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Fractal Conditional Correlation Dimension Infers Complex Causal Networks

Özge Canlı Usta, Erik M. Bollt

TL;DR

This work tackles causal discovery in complex dynamical networks by recasting information flow as a fractal, geometric quantity. It introduces the optimal conditional geometric information flow principle ($oGeoC$) and two algorithms (FORWARD GEOC and BACKWARD GEOC) to identify direct causal parents while removing indirect influences, using conditional correlation dimensions $GeoC_{J \rightarrow I | K}$. Through experiments on coupled logistic maps, the study shows that, with sufficient observations, the method achieves high true positive rates and low false positives, outperforming baselines in recovering network structure. The approach provides a model-free, geometry-based alternative for causal inference in nonlinear, potentially fractal systems, with potential extensions to real data and stochastic dynamics.

Abstract

Determining causal inference has become popular in physical and engineering applications. While the problem has immense challenges, it provides a way to model the complex networks by observing the time series. In this paper, we present the optimal conditional correlation dimensional geometric information flow principle ($oGeoC$) that can reveal direct and indirect causal relations in a network through geometric interpretations. We introduce two algorithms that utilize the $oGeoC$ principle to discover the direct links and then remove indirect links. The algorithms are evaluated using coupled logistic networks. The results indicate that when the number of observations is sufficient, the proposed algorithms are highly accurate in identifying direct causal links and have a low false positive rate.

Fractal Conditional Correlation Dimension Infers Complex Causal Networks

TL;DR

This work tackles causal discovery in complex dynamical networks by recasting information flow as a fractal, geometric quantity. It introduces the optimal conditional geometric information flow principle () and two algorithms (FORWARD GEOC and BACKWARD GEOC) to identify direct causal parents while removing indirect influences, using conditional correlation dimensions . Through experiments on coupled logistic maps, the study shows that, with sufficient observations, the method achieves high true positive rates and low false positives, outperforming baselines in recovering network structure. The approach provides a model-free, geometry-based alternative for causal inference in nonlinear, potentially fractal systems, with potential extensions to real data and stochastic dynamics.

Abstract

Determining causal inference has become popular in physical and engineering applications. While the problem has immense challenges, it provides a way to model the complex networks by observing the time series. In this paper, we present the optimal conditional correlation dimensional geometric information flow principle () that can reveal direct and indirect causal relations in a network through geometric interpretations. We introduce two algorithms that utilize the principle to discover the direct links and then remove indirect links. The algorithms are evaluated using coupled logistic networks. The results indicate that when the number of observations is sufficient, the proposed algorithms are highly accurate in identifying direct causal links and have a low false positive rate.

Paper Structure

This paper contains 10 sections, 16 equations, 7 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Preliminaries and Basic Definitions
  4. Geometric Causation of Information Flow in Networks
  5. Estimation of Correlation Dimension
  6. Results
  7. Discussion
  8. Estimation of correlation dimension
  9. Shuffle test to determine the zero
  10. Illustration of $\mathcal{D}_2$ estimations for the networks

Figures (7)

  • Figure 1: The performance of the proposed algorithm for the networks in Figures \ref{['fig:directed_7_nodes']} and \ref{['fig:bidirected_7_nodes']}. (a) A network with directed coupling consisting of $N=7$ nodes and 8 links. (b) A network with bidirectional coupling consisting of $N=7$ nodes and 12 links. In both networks, each circle represents a logistic map. (c) TPRs and FPRs are plotted against various sample sizes $(T)$ for the network in Figure \ref{['fig:directed_7_nodes']}. (d) We also illustrated TPRs and FPRs with respect to $(T)$ for the network in Figure \ref{['fig:bidirected_7_nodes']}. In both simulations, the number of permutations is $N_p =100$ and the significance threshold is $\theta=0.01$.
  • Figure 2: The performance of the proposed algorithm for the networks randomly coupled according to Erdős-Rényi (ER) model with the probability of $p= 0.1$. The simulations are repeated 10 times for different networks (a) An illustration of one of the networks. (b) Error bar points show the mean of TPRs and FPRs with respect to different sample sizes. The maximum and minimum points of the error bar indicate the highest and lowest values of TPRs and FPRs. The number of permutations is again $N_p =100$ with $\theta= 0.01$. (c) ROC curve for $T=500$ and $T=1000$. Here, $N_p =100$, but $\theta$ is varied from to $0.01$ to $0.99$ to plot ROC curve.
  • Figure A1: The curve of $\ln \hat{C}(\epsilon) / \ln(\epsilon)$ for a given dataset of (a) $X^{'(3)}$, (b) $(X^{'(3)}, X^{(2)})$, (c) $X^{(2)}$ for the network in Figure \ref{['fig:directed_7_nodes']}. The estimated correlation dimension is shown in the legend. The number of observations is $T=10000$.
  • Figure A2: The curve of $\ln \hat{C}(\epsilon) / \ln(\epsilon)$ for a given dataset of (a) $(X^{'(3)}, X^{(2)})$, (b) $X^{(2)}$, (c) $(X^{'(3)}, X^{(2)}, X^{(7)})$, (d) $(X^{(2)}, X^{(7)})$ for the network in Figure \ref{['fig:directed_7_nodes']}. The estimated correlation dimension is shown in the legend. The number of observations is $T=10000$.
  • Figure A3: The curve of $\ln \hat{C}(\epsilon) / \ln(\epsilon)$ for a given dataset of (a) $X^{'(7)}$, (b) $(X^{'(7)}, X^{(2)})$, (c) $X^{(2)}$ for the network in Figure \ref{['fig:twenty_nodes_logistic_network']}. The estimated correlation dimension is shown in the legend. The number of observations is $T=10000$.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Definition 3