Rényi transfer entropy (RTE) is a generalization of classical transfer entropy that replaces Shannon's entropy with Rényi's information measure. This, in turn, introduces a new tunable parameter $α$, which accounts for sensitivity to low- or high-probability events. Although RTE shows strong potential for analyzing causal relations in complex, non-Gaussian systems, its practical use is limited, primarily due to challenges related to its accurate estimation and interpretation. These difficulties are especially pronounced when working with finite, high-dimensional, or heterogeneous datasets. In this paper, we systematically study the performance of a k-nearest neighbor estimator for both Rényi entropy (RE) and RTE using various synthetic data sets with clear cause-and-effect relationships inherent to their construction. We test the estimator across a broad range of parameters, including sample size, dimensionality, memory length, and Rényi order $α$. In particular, we apply the estimator to a set of simulated processes with increasing structural complexity, ranging from linear dynamics to nonlinear systems with multi-source couplings. To address interpretational challenges arising from potentially negative RE and RTE values, we introduce three reliability conditions and formulate practical guidelines for tuning the estimator parameters. We show that when the reliability conditions are met and the parameters are calibrated accordingly, the resulting effective RTE estimates accurately capture directional information flow across a broad range of scenarios. Results obtained show that the explanatory power of RTE depends sensitively on the choice of the Rényi parameter $α$. This highlights the usefulness of the RTE framework for identifying the drivers of extreme behavior in complex systems.

Transfer entropy measures directed information flow in time series, and it has become a fundamental quantity in applications spanning neuroscience, finance, and complex systems analysis. However, existing estimation methods suffer from the curse of dimensionality, require restrictive distributional assumptions, or need exponentially large datasets for reliable convergence. We address these limitations in the literature by proposing TENDE (Transfer Entropy Neural Diffusion Estimation), a novel approach that leverages score-based diffusion models to estimate transfer entropy through conditional mutual information. By learning score functions of the relevant conditional distributions, TENDE provides flexible, scalable estimation while making minimal assumptions about the underlying data-generating process. We demonstrate superior accuracy and robustness compared to existing neural estimators and other state-of-the-art approaches across synthetic benchmarks and real data.

We consider the problem of reconstructing coupled networks (e.g., biological

Causal inference is crucial for humans to explore the world, which can be modeled to enable an agent to efficiently explore the environment in reinforcement learning. Existing research indicates that establishing the causality between action and state transition will enhance an agent to reason how a policy affects its future trajectory, thereby promoting directed exploration. However, it is challenging to measure the causality due to its intractability in the vast state-action space of complex scenarios. In this paper, we propose a novel Goal Discovery with Causal Capacity (GDCC) framework for efficient environment exploration. Specifically, we first derive a measurement of causality in state space, \emph{i.e.,} causal capacity, which represents the highest influence of an agent's behavior on future trajectories. After that, we present a Monte Carlo based method to identify critical points in discrete state space and further optimize this method for continuous high-dimensional environments. Those critical points are used to uncover where the agent makes important decisions in the environment, which are then regarded as our subgoals to guide the agent to make exploration more purposefully and efficiently. Empirical results from multi-objective tasks demonstrate that states with high causal capacity align with our expected subgoals, and our GDCC achieves significant success rate improvements compared to baselines.

Understanding causal relationships in time series is fundamental to many domains, including neuroscience, economics, and behavioral science. Granger causality is one of the well-known techniques for inferring causality in time series. Typically, Granger causality frameworks have a strong fix-lag assumption between cause and effect, which is often unrealistic in complex systems. While recent work on variable-lag Granger causality (VLGC) addresses this limitation by allowing a cause to influence an effect with different time lags at each time point, it fails to account for the fact that causal interactions may vary not only in time delay but also across frequency bands. For example, in brain signals, alpha-band activity may influence another region with a shorter delay than slower delta-band oscillations. In this work, we formalize Multi-Band V ariable-Lag Granger Causality (MB-VLGC) and propose a novel framework that generalizes traditional VLGC by explicitly modeling frequency-dependent causal delays. We provide a formal definition of MB-VLGC, demonstrate its theoretical soundness, and propose an efficient inference pipeline. Extensive experiments across multiple domains demonstrate that our framework significantly outperforms existing methods on both synthetic and real-world datasets, confirming its broad applicability to any type of time series data. Code and datasets are publicly available.

