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=== Early work on quantifying emergence ===
=== Early work on quantifying emergence ===
There have been some related works in the early stage that attempted to quantitatively analyze emergence. The computational mechanics theory proposed by Crutchfield et al. [27] considers causal states. This method discusses related concepts based on the division of state space and is very similar to Erik Hoel's causal emergence theory. On the other hand, Seth et al. proposed the G-emergence theory [28] to quantify emergence by using Granger causality. Computational mechanics The computational mechanics theory attempts to express the causal laws of emergence in a quantitative framework, that is, how to construct a coarse-grained causal model from a random process so that this model can generate the time series of the observed random process [27].
There have been some related works in the early stage that attempted to quantitatively analyze emergence. The computational mechanics theory proposed by Crutchfield et al. [27] considers causal states. This method discusses related concepts based on the division of state space and is very similar to Erik Hoel's causal emergence theory. On the other hand, Seth et al. proposed the G-emergence theory [28] to quantify emergence by using Granger causality.
==== Computational mechanics ====
The computational mechanics theory attempts to express the causal laws of emergence in a quantitative framework, that is, how to construct a coarse-grained causal model from a random process so that this model can generate the time series of the observed random process [27].
Here, the random process can be represented by <math>\overleftrightarrow{s}</math>. Based on time <math>t</math>, the random process can be divided into two parts: the process before time <math>t</math> and the process after time <math>t</math>, <math>\overleftarrow{s_t}</math> and <math>\overrightarrow{s_t}</math>. Computational mechanics records the set of all possible historical processes <math>\overleftarrow{s_t}</math> as <math> \overleftarrow{S}</math>, and the set of all future processes as <math> \overrightarrow{S}</math>.
Here, the random process can be represented by <math>\overleftrightarrow{s}</math>. Based on time <math>t</math>, the random process can be divided into two parts: the process before time <math>t</math> and the process after time <math>t</math>, <math>\overleftarrow{s_t}</math> and <math>\overrightarrow{s_t}</math>. Computational mechanics records the set of all possible historical processes <math>\overleftarrow{s_t}</math> as <math> \overleftarrow{S}</math>, and the set of all future processes as <math> \overrightarrow{S}</math>.
The causal emergence framework has many similarities with computational mechanics. All historical processes <math>\overleftarrow{s}</math> can be regarded as microscopic states. All <math>R \in \mathcal{R}</math> correspond to macroscopic states. The function <math>\eta</math> can be understood as a possible coarse-graining function. The causal state <math>\epsilon \left ( \overleftarrow{s} \right )</math> is a special state that can at least have the same predictive power as the microscopic state <math>\overleftarrow{s}</math>. Therefore, <math>\epsilon</math> can be understood as an effective coarse-graining strategy. Causal transfer <math>T</math> corresponds to effective macroscopic dynamics. The characteristic of minimum randomness characterizes the determinism of macroscopic dynamics and can be measured by effective information in causal emergence.
The causal emergence framework has many similarities with computational mechanics. All historical processes <math>\overleftarrow{s}</math> can be regarded as microscopic states. All <math>R \in \mathcal{R}</math> correspond to macroscopic states. The function <math>\eta</math> can be understood as a possible coarse-graining function. The causal state <math>\epsilon \left ( \overleftarrow{s} \right )</math> is a special state that can at least have the same predictive power as the microscopic state <math>\overleftarrow{s}</math>. Therefore, <math>\epsilon</math> can be understood as an effective coarse-graining strategy. Causal transfer <math>T</math> corresponds to effective macroscopic dynamics. The characteristic of minimum randomness characterizes the determinism of macroscopic dynamics and can be measured by effective information in causal emergence.
==== G-emergence ====
The G-emergence theory was proposed by Seth in 2008 and is one of the earliest studies to quantitatively quantify emergence from a causal perspective [28]. The basic idea is to use nonlinear Granger causality to quantify weak emergence in complex systems.
Specifically, if we use a binary autoregressive model for prediction, when there are only two variables A and B, there are two equations in the autoregressive model, each equation corresponds to one of the variables, and the current value of each variable is composed of its own value and the value of the other variable within a certain time lag range. In addition, the model also calculates residuals. Here, residuals can be understood as prediction errors and can be used to measure the degree of Granger causal effect (called G-causality) of each equation. The degree to which B is a Granger cause (G-cause) of A is calculated by taking the logarithm of the ratio of the two residual variances, one being the residual of A's autoregressive model when B is omitted, and the other being the residual of the full prediction model (including A and B). In addition, the author also defines the concept of G-autonomous, which represents a measure of the extent to which the past values of a time series can predict its own future values. The strength of this autonomous predictive causal effect can be characterized in a similar way to G-causality.
[[文件:G Emergence Theory.png|无|缩略图]]
As shown in the above figure, we can judge the occurrence of emergence based on the two basic concepts in the above G-causality (here is the measure of emergence based on Granger causality, denoted as G-emergence). If A is understood as a macroscopic variable and B is understood as a microscopic variable. The conditions for emergence to occur include two: 1) A is G-autonomous with respect to B; 2) B is a G-cause of A. The degree of G-emergence is calculated by multiplying the degree of A's G-autonomous by the degree of B's average G-cause.
