Avoiding Overfitting in Variable-Order Markov Models: a Cross-Validation Approach

TL;DR

This work addresses overfitting in higher-order Markov models by introducing DIVOP, a cross-validated framework that distinguishes informative higher-order paths from noise through $igtriangleup$ between $D_H$ and $D_L$ and path-variability metrics $V_{P_H}$ and $V_{P_L}$. A supervised HBGC classifier leverages these metrics to build sparse, reliable variable-order Markov chains, with synthetic data used to train thresholds. Across synthetic and real-world Orbis data, DIVOP achieves higher precision and recall than BuildHON+ and HYPA, and reveals that tax havens dominate significant higher-order dependencies, underscoring their impact on multinational corporate networks and enabling more trustworthy multi-step predictions. The approach combines cross-validation, informative-path detection, and mixed-order network construction to better capture hidden dynamics in complex sequences, with open-source data and code for broader applicability.

Abstract

Higher$\text{-}$order Markov chain models are widely used to represent agent transitions in dynamic systems, such as passengers in transport networks. They capture transitions in complex systems by considering not only the current state but also the path of previously visited states. For example, the likelihood of train passengers traveling from Paris (current state) to Rome could increase significantly if their journey originated in Italy (prior state). Although this approach provides a more faithful representation of the system than first$\text{-}$order models, we find that commonly used methods$-$relying on Kullback$\text{-}$Leibler divergence$-$frequently overfit the data, mistaking fluctuations for higher$\text{-}$order dependencies and undermining forecasts and resource allocation. Here, we introduce DIVOP (Detection of Informative Variable$\text{-}$Order Paths), an algorithm that employs cross$\text{-}$validation to robustly distinguish meaningful higher$\text{-}$order dependencies from noise. In both synthetic and real$\text{-}$world datasets, DIVOP outperforms two state$\text{-}$of$\text{-}$the$\text{-}$art algorithms by achieving higher precision, recall, and sparser representations of the underlying dynamics. When applied to global corporate ownership data, DIVOP reveals that tax havens appear in 82$\%$ of all significant higher$\text{-}$order dependencies, underscoring their outsized influence in corporate networks. By mitigating overfitting, DIVOP enables more reliable multi$\text{-}$step predictions and decision$\text{-}$making, paving the way toward deeper insights into the hidden structures that drive modern interconnected systems.

Figures (15)

• Figure 1: Summary of our DIVOP. (A) Sequence data representing transition between countries. (B) Example of the third-order path $(UK,IT,NL)$ and its counterpart second-order path $(IT,NL)$. The destinations of the path are the states reached after the Netherlands (NL). (C) Example division of the destinations of the path in validation and training sets using stratified cross-validation. (D) Calculation of the probability distribution of destinations. At each of the $4$-folds, a different 25% of the data is used for validation, with the rest used for training. The same procedure is applied to every path.
• Figure 2: Classifying paths as informative. Taking the example of the sequence data shown in Fig. \ref{['fig:division_lower']}, we illustrate the classification process of the path $(UK,IT,NL)$. Once we collect the destinations in the data of the paths $(UK,IT,NL)$ and $(IT,NL)$, we divide the patterns into $4$ folds (iterations). At every fold, we take a different 25% of the data as the validation set and the rest as the training set. At each iteration, we calculate the distributions in the validation and training set for both the lower and the higher-order. We use the distances of the distributions to calculate the variables $\Delta$, $V_{P_H}$ and $V_{P_L}$, as well as the empirical frequency of $P_H$ and $P_L$. We then calculate the mean and the standard deviation of the metrics on all the iterations. The values are then used as inputs of a classification algorithm to predict if the higher-order $(UK,IT,NL)$ is an informative path.
• Figure 3: Orbis real dataset DIVOP network: Representation of the reconstructed variable-order network through Xu et al. Xu2016 method. Nodes are the locations of firms in countries labeled with ISO-2 code. Higher-order nodes are informative paths and the ownership of firms goes from the left to the right, e.g. node (US,NL) represents the couples of firms in which the owner is located in the US, while its subsidiary is in The Netherlands. The edges represent ownership relations between nodes. Communities are obtained through modularity and highlighted through different colors. The used layout is ForceAtlas, Gephi.
• Figure 4: Partial dependence of (left) $\overline{\Delta}$ (right) $F_{P_H}$
• Figure 5: Partial dependence of (left) $\overline{V_{P_H}}$ (right) $\overline{V_{P_L}}$