · p. 45

3.4.1 Data-based Techniques

• Data-based techniques adapt augmentation, regularization, and subsampling to graph topology, addressing statistical heterogeneity and data isolation across partitioned graphs.They may recover neighborhoods, rewire or extend graphs, augment features and labels, synthesize graph instances, regularize representations, or selectively sample agents and nodes; these choices trade effectiveness against communication and privacy.Source: §3.4 · p. 45, §3.4.1 · pp. 46, 49

3.4.2 Model-based Techniques

• Open directions include characterizing topology-dependent heterogeneity and optimization, supporting evolving graphs and higher-order structures, pruning important layers, and integrating graph foundation models under collaborative constraints.Architecture decoupling can improve effectiveness but increase coordination costs, while embedding alignment and knowledge distillation expose intermediate representations and topology information requiring privacy protection.Source: §3.4.1 · p. 48, §3.4.2 · p. 51
• Model-based techniques personalize GNN parameters and architectures at granularities ranging from nodes and subgraphs to layers, edge types, and message-passing operations.Approaches include local adaptation, clustering, similarity-based personalization, model decoupling, pruning, multimodal models, knowledge distillation, and specialized processing for cross-boundary messages.Source: §3.4.2 · pp. 49–50

3.4.3 Training and Inference under Aligned Partitioning

• Aligned vertical and vertex-cut partitions split complementary views of the same entities across agents, requiring cross-agent view aggregation beyond parameter aggregation or cross-boundary message passing.This setting parallels vertical federated learning because agents hold complementary information about shared nodes or graphs rather than disjoint complete samples.Source: §3.4.3 · p. 51
• Aligned-partition methods aggregate node representations, transfer knowledge across agents, or combine graph-level representations to improve collaborative prediction.These families address shared nodes in vertex-cut partitions, aligned entities in vertical partitions, and multiple graph views of the same instance.Source: §3.4.3 · pp. 51–52
• Averaging node representations improves generalizability, while loss-restricted averaging and gradient-based updates can provide selective or more stable aggregation (Chen et al., 2021; 2022; Wu et al., 2022; 2021; Zhang et al., 2022; Qu et al., 2023; Han et al., 2024; Agrawal et al., 2024; Ni et al., 2021; Mai et al., 2024; Zheng et al., 2023).Personalization instead combines local and averaged representations using MLPs or learnable linear interpolation to address statistical shifts and task heterogeneity (Chen et al., 2021; Wang et al., 2024).Source: §3.4.3 · p. 52
• Transfer learning aligns source-agent and target-agent representations while the target trains locally, using adversarial training, domain discrimination, or knowledge distillation (Shen et al., 2020; Hinton et al., 2014).This enables adaptation of a frozen source model to the target domain without aggregating all node representations.Source: §3.4.3 · p. 52
• For vertically partitioned graph instances, averaging graph embeddings from agents representing the same item or person can improve graph-level prediction accuracy.The cited use case concerns health applications.Source: §3.4.3 · p. 52
• Open challenges include transfer across multimodal collaborators without negative transfer and selective aggregation that reduces communication overhead, privacy exposure, and leakage of topology or inference data (Frasca et al., 2024; Wang et al., 2025; Tian et al., 2025).Aligned entities may have different features and topology, while informative nodes vary by structural role; node or subgraph alignment itself also requires privacy guarantees.Source: §3.4.3 · pp. 52–53

3.5 Improving Inference and Learning Efficiency

• Graph learning efficiency must address training, inference, and communication used by gradient backpropagation, with existing work concentrating mainly on aggregation patterns shaped by graph partitioning.Unlike Euclidean settings, message passing creates efficiency costs for both model updates and node-embedding exchange.Source: §3.5 · p. 53

