SAIL: A Learnable Generalisation of Spatial Autocorrelation over Graph-Structured Data

A. Classical Spatial Autocorrelation Statistics

• Classical Moran’s I and Geary’s C quantify spatial autocorrelation for a predefined scalar variable using fixed, hand-crafted spatial weights.Spatial autocorrelation measures whether similar or dissimilar values tend to occur at neighbouring nodes.Source: §I · p. 1, A. Classical Spatial Autocorrelation… · p. 2

I. INTRODUCTION

• SAIL jointly learns feature projections and inter-node affinities to define an explicit, task-aware spatial autocorrelation operator for high-dimensional graph data.Its learned scalar fields provide interpretable combinations of features for measuring spatial structure.Source: §I · p. 1

1) Moran’s I: / 2) Geary’s C:

• Classical spatial autocorrelation statistics include Moran’s I for clustering or contrast, Geary’s C for local differences, Getis–Ord G⋆ for hotspots, and Morisita–Horn for neighbourhood overlap.Moran’s I values can indicate clustering or spatial contrast, while Geary’s C distinguishes similarity from dispersion.Source: 1) Moran’s I · p. 2, 2) Geary’s C · p. 2

B. Multivariate Extensions and Their Limitations

• Multivariate extensions often average per-feature statistics, while PCA-based approaches choose projections using variance or variance combined with Moran’s I rather than spatial coherence alone.A high-variance direction can outrank a weakly varying but highly coherent direction under MULTISPATI-PCA.Source: B. Multivariate Extensions and Their… · p. 3
• These approaches retain hand-crafted or fixed projections and predefined weights, limiting their ability to learn nonlinear feature combinations or identify spatial niches.They are also non-differentiable with respect to learned representations.Source: 2) Geary’s C · p. 2, B. Multivariate Extensions and Their… · p. 3

C. Graph Neural Networks

• Standard GNN message passing encodes spatial dependence implicitly, without directly measuring how strongly a representation is clustered or dispersed in space.GNNs also do not directly support significance testing of spatial structure without a supervised objective.Source: C. Graph Neural Networks · p. 3

D. Integration of Spatial Autocorrelation into Deep Learning

• Deep-learning approaches that incorporate spatial autocorrelation still use predefined indices, variables, and non-adaptive spatial weights, preventing discovery of task-relevant spatial modes.These approaches do not jointly adapt the feature combination and neighbourhood influence.Source: D. Integration of Spatial… · p. 3

III. METHODS

• SAIL makes feature representation and spatial weighting learnable, differentiable components of a modular spatial autocorrelation operator for graph data.The framework is intended for modalities represented as graphs, including images, spatial omics, point sets, and geospatial data.Source: §III · p. 3, A. Learning Multivariate Spatial… · p. 4

A. Learning Multivariate Spatial Autocorrelation with SAIL

• SAIL transforms each graph into local spatial-autocorrelation descriptors and can reduce node-level scores to graph-level descriptors.Its modular components include a feature mapping, affinity construction, local statistic, and order-invariant reduction.Source: A. Learning Multivariate Spatial… · p. 4

1) Input Feature Space Mapping:

• A learnable mapping projects node features into scalar fields, which are standardized so SAC scores reflect spatial coherence rather than differences in magnitude or variance.The mapping may be nonlinear, sparse, or orthogonalized to capture complex combinations, encourage compact feature sets, or promote diverse fields.Source: 1) Input Feature Space Mapping · pp. 4–5

2) Learnable Spatial Affinities:

• SAIL defines neighbourhood influence using spatial proximity and optionally learned feature relationships, with an optimizable parameter controlling their relative contribution.Feature-based affinity can be omitted to use purely spatial affinities.Source: 2) Learnable Spatial Affinities · p. 5

3) Local and Graph-Level SAIL Outputs:

• Local operators compare each projected field’s node values with affinity-weighted neighbours, and an order-invariant reduction produces graph-level descriptors.Local descriptors indicate where spatial structure occurs; global descriptors summarize coherence or heterogeneity for each field.Source: 3) Local and Graph-Level SAIL Outputs · p. 5
• Moran-style operators measure local agreement, Geary-style operators measure local dissimilarity, and alternative local operators support hotspot and neighbourhood-overlap statistics.For Moran-style scores, positive values indicate similar signed deviations and negative values indicate local contrast; for Geary-style scores, smaller values indicate greater local similarity.Source: 3) Local and Graph-Level SAIL Outputs · p. 5

B. Supervised Learning with SAIL

• SAIL descriptors can be used for supervised prediction in tasks where the target depends on spatial autocorrelation.The descriptors may be used at node or graph level and optimized end-to-end with a task-specific prediction loss.Source: B. Supervised Learning with SAIL · p. 5

C. Unsupervised Learning of Spatial Autocorrelation Modes

• Unsupervised SAIL Modes identify feature combinations with maximal spatial clustering, while Anti-Modes capture maximal spatial contrast.Jointly learning multiple directions can encourage diversity and sparsity for more interpretable feature combinations.Source: C. Unsupervised Learning of Spatial… · p. 6

D. Statistically Significant Spatial Hotspot Discovery

• A permutation test of Moran-style local scores, with multiple-testing correction, classifies significant nodes as hotspots, coldspots, or spatial outliers.The procedure holds affinities fixed while permuting field values across nodes; non-significant nodes remain unclassified.Source: D. Statistically Significant Spatial… · p. 6

Classical statistics as exact special cases

• With fixed choices of mapping, scoring, standardisation, and reduction, SAIL exactly recovers Moran’s I, Geary’s C, and standardised Getis–Ord G⋆_j for non-constant scalar fields under their stated conditions.The Getis–Ord recovery allows self-weights when the null variance is non-zero.Source: §IV · p. 6
• For non-negative multichannel fields, SAIL also recovers Morisita–Horn similarity between a node’s mapped representation and its affinity-weighted neighbourhood composition when local scoring acts jointly on channels.The local outputs are I_Moran(z; A), C_Geary(z; A), G⋆_j(z; A), and C_MH(u_j, v_j).Source: §IV · pp. 6–7

Inductive bias for SAC-driven learning

• Under the stated expressivity and loss assumptions, SAIL can approximate Bayes-optimal predictors for SAC-driven problems, with a high-probability risk bound for an approximate empirical-risk minimiser.The bound involves the hypothesis-class Rademacher complexity and the stated approximation and optimisation terms.Source: §IV · p. 7
• Finite T-hop linear graph filters with linear pooling and readout cannot generally represent quadratic SAC, while SAIL contains the required interaction; broader classes yield no smaller complexity-dependent bound solely through added expressivity.The no-larger excess-risk bound applies when SAIL and the broader class attain the same approximation and optimisation errors.Source: §IV · p. 7

B. Learning spatial autocorrelation beyond message passing

• r = 0.82 ± 0.01 for SAIL, the highest Pearson correlation across runs, versus r = 0.73 ± 0.02 for GCN on a complete graph.The improvement over all models is statistically significant (FDR-corrected p < 10−4).Source: Figure 3 · p. 8, B. Learning spatial autocorrelation… · p. 8

A. Recovering classical spatial autocorrelation with SAIL

• Moran’s I = −0.01 and Geary’s C = 1.12 for random values, versus Moran’s I = 0.99 and Geary’s C = 0.13 for an imposed gradient; LISA marks most random nodes non-significant and locates matching High–High and Low–Low clusters.These synthetic results empirically confirm recovery of classical SAC indices.Source: A. Recovering classical spatial… · p. 8, Figure 2 · p. 8

1) Datasets and Setup:

• On one 4i sample with 43 proteins across 270,876 locations, SAIL learns 20 Geary-objective modes and compares LISA hotspot masks with annotated compartments using Dice scores.Comparators include individual-protein Moran’s I, averaged univariate Moran’s I, MULTISPATI-PCA, and Hotspot.Source: 1) Datasets and Setup · p. 9

2) Results:

• In 4i, SAIL Modes capture distinct compartments and outperform univariate, mean, MULTISPATI-PCA, and Hotspot comparisons across compartments.Mode 8's Low–Low hotspots achieve Dice 0.84 with the nucleolus; Mode 1's achieve Dice 0.59 with Cell periphery 2.Source: 2) Results · p. 9, Table I · p. 9, Figure 4 · p. 10

1) Dataset and Setup:

• The cSCC analysis uses 140,633 cells from 11 responders and 10 non-responders, represented as patient graphs with cell-centroid coordinates and 1000-gene features.Separate tissue cores form separate k-nearest-neighbour components, with no edges between cores.Source: 1) Dataset and Setup · p. 10
• For cSCC, SAIL learns 20 modes without response labels across 28 biopsies from 21 patients, then tests per-patient scores against response.The unsupervised setup was chosen because a supervised model with only 21 patients and 1000 genes was considered highly prone to overfitting.Source: 1) Dataset and Setup · p. 10

2) Results:

• AUC = 0.86 for cSCC Mode 18, exceeding Favourable Niches and pre-treatment PD-L1; responders have higher Geary-style dissimilarity scores.The mode contrasts tumour–stromal programs with interferon and antigen-presentation genes, indicating sharper spatial juxtaposition in responders.Source: Table II · p. 9, Figure 5 · p. 10, 2) Results · p. 11

1) Datasets and Graph Construction:

• Survival analyses cover ORION-CRC cellular proteomics and TCGA-BRCA and TCGA-CRC whole-slide images; SAIL modes are learned without survival outcomes or clinical labels.Patients are split into high- and low-score groups by median SAIL score, then assessed using Kaplan–Meier analysis and survival metrics.Source: 1) Datasets and Graph Construction · p. 11

2) Results:

• C-Index = 0.74 for ORION Mode 17; it stratifies overall survival and exceeds the reported univariate, mean, MULTISPATI-PCA, and Hotspot baselines.The mode contrasts SMA and CD68 with CD163; high-risk patients show greater spatial intermixing of these signals.Source: 2) Results · p. 11, Table III · p. 12, Figure 6 · p. 12
• C-Index = 0.64 in TCGA-BRCA and 0.59 in TCGA-CRC; the best modes separate patients by disease-specific and overall survival, respectively.In BRCA, high SAIL scores are associated with worse prognosis; CRC Mode 17 gives the strongest overall-survival separation.Source: 2) Results · p. 12, Figure 6 · p. 12

E. Unsupervised SAC modes for prediction of patient survival

• Across cohorts, outcome is associated with spatial coherence, and ORION's signal is considerably weaker for individual features or their aggregate.The study concludes that outcome is linked to how molecular or morphological patterns are spatially organised across tissue.Source: 2) Results · pp. 11–12

1) Dataset and Setup:

• The SGH cohort comprised 234 tissue microarray samples from 90 patients, represented as cell graphs; evaluation compared SAIL with MesoGraph, PINS, and Max-MIL using the same graphs and features.The two feature sets were all 617 descriptors or 105 interpretable morphological, intensity, and texture descriptors; cross-validation held out one slide at a time.Source: 1) Dataset and Setup · p. 13

2) Results:

• In interpretable-feature models, Modes 11 and 12 had positive prediction weights toward non-epithelioid subtype, and high local Geary-style scores marked dissimilar neighbouring cells.The representative non-epithelioid sample showed scattered high-dissimilarity cells, whereas the epithelioid sample was largely uniform.Source: 2) Results · p. 13, Figure 7 · p. 14
• With all 617 features, SAIL achieved AUC= 0.902 ± 0.035 and AP= 0.849 ± 0.055; with 105 interpretable features, it achieved AUC= 0.866±0.042 and AP= 0.814 ± 0.040, ranking first in both settings.The all-feature AUC improvements over MesoGraph and PINS were significant (p = 0.011), as was the improvement over Max-MIL (p = 0.0002); with interpretable features, only the improvement over Max-MIL was significant (p = 0.0002).Source: Table IV · p. 13, 2) Results · p. 13