The G-emergence theory proposed by Seth is the first attempt to use causal measures to quantify emergence phenomena. However, the causal relationship used by the author is Granger causality, which is not a strict causal relationship. At the same time, the measurement results also depend on the regression method used. In addition, the measurement index of this method is defined according to variables rather than dynamics, which means that the results will depend on the choice of variables. These all constitute the drawbacks of the G-emergence theory.
The causal emergence framework also has similarities with the aforementioned G-emergence. The macroscopic states of both methods need to be manually selected. In addition, it should be noted that some of the above methods for quantitatively quantifying emergence often do not consider true interventionist causality.
==== Other theories for quantitatively characterizing emergence ====
In addition, there are some other quantitative theories of emergence. There are mainly two methods that are widely discussed. One is to understand emergence from the process from disorder to order. Moez Mnif and Christian Müller-Schloer [30] use Shannon entropy to measure order and disorder. In the self-organization process, emergence occurs when order increases. The increase in order is calculated by measuring the difference in Shannon entropy between the initial state and the final state. However, the defect of this method is that it depends on the abstract observation level and the initial conditions of the system. To overcome these two difficulties, the authors propose a measurement method compared with the maximum entropy distribution. Inspired by the work of Moez mif and Christian Müller-Schloer, reference [31] suggests using the divergence between two probability distributions to quantify emergence. They understand emergence as an unexpected or unpredictable distribution change based on the observed samples. But this method has disadvantages such as large computational complexity and low estimation accuracy. To solve these problems, reference [32] further proposes an approximate method for estimating density using Gaussian mixture models and introduces Mahalanobis distance to characterize the difference between data and Gaussian components, thus obtaining better results. In addition, Holzer and de Meer [33][34] and others proposed another emergence measurement method based on Shannon entropy. They believe that a complex system is a self-organizing process in which different individuals interact through communication. Then, we can measure emergence according to the ratio between the Shannon entropy measure of all communications between agents and the sum of Shannon entropies as separate sources.
Another method is to understand emergence from the perspective of "the whole is greater than the sum of its parts" [35][36]. This method defines emergence from interaction rules and the states of agents rather than statistically measuring from the totality of the entire system. Specifically, this measure consists of subtracting two terms. The first term describes the collective state of the entire system, while the second term represents the sum of the individual states of all components. This measure emphasizes that emergence arises from the interactions and collective behavior of the system.
=== Causal emergence theory based on effective information ===
In history, the first relatively complete and explicit quantitative theory that uses causality to define emergence is the causal emergence theory proposed by Erik Hoel, Larissa Albantakis, and Giulio Tononi [1][2]. This theory defines so-called causal emergence for Markov chains as the phenomenon that the coarsened Markov chain has a greater causal effect strength than the original Markov chain. Here, the causal effect strength is measured by effective information. This indicator is a modification of the mutual information indicator. The main difference is that the state variable at time $t$ is intervened by do-intervention and transformed into a uniform distribution (or maximum entropy distribution). The effective information indicator was proposed by Giulio Tononi as early as 2003 when studying integrated information theory. As Giulio Tononi's student, Erik Hoel applied effective information to Markov chains and proposed the causal emergence theory based on effective information.
=== Causal Emergence Theory Based on Information Decomposition ===
In addition, in 2020, Rosas et al. proposed a method based on information decomposition to define causal emergence in systems from an information theory perspective and quantitatively characterize emergence based on synergistic information or redundant information. The so-called information decomposition is a new method to analyze the complex interrelationships of various variables in complex systems. By decomposing information, each partial information is represented by an information atom. At the same time, each partial information is projected into the information atom with the help of an information lattice diagram. Both synergistic information and redundant information can be represented by corresponding information atoms. This method is based on the non-negative decomposition theory of multivariate information proposed by Williams and Beer. In the paper, partial information decomposition (PID) is used to decompose the mutual information between microstates and macrostates. However, the PID framework can only decompose the mutual information between multiple source variables and one target variable. Rosas extended this framework and proposed the integrated information decomposition method <nowiki><math>\Phi ID</math></nowiki>.
=== Resent Work ===
Barnett et al. [40] proposed the concept of dynamical decoupling by judging the decoupling of macroscopic and microscopic dynamics based on transfer entropy to judge the occurrence of emergence. That is, emergence is characterized as the macroscopic variables and microscopic variables being independent of each other and having no causal relationship, which can also be regarded as a causal emergence phenomenon. In 2024, Zhang Jiang et al. [26] proposed a new causal emergence theory based on singular value decomposition. The core idea of this theory is to point out that the so-called causal emergence is actually equivalent to the emergence of dynamical reversibility. Given the Markov transition matrix of a system, by performing singular value decomposition on it, the sum of the <nowiki><math>\alpha</math></nowiki> power of the singular values is defined as the reversibility measure of Markov dynamics (<nowiki><math>\Gamma_{\alpha}\equiv \sum_{i=1}^N\sigma_i^{\alpha}</math></nowiki>), where [math]\sigma_i[/math] is the singular value. This index is highly correlated with effective information and can also be used to characterize the causal effect strength of dynamics. According to the spectrum of singular values, this method can directly define the concepts of clear emergence and vague emergence without explicitly defining a coarse-graining scheme.