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The coarse-graining operation of this system is divided into two steps. The first step is to cluster the nodes in the Boolean network. As shown in figure b below, merge A and B to obtain the macroscopic node [math]\alpha[/math], and merge C and D to obtain the macroscopic node [math]\beta[/math]. The second step is to map the microscopic node states in each group to the merged macroscopic node states. This mapping function is shown in figure d below. All microscopic node states containing 0 are transformed into the off state of the macroscopic node, while the microscopic 11 state is transformed into the on state of the macroscopic. In this way, we can obtain a new macroscopic Boolean network, and obtain the dynamic mechanism of the macroscopic Boolean network according to the dynamic mechanism of the microscopic nodes. According to this mechanism, the state transition matrix of the macroscopic network can be obtained (as shown in figure e).

The coarse-graining operation of this system is divided into two steps. The first step is to cluster the nodes in the Boolean network. As shown in figure b below, merge A and B to obtain the macroscopic node [math]\alpha[/math], and merge C and D to obtain the macroscopic node [math]\beta[/math]. The second step is to map the microscopic node states in each group to the merged macroscopic node states. This mapping function is shown in figure d below. All microscopic node states containing 0 are transformed into the off state of the macroscopic node, while the microscopic 11 state is transformed into the on state of the macroscopic. In this way, we can obtain a new macroscopic Boolean network, and obtain the dynamic mechanism of the macroscopic Boolean network according to the dynamic mechanism of the microscopic nodes. According to this mechanism, the state transition matrix of the macroscopic network can be obtained (as shown in figure e).

Through comparison, we find that the effective information of macroscopic dynamics is greater than that of microscopic dynamics (<math>EI(f_M\)>EI(f_m\)</math>). Causal emergence occurs in this system.

Through comparison, we find that the effective information of macroscopic dynamics is greater than that of microscopic dynamics <math>EI(f_M\ )>EI(f_m\ ) </math>. Causal emergence occurs in this system.

=====Causal Emergence in Continuous Variables=====

=====Causal Emergence in Continuous Variables=====

Furthermore, in the paper "xxx", Hoel et al. proposed the theoretical framework of causal geometry, trying to generalize the causal emergence theory to function mappings and dynamical systems with continuous states. This article defines <math>EI</math> for random function mapping, and also introduces the concepts of intervention noise and causal geometry, and compares and analogizes this concept with information geometry. Liu Kaiwei et al. further gave an exact analytical causal emergence theory for random iterative dynamical systems.

Furthermore, in the paper "xxx", Hoel et al. proposed the theoretical framework of causal geometry, trying to generalize the causal emergence theory to function mappings and dynamical systems with continuous states. This article defines <math>EI</math> for random function mapping, and also introduces the concepts of intervention noise and causal geometry, and compares and analogizes this concept with information geometry. Liu Kaiwei et al. further gave an exact analytical causal emergence theory for random iterative dynamical systems.