Causal Network Inference by Optimal Causation Entropy

Abstract: The broad abundance of time series data, which is in sharp contrast to limited knowledge of the underlying network dynamic processes that produce such observations, calls for a rigorous and efficient method of causal network inference. Here we develop mathematical theory of causation entropy, an information-theoretic statistic designed for model-free causality inference. For stationary Markov processes, we prove that for a given node in the network, its causal parents forms the minimal set of nodes that maximizes causation entropy, a result we refer to as the optimal causation entropy principle. Furthermore, this principle guides us to develop computational and data efficient algorithms for causal network inference based on a two-step discovery and removal algorithm for time series data for a network-couple dynamical system. Validation in terms of analytical and numerical results for Gaussian processes on large random networks highlight that inference by our algorithm outperforms previous leading methods including conditioned Granger causality and transfer entropy. Interestingly, our numerical results suggest that the number of samples required for accurate inference depends strongly on network characteristics such as the density of links and information diffusion rate and not necessarily on the number of nodes.

• The paper introduces the optimal causation entropy principle to identify minimal direct causal sets in networked systems.
• It develops and validates two algorithms that aggregate and prune candidate causal nodes, outperforming traditional methods in accuracy.
• The study demonstrates sample size efficiency and potential applications in fields like neuroscience, epidemiology, and economics.

Causal Network Inference by Optimal Causation Entropy

The paper presents a method for causal network inference utilizing an information-theoretic framework called causation entropy. The method is designed to infer causal structures from multivariate time series data, particularly focusing on identifying causal relationships within complex networked systems. The authors develop and validate the theory of causation entropy, establishing it as a robust tool for model-free causality inference.

Summary of Key Contributions

1. Theoretical Foundation of Causation Entropy: The authors introduce the concept of causation entropy, a statistic based on conditional mutual information. For stationary Markov processes, they establish the "optimal causation entropy principle," which states that the causal parents of a node form the minimal set that maximizes causation entropy. This principle underlies the theoretical justification for identifying direct causal relationships while excluding indirect influences.
2. Algorithm Development: The authors propose two main algorithms—an aggregative discovery algorithm and a progressive removal algorithm. These are used to identify causal parents of each node efficiently. The first aggregates potential causal nodes, while the second prunes the non-causal nodes, achieving accurate causal network inference.
3. Validation and Comparison: Through analytical and numerical examinations, particularly on Gaussian processes applied to random networks, the causation entropy-based method significantly outperforms traditional methods such as transfer entropy and conditioning Granger causality. The study reveals that causation entropy is less susceptible to common issues encountered by other methods, such as indirect causality detection errors.
4. Sample Size Efficiency: A notable finding is that the number of samples required for accurate inference heavily depends on the network properties, such as link density and information diffusion rate, rather than sheer network size. This implies scalability and data efficiency for the proposed method, making it applicable to large-scale network problems despite limited observational data.

Implications and Future Directions

• Practical Applications: The successful implementation of causation entropy opens the door for advancements in diverse fields like neuroscience, epidemiology, and economics, where understanding causal networks is essential but direct experimentation might not be possible or ethical.
• Efficiency and Data Requirements: The findings suggest that leveraging causation entropy can lead to efficient causal discovery with fewer samples, critical for applications where data acquisition is costly or time-consuming.
• Extensions Beyond Gaussian Processes: While this study focused on Gaussian processes, the inherent power of information-theoretic measures like causation entropy hints at potential applications to non-Gaussian and even nonlinear processes, which represent a direction for future exploration.
• Algorithmic Enhancements: Future work could involve enhancing the computational efficiency of the proposed algorithms and integrating techniques to handle non-stationary environments, expanding their applicability to a broader range of dynamic systems.

This research underlines a pivotal step in automating the identification of causal relationships in networked systems, leveraging the strengths of information theory to transcend limitations faced by traditional statistical methods. The optimized approach for causation entropy not only enriches the toolkit available for researchers in the domain of network dynamics but also prompts further investigation into its application across other complex systems characterized by intricate interdependencies.