Purpose: This study introduces a novel framework for identifying and exploiting predictive lead-lag relationships in financial markets. We propose an integrated approach that combines advanced statistical methodologies with machine learning models to enhance the identification and exploitation of predictive relationships between equities. Methods: We employed a Gaussian Mixture Model (GMM) to cluster nine prominent stocks based on their mid-range historical volatility profiles over a three-year period. From the resulting clusters, we constructed a multi-stage causal inference pipeline, incorporating the Granger Causality Test (GCT), a customised Peter-Clark Momentary Conditional Independence (PCMCI) test, and Effective Transfer Entropy (ETE) to identify robust, predictive linkages. Subsequently, Dynamic Time Warping (DTW) and a K-Nearest Neighbours (KNN) classifier were utilised to determine the optimal time lag for trade execution. The resulting strategy was rigorously backtested. Results: The proposed volatility-based trading strategy, tested from 8 June 2023 to 12 August 2023, demonstrated substantial efficacy. The portfolio yielded a total return of 15.38%, significantly outperforming the 10.39% return of a comparative Buy-and-Hold strategy. Key performance metrics, including a Sharpe Ratio up to 2.17 and a win rate up to 100% for certain pairs, confirmed the strategy's viability. Conclusion: This research contributes a systematic and robust methodology for identifying profitable trading opportunities derived from volatility-based causal relationships. The findings have significant implications for both academic research in financial modelling and the practical application of algorithmic trading, offering a structured approach to developing resilient, data-driven strategies.

Understanding leadership dynamics in collective behavior is a key challenge in animal ecology, swarm robotics, and intelligent transportation. Traditional information-theoretic approaches, including Transfer Entropy (TE) and Time-Lagged Mutual Information (TLMI), have been widely used to infer leader-follower relationships but face critical limitations in noisy or short-duration datasets due to their reliance on robust probability estimations. This study proposes a method based on dynamic network inference using time-lagged correlations across multiple kinematic variables: velocity, acceleration, and direction. Our approach constructs directed influence graphs over time, enabling the identification of leadership patterns without the need for large volumes of data or parameter-sensitive discretization. We validate our method through two multi-agent simulations in NetLogo: a modified Vicsek model with informed leaders and a predator-prey model featuring coordinated and independent wolf groups. Experimental results demonstrate that the network-based method outperforms TE and TLMI in scenarios with limited spatiotemporal observations, ranking true leaders at the top of influence metrics more consistently than TE and TLMI.

The field of neuroscience has been revolutionized by the advent of brain imaging technologies, particularly functional magnetic resonance imaging in the resting state (rest fMRI) (Khalilullah et al, 2023; Vasilkovska et al, 2023; Liu et al, 2024). This powerful tool allows the measurement of blood-oxygen-level-dependent (BOLD) signals in predefined Regions of Interest (ROIs) within the brain, offering an unprecedented avenue for revealing information about potential diseases such as autism spectrum disorder (ASD) and schizophrenia (Philiastides et al, 2021; Kocak, 2021). Various brain atlases, including Harvard-Oxford (Makris et al, 2006) and Craddock 200 (Craddock et al, 2012) parcellations, have been used to define these ROIs. Furthermore, ROIs can be interestingly modelled as graph data, where the ROIs themselves represent nodes, and the connections between ROIs represent edges of graphs (Cui et al, 2022b). This graph-based data structure, inheriting the graph theory technique, has been instrumental in revealing meaningful relationships between ROIs in brain networks to diagnose brain diseases more effectively (Alsubaie et al, 2024; Ren and Xia, 2024). With the current popularity of deep learning, recent frameworks have developed graph neural networks (GNNs) (Xia et al, 2021; Febrinanto et al, 2023c) to extend the merits of modelling graph-structured data for detecting brain diseases with brain networks based on fMRI signals as input (Kan et al, 2022b; Li et al, 2021; Kan et al, 2022a; Cui et al, 2022a; ElGazzar et al, 2022; Febrinanto et al, 2023a). These techniques perform more accurately than typical machine learning or deep learning techniques. However, there is still a high consensus on how to construct or define an appropriate graph structure in brain networks in terms of two processes: 1) how do we generate the graphs?

Complexity science offers a wide range of measures for quantifying unpredictability, structure, and information. Yet, a systematic conceptual organization of these measures is still missing. We present a unified framework that locates statistical, algorithmic, and dynamical measures along three axes (regularity, randomness, and complexity) and situates them in a common conceptual space. We map statistical, algorithmic, and dynamical measures into this conceptual space, discussing their computational accessibility and approximability. This taxonomy reveals the deep challenges posed by uncomputability and highlights the emergence of modern data-driven methods (including autoencoders, latent dynamical models, symbolic regression, and physics-informed neural networks) as pragmatic approximations to classical complexity ideals. Latent spaces emerge as operational arenas where regularity extraction, noise management, and structured compression converge, bridging theoretical foundations with practical modeling in high-dimensional systems. We close by outlining implications for physics-informed AI and AI-guided discovery in complex physical systems, arguing that classical questions of complexity remain central to next-generation scientific modeling.

Recently, time series classification has attracted the attention of a large number of researchers, and hundreds of methods have been proposed. However, these methods often ignore the spatial correlations among dimensions and the local correlations among features. To address this issue, the causal and local correlations based network (CaLoNet) is proposed in this study for multivariate time series classification. First, pairwise spatial correlations between dimensions are modeled using causality modeling to obtain the graph structure. Then, a relationship extraction network is used to fuse local correlations to obtain long-term dependency features. Finally, the graph structure and long-term dependency features are integrated into the graph neural network. Experiments on the UEA datasets show that CaLoNet can obtain competitive performance compared with state-of-the-art methods.