3.5.1 Aggregation over a network

• Aggregation extends centralized, decentralized, and hierarchical patterns to both model parameters and embeddings, while computation and communication topologies may be coupled, independent, or distinct across training and inference.Embedding efficiency methods modify communication timing, topology, or exchanged content; model aggregation additionally depends on partition type and gradient communication costs.Source: §3.5.1 · pp. 53, 54, 57, 59
• Pre-aggregation with GCN reduces communicating an l-hop neighborhood to an (l−1)-hop neighborhood, while SGC reduces it to the target node’s data and can precompute local inference operations.Other strategies include pre-communication, stale embeddings, truncated message passing, computational-tree trimming, graph partitioning, and pseudo-label transmission.Source: §3.5.1 · pp. 55–57
• Efficiency techniques can degrade effectiveness, while highly decentralized inference, joint topology optimization, and modular post-processing remain open directions requiring further study.Pre-communication and embedding exchange also expose neighborhood structure or intermediate representations, motivating privacy-preserving schemes.Source: §3.5.1 · pp. 58–59

3.5.2 Asynchronous Aggregation and Delay Control

• Asynchronous aggregation addresses heterogeneous communication and computation delays that create stale models and embeddings, either by adapting protocols to staleness or by modifying MPNNs to tolerate or exploit delay.Solutions include skipped transmissions with convergence bounds, update chains, robust asynchronous architectures, and flood-and-echo propagation.Source: §3.5.2 · pp. 59–60
• Topological delays and accelerations can improve GNN expressiveness by addressing oversmoothing and oversquashing, using delayed or beyond-one-hop embeddings and learnable node actions.These mechanisms modify the aggregation step or control whether nodes listen, broadcast, or remain inactive.Source: §3.5.2 · p. 60
• Future asynchronous methods must adapt to dynamically changing agent capabilities and extreme staleness caused by prolonged delays, intermittent connectivity, and cyclical network patterns.Open problems include coordinating computational tree traversal across agents with varying resources and balancing recent distant embeddings against stale neighboring embeddings.Source: §3.5.2 · p. 60
• Asynchronous graph aggregation must manage staleness in both model parameters and node embeddings, using protocol modifications or delay-aware message passing to balance efficiency and effectiveness.Skipping transmissions or using stale embeddings reduces communication but can degrade effectiveness, especially for deep computational trees and high-degree nodes.Source: §3.5.2 · p. 60

3.5.3 Communication Efficiency

• Communication efficiency is central because collaborative GNNs exchange intermediate embeddings during both message passing and learning over heterogeneous, unreliable, and costly links.Unlike conventional data exchange, downstream tasks generally depend on aggregated embedding functions rather than exact embedding values.Source: §3.5.3 · pp. 61–62
• Transmission efficiency methods improve reliability or convergence through coding and retransmission, node subsampling, and failure-aware learning under bandwidth limits and stochastic communication disruptions.These approaches target inference accuracy, convergence time, or accurate inference despite failed message passing caused by wireless impairments or temporary disconnections.Source: §3.5.3 · p. 61
• Over-the-air embedding aggregation exploits wireless superposition to reduce communication costs and enable simultaneous updates, while scheduling and topology management adapt communication to network structure and resource constraints.Related work addresses channel impairments, variable delays, edge-server assignment, static and dynamic topologies, and adaptive communication-topology sampling (Liu et al., 2021; Zeng et al., 2022; Nazzal et al., 2024; Scardapane et al., 2020; Wang et al., 2025).Source: §3.5.3 · pp. 61–62
• Open directions include co-designing communication and computing, exploiting broadcast and multicast for communication-native inference, and ensuring safe decisions despite information loss and exposed embeddings.These directions must account for time-varying channel quality, complex topologies, communication impairments during training and inference, and radio-range eavesdropping risks.Source: §3.5.3 · p. 62

Overview

• Collaborative GNN privacy techniques target model parameters, exchanged node embeddings, and edge information, primarily in cross-silo applications involving sensitive personal data.They adapt blind computation and LDP from Euclidean federated learning to address parameter, embedding, and topology leakage.Source: §3.6 · p. 62

