We examine whether network representations of financial series produce interpretable predictive signals. Daily S&P 500 prices are mapped to Natural Visibility Graph (NVG), from which we extract multi-scale topological and spectral descriptors using overlapping windows of 100, 250, and 500 trading days. These features drive Random Forest (RF) models for two 7-day-ahead tasks: (i) directional classification (up/down) and (ii) magnitude regression of standardized forward returns, with training and evaluation conducted in temporal order. RFs are used for their robustness to heterogeneous inputs and their built-in mean decrease in impurity (MDI), enabling direct ranking of NVG features by contribution to performance. Out-of-sample, the classifier attains ROC-AUC = 0.62 and accuracy ≈ 0.584 on balanced classes statistically meaningful yet economically modest. Beyond point accuracy, the approach yields transparent importance profiles that identify which NVG attributes are most informative for short-horizon forecasts. Overall, the evidence indicates that VG features provide complementary, structure-aware information for stock-index prediction while preserving interpretability through RF-based importance analysis.
Financial indexes like the S&P 500 display rich, nonlinear behavior that challenges conventional
forecasting. A growing line of work uses complex network theory to represent time series as graphs,
enabling structural analysis across scales [
The rationale for network-based forecasting is to expose structural information that time-domain
methods can miss. Measures such as average degree, clustering, spectral radius, and path-length
phenomena like volatility clustering and
cyclicity [
At the same time, interpretability has become central in ML especially in finance, where
understanding model rationale is critical. Black-box models (e.g., deep networks) can obscure
decision drivers and raise risk concerns [
Motivated by these developments, we study RFs as interpretable rankers of VG-derived features
for financial forecasting. Using the S&P 500, we construct VGs on sliding windows of 100, 250, and
500 trading days to capture evolving structure [
In summary, we integrate complex network representations with interpretable ML to advance
financial time-series modeling. Representing an index as a VG yields a spectrum of structural features
[
2.1.
VGs map a time series into a network by linking samples that satisfy a line-of-sight criterion,
preserving salient geometric dynamics of the original signal [
Complementary to VGs, correlation-based market networks (and their minimum spanning trees,
MSTs) reveal hierarchical structures (e.g., sectoral clustering) and trace regime shifts: during crises
networks densify and lose modularity; in tranquil phases they are sparser and more fragmented [14
18]. Such topology shifts support regime detection and systemic-risk analysis, while spillover and
co-movement graphs help model contagion across markets [
Beyond description, forecasting the network itself (e.g., correlation-link addition/removal) via ML
with node/edge features improves predictive accuracy over raw correlations and enhances portfolio
rebalancing and risk control [
Recent work leverages graph neural networks (GNNs) to jointly learn temporal and cross-sectional dependencies. Spatio-temporal GAT variants (e.g., FSTGAT) have outperformed LSTM/XGBoost baselines, particularly in volatile regimes, and can anticipate turning points by learning dynamic inter-asset relations though interpretability remains a key concern prompting research on explainable GNNs [24 29].
Recent work integrates VG features ML models for time-series forecasting and classification,
leveraging network metrics such as degree, path length, and centrality as informative
representations of temporal patterns [
Among ML approaches, RFs are especially well-suited for VG-based forecasting due to their
flexibility and built-in feature importance metrics. Studies by Singh et al. and Bocaccio et al. have
demonstrated improved accuracy using RFs with VG inputs in contexts ranging from financial time series
to audio signal classification [
Network-based methods from correlation graphs to GNNs have proven effective in capturing
complex dependencies in financial data that traditional models often miss [
We extend this framework by pairing VG-derived features with RF, emphasizing interpretability alongside predictive performance. Unlike black-box models such as GNNs, RFs provide transparent insights via feature importance scores, supporting explainable AI in finance. Our approach bridges descriptive VG analysis and interpretable forecasting, showing how structural features of price series can improve predictions while revealing which patterns matter most. By doing so, we advance structure-aware modeling with a method that is both rigorous and accessible for practical decision-making.
3.1.
We generate VGs from time series data using the natural VG (NVG) algorithm, following the
approach introduced in the original study [
This condition ensures the straight line connecting points ( , ) and ( , ) lies above all intermediate points ( , ), establishing visibility. Applying this rule, we construct an undirected VG = ( , ), where | | = nodes correspond to the time series length. Due to the nature of the construction, each VG is fully connected across consecutive time steps. The resulting binary, symmetric adjacency matrix encodes the structural profile of the time series as a complex network. All further network-based analysis is conducted on these graphs. 3.2.
Once the VGs are constructed, we compute a suite of topological and spectral metrics to quantify their structural properties.
