Networks of modern industrial systems are increasingly monitored by distributed sensors, where each system comprises multiple subsystems that generate high-dimensional time-series. These subsystems are often interdependent, making it important to understand how temporal patterns observed at one subsystem relate to those observed across others. This problem is particularly challenging in decentralized settings where raw measurements cannot be shared, and client observations are heterogeneous. Moreover, in practice, each subsystem (client) operates a fixed, proprietary model that cannot be modified or retrained, limiting the applicability of existing approaches. Separately, nonlinear dynamics make cross-client temporal interdependencies difficult to characterize, as they are embedded implicitly within nonlinear state-transition functions. This paper presents a vertical federated learning framework for learning one-step-ahead cross-client temporal interdependencies under these constraints. Each client maps high-dimensional local observations to low-dimensional latent states using a nonlinear state-space model, while a Graph Attention Network (GAT)-based central server learns a state-transition model over the communicated states. Given a candidate graph skeleton, the server learns state-dependent attention coefficients that capture cross-client interactions. We analytically relate the GAT attention coefficients to the Jacobian of the learned state-transition model, yielding a first-order characterization of cross-client interdependencies. We further establish bounded closeness of the learned dynamics and Jacobians to a corresponding centralized oracle. Synthetic experiments validate the theoretical results on controlled systems with known graph skeletons. Experiments on real-world datasets with unknown graph skeletons demonstrate superior federated root cause analysis over baselines.
Figures & tables
Figure 1. Schematic of our proposed framework with communication between client m and the server
Figure 2. Training loss curves on the nonlinear SSM synthetic dataset for clients (left) and server (right) for a M=3 clients.
Method
Linear SSM
Nonlinear SSM
Lorenz-96
Centralized Oracle
0.036
0.005
0.001
Proprietary
0.654
0.117
0.932
Ours
0.101
0.010
0.011
Reduction (%)
84.6
91.4
98.9
Table 1. Prediction accuracy (test MSE) on synthetic datasets.
Figure 3. Average client residuals on the nonlinear SSM.
Method
Linear SSM
Nonlinear SSM
Lorenz-96
FedPC
0.072
0.586
0.499
FedCSL
0.072
0.585
0.499
NOTEARS-ADMM
0.068
0.583
0.497
FDBNL
0.063
0.587
0.500
FedDAG
0.027
0.415
0.477
Ours
0.965
0.956
0.732
Table 2. Cosine similarity: learned vs ground truth Jacobians.
Figure 4. Synthetic validation: Correlation between the learned model and the ground truth for (a) attention coefficients and (b) Jacobian operators. (c) Correlation between attention coefficients and Jacobians in the learned model.
HAI
TEP
SMD
Method
AC@1
AC@2
AC@1
AC@2
AC@1
AC@2
FedPC
0.41
0.76
0.19
0.40
0.41
0.60
FedCSL
0.41
0.65
0.20
0.39
0.43
0.63
NOTEARS-ADMM
0.35
0.76
0.18
0.35
0.42
0.61
FDBNL
0.41
0.71
0.19
0.37
0.44
0.62
FedDAG
0.35
0.65
0.18
0.39
0.41
0.62
Table 3. RCA performance (AC@ k ) on real-world datasets.
Method
HAI
TEP
SMD
Centralized (Oracle)
0.240
0.635
0.377
Proprietary
0.873
0.720
0.712
Ours
0.508
0.699
0.558
Reduction (%)
41.8
2.9
21.6
Table 4. Prediction accuracy (test MSE) on real-world datasets.
Dataset
Corrupted Clients
AC@1
AC@2
HAI
None
0.76
0.94
1 of 3 (33%)
0.59
0.94
2 of 3 (67%)
0.59
0.94
All
0.47
0.82
TEP
None
0.45
0.65
1 of 5 (20%)
0.40
0.63
Table 5. Robustness under corrupted proprietary models.
Dataset
Server
Client Loss ↓
Mass on Absent Edges ↓
Linear SSM
GAT
0.101
0.000
LSTM
0.101
0.146
MLP
0.101
0.129
Transformer
0.101
0.224
Nonlinear SSM
GAT
0.0100
0.000
LSTM
0.0112
0.457
Table 6. Comparison of server architectures.
Dataset
Server Graph
Client Loss ↓
Cosine Similarity ↑
Linear SSM
True Graph
0.1010
0.965
ρ=10%
0.1010
0.880±0.036
ρ=20%
0.1010
0.817±0.026
ρ=30%
0.1010
0.721±0.019
Fully Connected
0.1010
0.932
Diagonal Only
0.1000
0.067
Table 7. Graph misspecification study on synthetic datasets.
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 5. Toy example ( M=3 nonlinear SSM, same oracle and server GAT as Section 7 ): edge-by-edge temporal tracking of the Jacobian magnitudes ∥Jmn(t)∥ (standardized), learned server vs. ground truth, over the 3×3 grid of client pairs.
Figure 6. Toy example ( M=3 ): edge-by-edge temporal tracking of the attention coefficients αmn(t) (standardized) over the 3×3 grid of client pairs.
Figure 7. Toy example ( M=3 ): per-edge temporal correlation between attention coefficients and Jacobian magnitudes.