G. Additional Results

• On synthetic signals and a 3D point cloud, SAIL recovered expected positive or negative autocorrelation patterns; in simulated gas mixing, a species-identity mode score decayed faster when the partition was removed than through a small hole.The score also stayed constant when the partition was closed.Source: G. Additional Results · pp. 13–14
• An unsupervised mode contrasting stromal and immune cells stratified BRAF-mutant colorectal cancer patients by overall survival (log-rank p = 0.002, HR = 1.83) and outperformed univariate cell-type baselines..Source: G. Additional Results · p. 14
• SAIL discovered cross-country economic coherence modes evolving from 2000 to 2023 and recurring regional synchronisation of COVID-19 case rates, with hotspots concentrated in Western and Central Europe.Source: G. Additional Results · p. 14
• SAIL outperformed message-passing architectures when targets depended on SAC, but not when they depended on feature content; ablations found robustness to most choices, while strong decorrelation degraded performance.Geary’s C and Moran’s I consistently outperformed Getis–Ord G⋆, and a learnable spatial temperature helped when analysing many graphs jointly; sparse-kNN training costs were linear in node count, with the Stiefel constraint adding O(d^3).Source: G. Additional Results · pp. 13–14

VI. CONCLUSION

• Across synthetic and real-world datasets, SAIL made SAC explicit and interpretable, with unsupervised modes recovering biological niches and supervised descriptors alone discriminating mesothelioma subtype.The conclusion notes that future work will explore cross-channel coherence statistics.Source: §VI · p. 14

A. Spatial Omics Modalities; B. Tissue Structure and Spatial Niches

• Spatial omics pairs molecular readouts with their tissue locations, while a spatial niche is a region with characteristic molecular or cellular composition.Tissue function depends on cell types and their spatial arrangement.Source: A. Spatial Omics Modalities · p. 17, B. Tissue Structure and Spatial Niches · p. 17

C. Histology and Whole-Slide Images; D. Reference Annotations and Overlap (Dice)

• A whole-slide image is tiled into patches, each represented by features from a pretrained model; Dice measures overlap between a discovered mode and a reference region from 0 to 1.Source: C. Histology and Whole-Slide Images · p. 17, D. Reference Annotations and Overlap… · p. 17

E. Survival Analysis

• Survival analysis addresses censored outcomes using median-split groups, Kaplan–Meier curves, log-rank tests, C-index, hazard ratios, and restricted mean survival time.The C-index measures pairwise ordering of survival, while ∆RMST reports the difference in average event-free time over a fixed window.Source: E. Survival Analysis · p. 17

A. Getis–Ord G⋆

• Getis–Ord G⋆ is a local z-score whose positive values indicate potential hotspots and negative values indicate coldspots.Source: A. Getis–Ord G⋆ · p. 18

B. Morisita–Horn Index

• The Morisita–Horn index compares a node’s feature composition with its affinity-weighted neighbourhood composition, with higher values indicating similarity and lower values heterogeneity.Source: B. Morisita–Horn Index · p. 18

A. Linear Projection; B. Orthogonal Projection

• A linear projection learns scalar fields as weighted combinations of input features, while an orthogonal projection encourages independent spatial signals.Source: A. Linear Projection · p. 18, B. Orthogonal Projection · p. 18

C. Gumbel-Softmax Relaxation

• Gumbel-Softmax provides a differentiable relaxation of discrete concept assignments that approaches a one-hot vector as its temperature tends to zero.Source: C. Gumbel-Softmax Relaxation · p. 18

D. Concept-Based (One-Hot) Projection

• Each node is hard-assigned to one of m learned concepts by one-hot projection, with gradients flowing through H while the forward pass uses the discrete encoding.The projection uses a straight-through estimator.Source: D. Concept-Based (One-Hot) Projection · p. 18

APPENDIX D TRAINING ALGORITHMS

• Supervised task-loss training and unsupervised discovery of SAIL Modes both use the SAIL forward pass from the main-paper algorithm.Source: Appendix D · p. 18