Clustering Metrics: Global Clustering Coefficient ( ) captures the likelihood of triangle formation by averaging local clustering over all nodes: = −1 ∑ =1 2 ⁄ ( − 1) where is the number of connections among node Transitivity ( ) measures the ratio of closed triplets to all connected triplets, indicating overall triangle density. Square Clustering evaluates the frequency of 4-node cycles, reflecting square-like substructures in the graph.
Efficiency and Path Length: Global Efficiency is the average inverse shortest-path length across all node pairs, indicating network-wide navigability. Local Efficiency captures the effi-world networks and low average path length, combining local clustering with global contypically exhibit high nectivity.
Assortativity ( ): Measures the correlation of node attributes (e.g., degree) at both ends of an edge. Positive implies similar nodes connect; negative implies dissimilarity.
Centrality and Hubs: Maximum Degree indicates the most connected node, suggesting dominant time points. Betweenness Centrality quantifies how often a node lies on shortest paths,
Small-Worldness ( ): Defined as ( ⁄ )⁄( ⁄ ), where and are clustering and path length in a random graph. > 1 indicates small-world structure high local clustering with short global paths.
Spectral Measures: Graph Index Complexity ( ) uses the spectral radius of the adjacency matrix to capture structural complexity: = 1 − (2 − 1)2, where is the normalized spectral radius. peaks for intermediate connectivity, distinguishing graphs that are neither too sparse nor too dense. Algebraic Connectivity 2 (second-smallest Laplacian eigenvalue) reflects overall network cohesion and robustness. Adjacency Spectral Gap = 1 − 2 assesses the dominance of the leading eigenmode; a large gap suggests integration, while a small gap indicates community structure.
Together, these metrics characterize both local motifs and global architecture of the VG, offering a multi3.3.
RF are ensemble models that combine multiple decision trees to enhance prediction accuracy and
reduce overfitting. In classification, each tree votes, and the majority class is chosen; in regression,
predictions are averaged [
Each tree is built using recursive binary splits, selecting the feature and threshold that maximize impurity reduction (e.g., Gini index or variance). Splitting continues until a stopping criterion is met (e.g., max depth or minimum node size). This results in diverse trees that capture complex patterns while ensemble averaging controls overfitting.
Key RF hyperparameters include: • • • • • n_estimators: Number of trees (e.g., 100 500). More trees generally reduce variance, with diminishing returns beyond a point. max_features: Number of features considered at each split. Smaller values increase digression. max_depth: Maximum tree depth. Fully grown trees (no limit) are common, but limiting depth can reduce complexity and overfitting. min_samples_split/leaf: Minimum number of samples to split or form a leaf, used to regularize overly deep trees. bootstrap: Whether to use bootstrapped samples. When enabled (default), it increases diversity and allows for out-of-bag error estimation.
Additional parameters like max_leaf_nodes or n_jobs aid in controlling tree size or parallelizing
training. In this study, RF is implemented using Scikit-learn, and hyperparameters are tuned
empirically via validation or randomized search [
RFs provide embedded feature importance via the MDI
often called Gini importance when using
the Gini index [
Imp( ) = 1 ∑
∑ =1 ∈ : ( )= Δ ( ) .
Importances are non-negative and typically normalized to sum to 1. MDI is fast and useful for
ranking predictors and for feature selection; averaging across many randomized trees also stabilizes
estimates [
MDI is computed on training splits and can overstate importance; validate selections with
crossvalidation. It also favors high-cardinality/continuous features [
We address long-horizon nonstationarity with an overlapping sliding-window scheme. For daily S&P 500 prices and window lengths
∈ {100, 250, 500}, each segment , = { − +1, … , } for = , … , is mapped via the natural-visibility rule to a VG , . From each , we compute a descriptor vector ( , ) ∈ ℝ ; concatenating scales yields a multi-scale feature stream Τ
Τ = [ ( ,100) , ( ,250) , ( ,500) ] .
Τ Τ
Overlapping windows smooth feature evolution and enable high-resolution tracking of structural change. Crucially, features at time use only , (no look-ahead). The sample spans 23 Dec 1981 21 Aug 2025.
Targets use a 7-day horizon ℎ = 7. The forward return is standardized with a trailing 50-day window:
,ℎ = ( +ℎ − )⁄ . = 1 ∑ −1 50 = −50 ,ℎ, = √
2 1 ∑ −1 49 = −50( ,ℎ − ) , ,ℎ = ( ,ℎ − )⁄ .
We use ,ℎ for regression and ,ℎ = sign( ,ℎ) for classification. Standardization mitigates drift/volatility shifts and helps balance Up/Down classes. 3.6.
