Dynamical Reversibility and A New Theory of Causal Emergence based on SVD

TL;DR

This work introduces a robust causal emergence (CE) framework rooted in approximate dynamical reversibility, using a singular-value–based analysis of the TPM to identify redundant information pathways. By defining the reversibility measure $\Gamma_\alpha$ and relating it to effective information $EI$, the authors show that CE can be quantified as potential improvements in information transmission and reversibility, independent of coarse-graining schemes. They prove theoretical connections between $EI$ and $\log\Gamma_\alpha$, validate the approach with Boolean networks, cellular automata, and stochastic block models, and propose a practical SVD-based coarse-graining procedure that yields comparable CE assessments to EI maximization while avoiding coarse-graining ambiguities. This framework provides a principled, coarse-graining–agnostic lens on emergence, with implications for understanding macroscopic causality and for constructing reversible macro-dynamics in complex systems.

Abstract

The theory of causal emergence (CE) with effective information (EI) posits that complex systems can exhibit CE, where macro-dynamics show stronger causal effects than micro-dynamics. A key challenge of this theory is its dependence on coarse-graining method. In this paper, we introduce a fresh concept of approximate dynamical reversibility and establish a novel framework for CE based on this. By applying singular value decomposition(SVD) to Markov dynamics, we find that the essence of CE lies in the presence of redundancy, represented by the irreversible and correlated information pathways. Therefore, CE can be quantified as the potential maximal efficiency increase for dynamical reversibility or information transmission. We also demonstrate a strong correlation between the approximate dynamical reversibility and EI, establishing an equivalence between the SVD and EI maximization frameworks for quantifying CE, supported by theoretical insights and numerical examples from Boolean networks, cellular automata, and complex networks. Importantly, our SVD-based CE framework is independent of specific coarse-graining techniques and effectively captures the fundamental characteristics of the dynamics.

Figures (11)