Anonymizing model parameters

• Model parameters and gradients can be anonymized with federated-learning techniques, including adaptive noise scaling that adjusts perturbations to gradient norms.Adaptive scaling is intended to improve the stability–privacy trade-off during training.Source: §3.6 · p. 63

Anonymizing exchanged embeddings

• Intermediate embeddings require protection because exchanging them across subgraph boundaries enables feature extraction and membership inference, even when raw node features remain local.Blind computation securely aggregates embeddings, while LDP uses noise, clipping, subsampling, quantization, and privacy-budget adjustment; these methods can reduce learning effectiveness.Source: §3.6 · p. 63

Anonymizing edge information

• Topology protection addresses link stealing through Edge LDP, blind subgraph expansion, and private graph-statistics aggregation, but topological noise can densify graphs and cause oversmoothing.Mitigations include regularization, degree preservation, and allocating privacy budget to maintain adjacency sparsity.Source: §3.6 · pp. 63–64

Graph-level privacy

• Graph-specific privacy can jointly protect features and topology through node- or subgraph-level privacy, ego-graph blending, pseudo-edge inference, and selectively combined protections across hierarchy levels.These mechanisms anonymize local graph data and can be tailored to device–edge–server communication hierarchies.Source: §3.6 · p. 64

Open challenges

• The field faces a privacy–efficiency–effectiveness trilemma: blind computation is costly, whereas LDP can impair message passing; multi-agent privacy amplification and unified privacy–defense frameworks remain unexplored.Amplification could preserve privacy along sequential computation chains with less information loss, while unified defenses must address both honest-but-curious and Byzantine adversaries.Source: §3.6 · p. 65

• Collaborative learning on graph-structured data opens opportunities beyond centralized graph ML by mining distributed data across domains such as sensor networks and molecular biology.Source: §4 · p. 65
• The survey systematically maps problem formulations, data and system heterogeneity, and the effectiveness–efficiency–privacy trilemma from Euclidean to graph-structured collaborative learning.Source: §4 · p. 65
• Graph-structured collaborative learning forms a nascent, rich landscape organized around isolated inference and collaborative inference, with collaborative inference more central because GNNs rely on message passing.Source: §4 · p. 65
• Combining distributed relational data with collaborative training or inference introduces unique constraints requiring joint consideration of topologies, communication, privacy, and heterogeneities.Source: §4 · pp. 65–66
• The survey establishes a coherent vision for developing collaborative graph learning methods and enabling practical impact in drug discovery, healthcare, sustainable industries, and smart cities.Source: §4 · p. 66

Organization of the solutions

• The solution taxonomy spans effective, efficient, and privacy-preserving collaborative learning for Euclidean and graph data.Source: Figure 21 · p. 94

Distributed structures

• Distributed systems spread data, computation, or control across agents, including data parallelism and model parallelism.Data parallelism distributes centrally collected data across agents, whereas model parallelism distributes centrally maintained model parts.Source: Glossary · pp. 94–95

Collaborative systems

• A collaborative system is a distributed system of agents cooperating toward shared objectives, although local constraints or incentives may limit their willingness to collaborate.Agents have local data, computation, and communication capabilities; a server centrally supervises agents, while federated systems use central supervision and decentralized systems do not.Source: Glossary · pp. 94–95

Scope and topology

• Local properties concern one agent, whereas global properties are shared across or computed from many agents without implying that every agent has a complete view.Topology describes the communication or computational relationships between agents, typically as a graph.Source: Glossary · p. 95

Learning concepts

• A sample has features and a target label; training optimizes model parameters to reduce prediction errors, while inference predicts labels from features using the learned model.Learning encompasses the methodologies, algorithms, and theoretical foundations for effective training and inference.Source: Glossary · p. 95

Data structures

• Euclidean data represents samples as vectors in Rn with the standard inner product, whereas non-Euclidean data has relational structure not faithfully representable in that space.Graphs are the most common example of discrete non-Euclidean structures.Source: Glossary · p. 95

Notations

• Table 3 organizes the paper’s notations into comprehensive categories and alphabetical order.Source: Table 3 · p. 96