We tune hyperparameters ∈ Θ via randomized search, which explores high-impact ranges efficiently without the combinatorial cost of grid search. Evaluation is time-aware: a 5-split purged, expanding cross-validation with an embargo = 500 days (equal to the maximum feature window
= 500 prevents leakage from overlapping windows.
Let 1 < 1 < ⋯ < < . For split : train = [1, ], embargo = ( , + ], test = ( + , +1]. For each candidate ( ) and split , we fit on the train set and score on the test set using the task-appropriate metric. The cross-validated objective is and we select ∗ = argmax ∈Θ ̂( ) (or argmin, depending on the metric).
This procedure emulates real-time deployment and strictly enforces temporal separation. We run it independently per model class (e.g., RF with varied , max_depth, max_features, etc.). 3.7.
We use time-ordered samples {( , ,ℎ)} =− ℎ0, where ∈ ℝ are VG descriptors and ,ℎ is the ℎday standardized return (ℎ = 7). For direction, labels are ,ℎ = 1{ ,ℎ > 0} ∈ {+1, −1}.
Modeling & tuning. Regression fits ,ℎ ≈ ( ); we tune hyperparameters via purged, forward-chaining CV by minimizing the median absolute error (MedAE) across splits, then refit on the full training range. For classification, we maximize mean ROC AUC across splits.
Error metrics (regression), with = − ̂ : • • • • • • •
MAE = −1 ∑
=1| |; MSE = −1 ∑
2 =1 ;
RMSE = √MSE; • 2 = 1 − ∑ =1( − ̂ )2⁄∑
=1( − ̅)2.
Classification metrics (TP, TN, FP, FN):
ACC = (TP + TN)⁄(TP + TN + FP + FN); Prec = TP⁄(TP + FP); Rec = TP⁄(TP + FN);
F1 = (2 Prec⋅Rec)⁄(Prec + Rec).
Report per-class scores and summarize by macro average −1 ∑ Metric or weighted average ∑ ( ⁄ )Metric .
This section evaluates the short-horizon predictive content of VG features for the S&P 500. We consider two 7-day tasks: (i) directional classification (Up/Down) and (ii) magnitude regression of standardized forward returns. Models are trained and tested in temporal order using a rolling-origin setup with non-overlapping test blocks; all results are out-of-sample. Unless noted, the classifier uses a 0.5 decision threshold, and regression is summarized by median/mean errors.
The classifier shows moderate ranking ability: AUC = 0.62, i.e., a random Up case outranks a Down case ~62% of the time. Class-wise scores are tightly aligned Up: Precision = 0.5842, Recall = 0.5857, F1-score = 0.5850; Down: Precision = 0.5836, Recall = 0.5822, F1-score = 0.5829 indicating no material class bias at the default threshold and roughly symmetric type-I/II errors. Because the test set is essentially balanced, Accuracy = 0.5839, Macro = 0.5839, and Weighted = 0.5839 coincide. Interpreted probabilistically, the model is correct ~58.4% of the time better than chance (50%) but economically modest without further tuning.
Performance could likely improve with (i) threshold optimization, (ii) probability calibration, and (iii) cost-sensitive training when false-positive/false-negative costs differ. Overall, the results reflect a balanced, threshold-dependent signal: the model captures useful structure, but converting ranking skill into higher decision accuracy requires careful operating-point selection and/or feature/model refinements.
Figure 1 demonstrates top-20 impurity-based feature importances for RF classifier.
Predictive weight is concentrated in meso-scale VG measures. Features from the 250-day window lead, 500-day contribute secondarily, and 100-day add little for a 1-week horizon. Node-centric extremes (DegreeMax) and path brokerage (GlobalBetweennessCentrality) rank near the tail, indicating reliance on global/topological organization rather than local hubs.
High efficiency (short paths) and transitivity (triadic closure) signal globally navigable yet locally cohesive structures that precede directional moves. Repeated GIC entries suggest that distance from path-like or clique-like extremes is systematically informative (or that correlated proxies capture the same regime).
MDI is relative and sensitive to feature correlation; importance can disperse across similar features and provides no direction of effect.
Features summarizing global navigability, local cohesion, and intermediate connectivity are most promising for week-ahead direction. RMSE = 1.1198. The negative 2 indicates the RandomForestRegressor underperforms a mean-only baseline. Because targets are standardized, an RMSE clearly above 1 means the model does worse than a naive zero forecast. Overall, the forest captures broad, low-frequency patterns while shrinking predictions toward zero and underestimating extremes consistent with VG features being more useful for directional ranking than for accurate magnitude prediction at a 1-week horizon. never enter the top-20, implying the regressor relies on distributed structure, not isolated hubs or single bottlenecks, to forecast week-ahead magnitudes.