APPENDIX E ADDITIONAL THEORETICAL ANALYSIS

• Under fixed operator choices, SAIL exactly recovers Moran’s I, Geary’s C, standardized Getis–Ord G⋆, and Morisita–Horn similarity.The stated cases use population standardisation for Moran’s I and Getis–Ord, sample standardisation for Geary’s C, and non-negative unstandardised channels for Morisita–Horn; the Getis–Ord case requires non-zero null variance.Source: Appendix E · pp. 19–20
• With sufficiently expressive components that implement the target SAC functional, SAIL can approximate Bayes-optimal predictors for SAC-driven problems.The stated finite-sample result assumes bounded Lipschitz loss and an approximate empirical-risk minimiser.Source: Appendix E · pp. 20–21
• Linear graph filters with linear pooling and readout cannot generally represent quadratic SAC targets, whereas SAIL explicitly contains the required second-order interaction.Source: Appendix E · pp. 21–22
• If SAIL and a broader class have equal approximation and optimisation errors on an SAC-driven problem, SAIL’s excess-risk bound is no larger.Greater generic expressivity alone does not reduce the Rademacher-complexity term in the standard bound.Source: Appendix E · pp. 21–22

• Each architecture is assessed at its best configuration under an identical optimisation budget, using Adam, MSE loss, validation-correlation early stopping, and 600/250/600 train-validation-test graphs.Source: Appendix F · p. 22
• Stage 1 selects hyperparameters on one sweep seed, while Stage 2 retrains each selected model across 1000 independent seeds.Source: Appendix F · p. 22
• Test Pearson correlation and MSE distributions are reported, with significance assessed by paired Wilcoxon tests and false-discovery-rate correction.Source: Appendix F · p. 22

• Each independently generated graph has 50–100 nodes with coordinates sampled independently from the two-dimensional unit square.Source: A. Dataset Construction · p. 22
• The target is constructed from a global spatial-autocorrelation functional of a hidden projected field on a hidden neighbourhood, with the projection and spatial scale varying across graphs.The generating neighbourhood is a symmetric 8-nearest-neighbour graph with distance-based RBF weights, and it is not exposed to models.Source: A. Dataset Construction · pp. 22–23
• Eight Gaussian-process feature channels use graph-specific lengthscales, are mixed through a rank-3 matrix, and are standardised across nodes.The lengthscale is sampled log-uniformly from 0.15 to 1.2.Source: A. Dataset Construction · p. 22

B. Neighbourhood Condition

• The synthetic comparison gives every model access to all node pairs, with distance-weighted edges or pairwise distances, so models must learn spatial weighting rather than receive the generating operator.SAIL learns spatial temperature τ.Source: B. Neighbourhood Condition · p. 23

C. Models and Architectural Choices

• Models share a node encoder, global mean pooling, and linear prediction head, but differ in graph-structure handling; SAIL uses a dense distance-based Moran-style index.Baseline architectures include single-layer GCN and GAT variants and an MLP control.Source: C. Models and Architectural Choices · p. 23

D. Two-Stage Selection and Evaluation

• A single-seed hyperparameter sweep selects each model configuration before evaluation on 1000 independent regenerated datasets, with test Pearson and MSE recorded.Source: D. Two-Stage Selection and Evaluation · p. 23

SPATIALLY COHERENT STRUCTURES IN SPATIAL OMICS

• On 4i imaging, SAIL’s unsupervised hotspot masks attain the highest Dice on 7 of 10 compartments and outperform four listed baselines across compartments.The reported paired-test p-values are 0.003 versus Univariate, 0.003 versus Mean, 0.014 versus MULTISPATI-PCA, and 0.010 versus Hotspot.Source: SPATIALLY COHERENT STRUCTURES IN… · p. 23

APPENDIX H ADDITIONAL DETAILS ON DISCOVERING SPATIAL COHERENCE BIOMARKERS OF TREATMENT RESPONSE

• For cSCC response discovery, SAIL learns 20 unsupervised Geary-style modes across patients and tests each mode’s subject-level score against responder status.The analysis uses baseline biopsies from 11 responders and 10 non-responders; each patient is represented by a graph with within-core kNN edges.Source: Appendix H · pp. 23–24

