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Algorithmic Identification of Essential Exogenous Nodes for Causal Sufficiency in Brain Networks

Abdolmahdi Bagheri, Mahdi Dehshiri, Babak Nadjar Araabi, Alireza Akhondi Asl

TL;DR

This work tackles causal sufficiency in brain networks by proposing an algorithmic pipeline to identify essential exogenous nodes that must be included to avoid spurious causal links. The method combines PC-like independence testing, KS-based discrimination of candidate confounders across networks, and NF-iVAE with Correlation Coefficient index to confirm confounding variables, all in a population-based framework. Applied to the HCP movie-watching dataset with Schaefer-based parcellation plus subcortical regions, the approach identifies dorsal regions as confounders for visual networks and validates findings against neuroscientific expectations, with reliability shown across 30 NF-iVAE initializations. The results demonstrate that incorporating identified exogenous nodes enables more accurate causal discovery in larger brain networks and provides a practical pathway for applying existing causal methods to smaller, confounded subgraphs while preserving causal_sufficiency.

Abstract

In the investigation of any causal mechanisms, such as the brain's causal networks, the assumption of causal sufficiency plays a critical role. Notably, neglecting this assumption can result in significant errors, a fact that is often disregarded in the causal analysis of brain networks. In this study, we propose an algorithmic identification approach for determining essential exogenous nodes that satisfy the critical need for causal sufficiency to adhere to it in such inquiries. Our approach consists of three main steps: First, by capturing the essence of the Peter-Clark (PC) algorithm, we conduct independence tests for pairs of regions within a network, as well as for the same pairs conditioned on nodes from other networks. Next, we distinguish candidate confounders by analyzing the differences between the conditional and unconditional results, using the Kolmogorov-Smirnov test. Subsequently, we utilize Non-Factorized identifiable Variational Autoencoders (NF-iVAE) along with the Correlation Coefficient index (CCI) metric to identify the confounding variables within these candidate nodes. Applying our method to the Human Connectome Projects (HCP) movie-watching task data, we demonstrate that while interactions exist between dorsal and ventral regions, only dorsal regions serve as confounders for the visual networks, and vice versa. These findings align consistently with those resulting from the neuroscientific perspective. Finally, we show the reliability of our results by testing 30 independent runs for NF-iVAE initialization.

Algorithmic Identification of Essential Exogenous Nodes for Causal Sufficiency in Brain Networks

TL;DR

This work tackles causal sufficiency in brain networks by proposing an algorithmic pipeline to identify essential exogenous nodes that must be included to avoid spurious causal links. The method combines PC-like independence testing, KS-based discrimination of candidate confounders across networks, and NF-iVAE with Correlation Coefficient index to confirm confounding variables, all in a population-based framework. Applied to the HCP movie-watching dataset with Schaefer-based parcellation plus subcortical regions, the approach identifies dorsal regions as confounders for visual networks and validates findings against neuroscientific expectations, with reliability shown across 30 NF-iVAE initializations. The results demonstrate that incorporating identified exogenous nodes enables more accurate causal discovery in larger brain networks and provides a practical pathway for applying existing causal methods to smaller, confounded subgraphs while preserving causal_sufficiency.

Abstract

In the investigation of any causal mechanisms, such as the brain's causal networks, the assumption of causal sufficiency plays a critical role. Notably, neglecting this assumption can result in significant errors, a fact that is often disregarded in the causal analysis of brain networks. In this study, we propose an algorithmic identification approach for determining essential exogenous nodes that satisfy the critical need for causal sufficiency to adhere to it in such inquiries. Our approach consists of three main steps: First, by capturing the essence of the Peter-Clark (PC) algorithm, we conduct independence tests for pairs of regions within a network, as well as for the same pairs conditioned on nodes from other networks. Next, we distinguish candidate confounders by analyzing the differences between the conditional and unconditional results, using the Kolmogorov-Smirnov test. Subsequently, we utilize Non-Factorized identifiable Variational Autoencoders (NF-iVAE) along with the Correlation Coefficient index (CCI) metric to identify the confounding variables within these candidate nodes. Applying our method to the Human Connectome Projects (HCP) movie-watching task data, we demonstrate that while interactions exist between dorsal and ventral regions, only dorsal regions serve as confounders for the visual networks, and vice versa. These findings align consistently with those resulting from the neuroscientific perspective. Finally, we show the reliability of our results by testing 30 independent runs for NF-iVAE initialization.
Paper Structure (17 sections, 2 theorems, 5 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 17 sections, 2 theorems, 5 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Proposed method
  3. Definitions and preliminaries
  4. Causal sufficiency
  5. Identification of Essential exogenous nodes for causal sufficiency
  6. Data, preprocessing, and parcellation
  7. Experiments
  8. Experiment design
  9. Results
  10. Conclusion
  11. Definitions and preliminaries required for the proposed method
  12. Definitions and preliminaries
  13. Causal sufficiency
  14. Peter-Clark algorithm
  15. Unconditional and conditional independence test
  16. ...and 2 more sections

Key Result

lemma thmcounterlemma

zhang2011kernel Under the null hypothesis that $v_1$ and $v_2$ are statistically independent, the statistic has the same asymptotic distribution as i.e., $T_{UI} \overset{d}{=} \hat{T}_{UI}$ as $n \to \infty$. In this lemma, $\hat{K}_{v_1}$ and $\hat{K}_{v_2}$ are the corresponding centralized kernel matrices. Given the samples $v_1$ and $v_2$, $\lambda_{v_1,i}$ and $\lambda_{v_2,j}$ are the eig

Figures (3)

  • Figure 1: a.The true underlying network with variables $z_1, ..., z_5$, and confounders, $c_1$ and $c_2$ b. starting PC algorithm with fully connected graph, c. result of PC algorithm applying d. Identifying essential confounders based on the proposed algorithm i.e., $s_2=c_1$ and $s_4=c_2$, e. applying PC algorithm to $z_1,..., z_5$ and $c_1$ and $c_2$, F. result of applying causal discovery method with confounders
  • Figure 2: The CCI values for identified regions based on NF-iVAE and iVAE. a. results for the case that attention is confounder b. results for the case that visual is confounder c. the relation between dorsal, ventral and visual regions
  • Figure 3: Probability of Visual Cortex Nodes Ranked Among Top 5 by CCI Across 30 Runs. The y-axis denotes the frequency of inclusion in the top 5 rankings, and the x-axis lists the nodes.

Theorems & Definitions (4)

  • definition thmcounterdefinition: A general definition of causal sufficiency
  • definition thmcounterdefinition: A general definition of causal sufficiency
  • lemma thmcounterlemma: Independence test
  • lemma thmcounterlemma: Conditional independence test