An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems

TL;DR

The paper develops an exact theory of causal emergence for linear stochastic iteration systems with continuous state spaces and Gaussian noise, deriving an analytical expression for effective information and a dimension-normalized measure $\mathcal{J}$ to compare micro- and macro-dynamics. It provides an explicit formula for the causal-emergence increment $\Delta\mathcal{J}$, decomposed into degeneracy and determinism emergence, and establishes upper bounds and an optimal coarse-graining set $\mathcal{W}^*$ that maximize $\Delta\mathcal{J}$ under an information-loss constraint. The maximal emergence is governed by the system’s eigenstructure, with results showing that retaining the top $k$ eigenvalues (and corresponding eigenvectors) of $A$ (or $A\Sigma^{-1/2}$ in the $W^†=W^T$ case) yields the greatest causal leverage, while the noise covariance $\Sigma$ and the singular values of $W$ tune the balance between determinism and degeneracy. The framework is validated against three simplified physical systems (random walk, energy dissipation, and spiral rotation), with analytical predictions matching numerical simulations and revealing how coarse-graining choices shape emergent causality and predictive accuracy.

Abstract

After coarse-graining a complex system, the dynamics of its macro-state may exhibit more pronounced causal effects than those of its micro-state. This phenomenon, known as causal emergence, is quantified by the indicator of effective information. However, two challenges confront this theory: the absence of well-developed frameworks in continuous stochastic dynamical systems and the reliance on coarse-graining methodologies. In this study, we introduce an exact theoretic framework for causal emergence within linear stochastic iteration systems featuring continuous state spaces and Gaussian noise. Building upon this foundation, we derive an analytical expression for effective information across general dynamics and identify optimal linear coarse-graining strategies that maximize the degree of causal emergence when the dimension averaged uncertainty eliminated by coarse-graining has an upper bound. Our investigation reveals that the maximal causal emergence and the optimal coarse-graining methods are primarily determined by the principal eigenvalues and eigenvectors of the dynamic system's parameter matrix, with the latter not being unique. To validate our propositions, we apply our analytical models to three simplified physical systems, comparing the outcomes with numerical simulations, and consistently achieve congruent results.

Figures (6)

• Figure 1: Experimental results of the simulation about the lemma and the theorem presented in Section 3. (a) For matrix $A$ with eigenvalues of $\lambda_1=2.540, \lambda_2=1.380, \lambda_3=-0.4899, \lambda_4=0.1149$, we randomly generate matrix $W$. We can see that there is an upper bound $\lambda_1\lambda_2$ on the corresponding $\det(A_M)$ value of each $W$. (b) If $A$ is also randomly generated, we can see the scatter plot that conforms to Lemma.\ref{['thm.Maximizing-determinant']}. (c) We validate the inequality using randomly generated $\Sigma$ and $W$. We can control the range of $\det(W\Sigma W^{T})$ by adjusting the magnitude of the singular value of $W$. (d) The smaller the mean singular value of $W$ we generate, the greater the degree of causal emergence $\Delta\mathcal{J}$.
• Figure 2: The visualization of the set for the optimal solutions of the coase-graining strategy $W$. When $w_1w_2^{T}=0$, $\Sigma=\sigma^2I_3$. Although $W\in \mathcal{R}^{2\times 3}$ is in a six-dimensional space, we can draw the range of values for $w_1$ while limiting $w_2$, $w_{11}v_{13}+w_{12}v_{23}+w_{13}v_{33}=0$ and $w_{11}^2+w_{12}^2+w_{13}^2=R^2,$ in which $R^2=\epsilon/(w_2w_2^{T})$. When causal emergence $\Delta\mathcal{J}(W^*)=\Delta\mathcal{J}^*$, the solution set of $w_i$ is the intersection of a plane (blue) and a sphere (red) in 3D space $\mathcal{R}^3$, which is a circle
• Figure 3: Experimental results for random walks model. (a) The trajectories $x_t$ of a random walker in different dimensions. (b) The probability density functions for the four dimensions of micro-state noises $\varepsilon_t$ and the macro-state noise $\varepsilon_{M,t}$. (c) The degree of causal emergence under different macro-state dimension $k$. Under the condition that the singular values of $W$ are all 1, the higher $k$, the stronger the uncertainty of the system, and the smaller the degree of causal emergence. (d) $\Delta\mathcal{J}$ and $std(\kappa)$. The larger the variance of the eigenvalues $\kappa_i$ of $\Sigma$, $i=1,\dots,n$, the more likely the occurrences of causal emergence with higher degrees.
• Figure 4: Experimental results for heat dissipation model. (a) The illustration of heat transfer and dissipation for a one-dimensional chain with four vertices. (b) The micro-state data of temperature at different times $t$ for 4 nodes. (c) The comparison between the macro-state data directly obtained by coarsening the micro-state data, $y_t$ to obtain $y_t=Wx_t$ for $t=0,1,2,\dots$, and $\hat{y}_t$ which is generated by the macro-state dynamics to iterate the initial macro-state $y_0$. (d) The comparisons for the degrees of causal emergence under different dimensions of macro-states $k$. when $k=1$ and $\epsilon=1$, maximized degree of causal emergence occurs as $\Delta\mathcal{J}=0.6656$. (e) The comparison between $\Delta\mathcal{I}$ and $\Delta\mathcal{J}$. As the sample size increases, the numerical solutions for causal emergence $\Delta\mathcal{I}$ gradually approach the analytical solutions $\Delta\mathcal{J}$.
• Figure 5: Experimental results of spiral rotating. (a) When $\Psi={\rm diag}(0.94,0.94,0.99)$ and $x_0=(1,1,3)^T$, $x_t$ represents a point in space that rotates around the axis of rotation and contracts towards the axis of rotation. By the optimal coarse-graining $W^*$, we can obtain macro-states that move along the axis of rotation. (b) The degree of causal emergence and its dependence on the dimension of macro-states for the example in (a), which reaches its maximum value as $\Delta\mathcal{J}=0.0341$ when $k=1$. (c) The spiral curve when $\Psi={\rm diag}(0.99,0.97,0.2)$ and $x_0=(1,1,1)^T$, the trajectory of $x_t$ is compressed to a plane perpendicular to $u$ at the initial stage. (d) We can project $x_t$ on the plane as a macro-state, where $k=2$, $\Delta\mathcal{J}=0.5295$. (e) When $\pi/2<\theta<3\pi/2$, $x_t$ tends to oscillate more than rotate, while $\theta<\pi/2$ or $\theta>3\pi/2$$x_t$ tends to be a rotate model. (f) When $\theta$ approaches $\pi$, the degree of causal emergence is smaller, while when it approaches $0$ or $2\pi$, the degree of causal emergence is larger.

Theorems & Definitions (28)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 3.1
• Definition 3.2
• Theorem 3.1
• Lemma 3.1
• Lemma 3.2
• Theorem 3.2
• proof