FOR PREDICTION OF PATIENT SURVIVAL

• The survival analyses use unsupervised modes and patient-level global SAIL scores; ORION-CRC uses 20 modes, while BRAF-CRC uses 30.The ORION setup lowers the sparsity penalty because its default produced single-marker modes; the BRAF-CRC mode uses one-hot cell-type features.Source: FOR PREDICTION OF PATIENT SURVIVAL · p. 24, A. ORION-CRC · p. 24, C. BRAF-Mutant Metastatic Colorectal… · p. 25

B. TCGA-BRCA and TCGA-CRC

• For TCGA-BRCA and TCGA-CRC, 20 modes are learned per analysis, and Supplementary Table IX reports each mode’s survival association.Both cohorts use patch graphs with kNN = 32.Source: B. TCGA-BRCA and TCGA-CRC · p. 25

C. BRAF-Mutant Metastatic Colorectal Cancer (BRAF-CRC)

• The best BRAF-CRC SAIL mode contrasts stromal with immune cells and stratifies overall survival (log-rank p = 0.002, HR = 1.83), exceeding univariate Geary’s C and mean aggregation.Per-image scores are averaged to produce one score per patient when patients contribute multiple images.Source: C. BRAF-Mutant Metastatic Colorectal… · p. 25

A. Cross-Validation Protocol

• Mesothelioma subtype evaluation uses leave-one-slide-out test folds, with the remaining cores split 75%/25% into training and validation sets.The split is stratified by core-level label, and all models share the same fixed splits.Source: A. Cross-Validation Protocol · p. 26

B. Model Architecture

• The mesothelioma model evaluates one shared SAIL descriptor per core and feeds it to separate epithelioid and non-epithelioid heads trained jointly with pairwise margin ranking.The final score is the non-epithelioid head output minus the epithelioid head output.Source: B. Model Architecture · p. 26, C. Training and Model Selection · p. 26

B. Benchmark Design

• The benchmark varies node count, feature dimension, mode count, and neighbourhood size from a shared synthetic-graph reference configuration, measuring time and GPU memory across model variants.Measurements use five repeats of ten timed training steps after three warm-ups; the hardware is one NVIDIA A100-80GB GPU.Source: A. Empirical Computational Complexity… · p. 27, B. Benchmark Design · pp. 27–28, C. Measurement Protocol · p. 28

D. Results

• 27 GB of peak memory is reached at n = 3 × 10^6; the orthogonal map’s matrix exponential scales approximately cubically with d, while learned affinities add substantial memory cost.At d = 10,000, the orthogonal map takes 30 s per step versus 13 ms for the linear map; learned affinity uses 13.0 versus 0.9 GB at the reference size.Source: D. Results · p. 28

WITH SAIL IN ADDITIONAL SYNTHETIC DATASETS

• Synthetic examples span positive, negative, edge-localized, and alternating local spatial dependence, plus a 3D Perlin terrain point set without explicit edges.The examples include a checkerboard with strongly negative expected Moran’s I and concentric rings alternating locally positive and negative dependence.Source: Appendix L · pp. 28–29

A. 4i Spatial Proteomics

• On 4i, strong decorrelation reduces mean Dice to 0.36 at λcorr = 10 and 0.32 at λcorr = 100, while sparsity up to λsparse = 1 leaves it at 0.44.Unconstrained projections score 0.47 versus 0.44 for Stiefel projections, but can yield redundant modes; learned feature affinities reduce Dice to 0.33 or 0.36.Source: A. 4i Spatial Proteomics · p. 29

B. cSCC PD-1/PD-L1 Blockade Response

• For cSCC response, larger neighbourhoods improve AUC from 0.79 at kNN = 8 to 0.91 at kNN = 64; fixing τ = 1 lowers AUC from 0.86 to 0.79.Strong decorrelation, learned feature affinity, and sum reduction also produce notable drops; mean and max reduction both yield 0.86.Source: B. cSCC PD-1/PD-L1 Blockade Response · pp. 29–30