Figure 8. State dependence of the learned operator (Nonlinear SSM). Left of the dotted line: window-averaged ∥Jmn∥ in an early and a late window, ground truth (top) vs. ours (bottom); each map matches the truth (match cosine annotated). Right: the between-window change map: the learned operator changes in the same places and directions as the truth (change-match annotated). Gray = no edge; color scales are per row.
HAI
TEP
SMD
Operator
AC@1
AC@2
AC@1
AC@2
AC@1
AC@2
Fixed identity J
0.43
0.77
0.20
0.40
0.44
0.64
Fixed all-ones J
0.56
0.85
0.40
0.63
0.52
0.70
Ours, learned J(t)
0.76
0.94
0.45
0.65
0.53
0.70
Appendix
Table 8. RCA with fixed reference operators in place of the learned J(t) , under the same detector, timing, and ranking as Table 3 .
Scale w.r.t.
Value
Server loss Ls
Read-out (s/step)
8
0.0022±0.0001
0.0003
16
0.0023±0.0001
0.0003
Observation dim. dm
32
0.0025±0.0002
0.0003
64
0.0028±0.0003
0.0003
128
0.0031±0.0004
0.0003
2
0.0023±0.0001
0.0003
Appendix
Table 9. Scalability: server loss on the test window (mean ± s.d. over 3 seeds; base M=3 , pm=2 , dm=16 ) and wall-clock time of the Jacobian read-out per time step. The M rows keep the cross-client graph density fixed (Table 10 ).
M
Cross edges
Possible
Density
3
4
6
0.667
4
8
12
0.667
8
37
56
0.661
16
160
240
0.667
32
661
992
0.666
64
2688
4032
0.667
Appendix
Table 10. Realized adjacency statistics of the M sweep: the cross-edge density is 2/3 at every M (up to integer rounding of the edge count).
Figure 9. Server loss on the test window versus ε for the two communication directions (one curve per δ ), normalized per client and state dimension. Left: client-to-server latent-state perturbation. Right: server-to-client gradient perturbation. Dashed: non-private reference. Shaded bands: ±1 standard deviation over seeds.
Figure 10. Interpretability under privacy on the M=6 nonlinear SSM: cosine between the learned interdependency structure and the ground-truth interaction structure.
Granger causality identifies predictive dependencies in multivariate time series. In distributed settings where parties cannot share data, federated causal learning enables joint analysis. Most federated causal methods assume that clients observe the same features and infer causal relationships as point estimates, with little formal uncertainty quantification. These assumptions do not hold in many industrial systems, where clients observe different features, and the objective is to estimate cross-client dependencies (edges). These dependencies must be estimated indirectly through repeated client-server iterations. Uncertainty from client data and model parameters propagates through this process, making point estimates alone insufficient for assessing cross-client edges. This paper characterizes this uncertainty propagation and uses edge-specific variances to distinguish genuine cross-client dependencies from spurious estimated edges. We consider aleatoric uncertainty from client data variability and epistemic uncertainty from model parameters. We derive closed-form variance recursions and steady-state variances for the client-server iterations. We prove that the propagated contribution of the initial model-parameter uncertainty vanishes asymptotically. These variances enable statistically principled selection of cross-client edges. Synthetic experiments show that our approach improves cross-client edge recovery over competing baselines. On real-world industrial datasets, it achieves high root-cause identification accuracy while yielding interpretable dependency structures.
Spatiotemporal prediction on graph-structured data is central to traffic forecasting and environmental monitoring, yet decentralized and heterogeneous data complicate both sequence modeling and collaborative training. We propose F2STNet, a federated forecasting framework that combines truncated graph-Fourier features, a lightweight diagonal state-space temporal encoder, graph convolution, and Fairness-aware Federated Aggregation (FFA). The spectral branch exposes graph-frequency structure, while the state-space layer models long temporal dependencies with linear complexity in the sequence length. FFA adjusts the FedAvg prior using client validation losses and an increasing fairness schedule. Experiments on PeMS04, HZMetro, and KnowAir show favorable forecasting accuracy relative to the evaluated baselines; federated experiments on PeMS04 additionally improve worst-client and client-dispersion metrics.
Jiayi Zhang, Jinfeng Xu, Hewei Wang +7
University of Nottingham · The University of Hong Kong · Carnegie Mellon University +1
Subgraph pattern detection aims to uncover complex interaction structures in graphs. However, state-of-the-art graph neural network (GNN)-based solutions assume centralized access to the entire graph. When graphs are instead distributed across multiple parties, client-local GNN computations diverge from those of a centralized model, resulting in a representation-equivalence gap. We formalize this as a structural observability problem, where subgraph patterns crossing partition boundaries become locally unidentifiable. To bridge this gap, we propose a per-step, layer-wise embedding exchange framework in which clients synchronize intermediate node representations at each layer of the forward pass, without exposing raw features or labels. Under an extended-subgraph assumption and shared model parameters across clients, this framework recovers the same node representations as a centralized GNN over the full graph. Experiments on synthetic directed multigraphs with cycles, bicliques, and scatter-gather patterns show that embedding exchange and federated parameter aggregation are complementary rather than interchangeable: their combination recovers most of the representation gap, provided exchanged embeddings are fresh per-step rather than stale per-epoch.
Selin Ceydeli, Rui Wang, Kubilay Atasu
Delft University of Technology, Department of Software Technology Van Mourik Broekmanweg 6, 2628 XE Delft, The Netherlands