• Figure 1: Examples of transitional probability matrices (TPMs) used in different Markov chains, along with measures of effective information (EI and normalized one $\mathit{eff}$) for causality, and the measure of $\Gamma_{\alpha}$ (and normalized one $\gamma_{\alpha}$ by setting $\alpha=1$) for approximate reversibility of dynamics.
• Figure 2: Comparison between $EI$ and $\Gamma$ for various generated TPMs. (a) shows the approximate relationship $EI\sim \Gamma$ for randomly generated TPMs which are softening of permutation matrices controlled by different $\sigma$ (detailed could be found in Section \ref{['sec:perturbation_permutation']}). The theoretical and empirical upper bounds are also shown as dashed lines. The lower bound is omitted because it is size-dependent. Each curve is obtained by tuning up the magnitude of softening on a randomly generated permutation matrix with different sizes. (b) demonstrates the same relationship between $EI$ and $\Gamma$ for the TPMs which are generated by the similar softening method, but which are based on a variance of the $N\times N$ identity matrix. The variance is to change $N-r$ row vectors of identity matrix to the same one-hot vectors (with the value one as the first element), where $r$ is the rank of the matrix, $N=50$. And the number $N-r$ can be treated as a control of the degeneracy of the TPM. On this figure, all the upper bounds and lower bounds are shown as dashed lines. (c) shows the same relationship for the combination of randomly sampled normalized row vectors for various sizes ($N\in \{2,3,\cdots,100\}$). On each size, 100 such random matrices are sampled to get the scatter points. The scatter points for particular sizes $N=20,30,50$ are rendered with red to show the nearly logarithmic relation between $EI$ and $\Gamma$. The empirical upper bound is also shown as the dashed line. (d) demonstrates the dependence of the difference between $EI$ and $\Gamma$ ($\log\Gamma-EI$) on the softening magnitude $\sigma$ for the matrices generated. (e) and (f) shows the density plots of $EI$(c) and $\Gamma$(d) with different parameters $p$ and $q$ computed for a parameterized simplest TPM: $P=p1-p1-qq$ with size $2\times 2$
• Figure 3: Examples of clear causal emergence on a Boolean network and a one-dimensional cellular automaton. (a) is the Boolean network model with 6 nodes and 12 edges, the micro-mechanism can be referred to ref.Hoel2013. (b) is the coarse-grained Boolean network model according to the coarse-grained TPM of (e). (c) is the corresponding TPM of (a). (d) is the spectrum for the singular values of (c) with only 8 non-zero values. (e) is the coarse-graining of (c). (f) is the projection matrix from the micro-states to the macro-states obtained according to our coarse-graining method based on SVD. (g) is the evolution of the 40th elementary cellular automaton (the rule is $000\rightarrow 0, 001\rightarrow 0, 010\rightarrow 1, 011\rightarrow 0, 100\rightarrow 1, 101\rightarrow 0, 110\rightarrow 0, 111\rightarrow 0$). (h) shows the two spectra of the singular values for four distinct local TPMs of the same cellular automaton. The local TPM elucidates the process by which a cell moves from its present state to the subsequent state within a specified environment, encompassing the states of the focal cell and its two neighboring cells. (i) shows the quantification of CE for local TPM ($\Delta \Gamma$, the red dots indicate the cells where the quantities of CE are non-zeros.) as well as the original evolution of the original automaton (the background).
• Figure 4: Examples of clear and vague CE and their coarse-grained models based on SVD method for a Boolean network and complex networks generated by the stochastic block models. (a) The original stochastic Boolean network model, each node can only interact with its network neighbors; (b) Shared node dynamics for all nodes in (a). Each row corresponds to the states combination of one node's all neighbors in previous time step, and each column is the probability to take 0 or 1 of the node at current time step. (c) The coarse-grained Boolean network of (a) which is extracted from the TPMs and the relations between micro- and macro nodes illustrated in (f) and (i) by identifying the macro-state $\alpha=0,\beta=0$ as the micro-states for $(0,0,0,0),(0,0,0,1),(0,0,1,0),(0,1,0,0),(0,1,0,1),(0,1,1,0),(1,0,0,0),(1,0,0,1),(1,0,1,0)$, $\alpha=0,\beta=1$ as the micro-states for $(0,0,1,1), (0,1,1,1),(1,0,1,1)$, $\alpha=1,\beta=0$ as the micro-states for $(1,1,0,0),(1,1,0,1),(1,1,1,0)$, and $\alpha=0,\beta=1$ as the micro-states for $(1,1,1,1)$. (d) The corresponding TPM of (a) and (b). (e) The singular value spectrum for (d). (g) A perturbed TPM from (d). (h) The singular value spectrum for (g). (f) and (i) are the reduced TPMs and the projection matrices (below) after the application of our coarse-graining method on the original TPMs in (d) and (g), respectively. (j) is the visulization of the original network sampled from the stochastic block model with p = 0.9(the probability for inner community connections) and q = 0.1(the probability for inter community connections), and the nodes are colored with different hues to distinguish the blocks to which they belong. There are 5 blocks in total. The edges are undirected and binary. The TPM is obtained by normalizing the adjacency matrix by dividing by each node's degree. (k) The singular value spectrum of three samples of the stochastic block model network with different p and q. (l) is the reduced network of (j) obtained through our coarse-graining method, with the node grouping results aligning with the initial block settings.
• Figure Supplementary Figure 1: The singular vectors and the vectors representing the coarse-graining strategies. The red arrows are the singular vectors in $U$(the magnitude is multiplied by the corresponding singular values, the light red arrow represents the singular vector with the smallest singular value), the black arrows represent the optimal coarse-graining strategy with EI maximization, $\{\{1,2\},3\}$. The gray arrows represent the coarse-graining strategy $\{\{2,3\},1\}$, and the light gray arrows represent the strategy $\{\{1,3\},2\}$.

Theorems & Definitions (25)

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