C. TCGA-BRCA Survival

• In TCGA-BRCA, the default mode has C-Index 0.64, while removing decorrelation or using max reduction reaches 0.67; strong decorrelation removes significant separation.In TCGA-CRC, the default C-Index is 0.60, and learned affinities or smaller neighbourhoods reach 0.63; fixing τ = 1 has little effect.Source: C. TCGA-BRCA Survival · p. 30, D. TCGA-CRC Survival · p. 30

E. Summary

• Across cohorts, strong decorrelation degrades performance, Geary’s C and Moran’s I perform similarly, and the effect of learned affinity depends on the dataset.Learning τ helps on cSCC and TCGA-BRCA but has little effect on 4i and TCGA-CRC.Source: E. Summary · p. 30

TEMPORAL GEOSPATIAL DATA

• The macroeconomic analysis builds annual country graphs from 145 countries and 25 standardized indicators selected from a cleaned 2000–2023 panel.The retained feature set excludes non-comparable local-currency levels and other specified variables; graph edges use six nearest neighbours by great-circle distance.Source: A. Data Source and Cleaning · pp. 30–31, B. Feature Set · p. 31, C. Graph Construction · p. 31

D. Training Configuration

• One shared set of 8 modes is trained jointly across yearly country graphs with a Geary-style objective, decorrelation and sparsity regularization, and is tracked through global scores, projected fields, and LISA assignments.Training uses Adam at learning rate 0.05 for 1000 epochs.Source: D. Training Configuration · pp. 31–32

E. Results

• SAIL discovers 8 sparse economic-coherence modes built from recurring trade, government-finance, and economic-size indicators, often capturing contrasts between related quantities.Across 2000–2023, some mode scores remain stable, while others change gradually or show short-lived shifts coinciding with the global financial crisis or COVID-19 pandemic.Source: E. Results · p. 32

A. Data Source and Preprocessing

• For COVID-19 analysis, daily new confirmed cases per million are aggregated to weekly country means from January 2020 to May 2023, then represented on country-centroid kNN graphs with kNN = 16.Weeks with missing values for matched countries are dropped.Source: A. Data Source and Preprocessing · p. 32, B. Graph Construction · p. 32

C. Results

• Weekly COVID-19 Moran’s I varies substantially, with peaks indicating stronger synchronization in neighbouring countries and recurring High–High hotspots in Western and Central Europe across multiple waves.The pattern suggests regionally synchronized outbreaks rather than isolated events.Source: C. Results · pp. 32–33

A. Dataset Setup

• The gas-mixing simulations compare a closed partition, a 120-unit hole, and a removed partition across 5 seeds, with a separate partition-removed run reserved for SAIL training.The simulated system contains 600 equal-mass particles in a 1000×540 box.Source: A. Dataset Setup · p. 33

B. Graph Construction and SAIL Model

• For gas mixing, particle graphs use kNN = 8 edges and features for species identity, speed, and four nuisance variables; unsupervised SAIL learns one Moran-style mode with a learnable spatial temperature and no learned feature affinity.The trained mode is frozen and applied to evaluation snapshots.Source: B. Graph Construction and SAIL Model · p. 33

C. Results

• With the partition removed, the coherence score falls from 0.999 at t = 0 to 0.04 ± 0.03 at t = 80 s; with a small hole it is 0.21 ± 0.01, while the closed-partition control remains at 0.982 ± 0.005.The learned mode loads almost entirely on species identity (1.00), and LISA maps localize remaining clusters as mixing proceeds.Source: C. Results · p. 33

B. Results

• At γ = 0, SAIL achieves the highest Pearson correlation (0.831 ± 0.009); at γ = 1, MLP (0.998±0.001) and GAT (0.993±0.004) recover the feature-content target, with a model crossover near γ ≈0.3–0.5.SAIL’s advantage is confined to targets governed by spatial autocorrelation and is not expected when the target depends on feature content alone.Source: Appendix Q · p. 33, Table XVIII · p. 34, B. Results · p. 34