Features built on the 250-day window dominate; 500-day GICs add secondary signal, while 100day metrics are largely absent. This aligns with a one-year context offering the best bias variance trade-off for a 7-day horizon
short windows are noisy, very long ones dilute regime information.
The few 500-day entries suggest a slow background state still matters after medium-term structure is captured. −
The mix of contemporaneous ( ) and short-lag ( − 1 … − 7) versions of the same measures indicates persistence over roughly a trading week and gives the forest multiple, near-collinear split options around the forecast origin helpful against small timing jitters.
Together, transitivity (clustering) and global efficiency (short average paths) emphasize regimes that are locally cohesive yet globally navigable VG configurations that empirically co-move with next-week return magnitude more than node-local centralities do. Repeated GIC appearances (maximal at intermediate connectivity) further suggest that distance from trivial structures (path-like vs. clique-like) is systematically tied to return scale; the breadth of GIC lags signals either a robust link or correlated proxies of the same latent regime.
Overall, the figure indicates that medium-horizon, meso-scale organization captured by clustering, efficiency, and spectral complexity carries the main explanatory power for week-ahead return magnitudes, while purely local centralities are secondary. This is structurally consistent with the VG framework and guides feature engineering and robustness checks.
This study offers useful insights but has notable constraints. First, the mutual-information prefilter may miss higher-order feature interactions, biasing selection. Second, we evaluated only one learner (RF), so we cannot judge how alternative models would capture patterns or reorder feature importance. Third, results may depend on the VG window length: we tested a few fixed sizes (100/250/500 days) and found the strongest signal near ~250 days, while shorter windows were noisier. Likewise, we fixed the forecast horizon at 7 trading days and targets to either a binary direction or a single 7-day return, leaving other horizons and targets unexplored.
Our evidence is also dataset-specific: we used daily S&P 500 closes only. Other indices, single stocks, and non-equity assets (bonds, commodities, FX, crypto), as well as different sampling frequencies (intraday or lower-frequency macro series), may exhibit different VG behavior; thus generalization remains an open question.
Future work should enrich the feature set with additional complexity and nonlinear descriptors
e.g., non-extensive statistics [
MDI is only one importance metric and is known to be biased toward high-cardinality features. Follow-up studies should pair it with permutation importance or SHAP, and consider wrapper or regularization-based selection (e.g., LASSO, embedded tree-ensemble methods) to validate which features are truly predictive.
Finally, broaden the task design: test multiple horizons (1-day to monthly, multi-horizon setups) and alternative targets (magnitude buckets, volatility/drawdowns, multi-output objectives). Extending across assets, markets, and frequencies including intraday data will determine how robust VG-based features are and where domain-specific adaptations are needed.
We pair complex network analysis with an interpretable ML approach to enhance financial timeseries forecasting. Converting the S&P 500 into VG and extracting diverse topological and spectral as an interpretability tool, we identify which network features most influence short-horizon outcomes. Our goal is insight rather than state-of-the- -in importance measures make the drivers transparent. The evidence shows that combining network-based features with an interpretable ensemble offers fresh perspective on market dynamics.
The most informative predictors are meso-scale connectivity and clustering. In both classification and regression, global efficiency and transitivity computed over roughly one year consistently dominate, indicating that highly navigable, locally clustered VGs tend to precede sizable index moves. A spectral complexity score ( ) also ranks highly, suggesting that distance from trivial structures (chains or cliques) carries signal. By contrast, node-local metrics (e.g., maximum degree, betweenness) contribute little, implying that distributed structural patterns not isolated extremes primarily guide forecasts.
Empirically, the RF achieves about 58% weekly direction accuracy (vs. a 50% baseline) and the regression model tracks low-frequency drift while underestimating extremes consistent with our emphasis on understanding rather than optimizing raw performance. Crucially, the importance profiles clarify why the model forecasts as it does, a key requirement in finance.
Overall, we show that graph-derived measures of connectivity, clustering, and complexity can act as leading indicators of short-term behavior and complement traditional signals. The framework is general: it can be applied to other indices or non-financial series, or combined with more sophisticated learners while retaining interpretability. Uniting complex network science with ML thus offers a path to models that are both data-driven and structurally explainable.
The article was prepared as part of the state-funded research project mation in the post(State Registration No. 0125U000541), conducted at Kyiv National Economic University named after Vadym Hetman.
Moreover, the authors express their profound gratitude to the Armed Forces of Ukraine, whose service and sacrifice made it possible to carry out this research during wartime.
During the preparation of this work, the authors used GPT-5 for grammar and spelling assistance. After using this tool, they reviewed and edited the content as needed and take full responsibility for the content of the publication.