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In 2020, Klein and Hoel improved the method of quantifying causal emergence on Markov chains to apply to [[complex networks]] <ref>Klein B, Hoel E. The emergence of informative higher scales in complex networks[J]. Complexity, 2020, 20201-12.</ref>. The authors defined the [[Markov chain]] in the network with the help of [[random walkers]]. Placing random walkers on nodes is equivalent to intervening on nodes. They then defined the transition probability matrix between nodes based on the random walk probability. At the same time, the authors establish a connection between EI and the connectivity of the network. Connectivity is characterized by the uncertainty in the weights of outgoing and incoming edges of the nodes. Based on this, the EI in complex networks is defined. For detailed methods, refer to [[causal emergence in complex networks]].

In 2020, Klein and Hoel improved the method of quantifying causal emergence on Markov chains to apply to [[complex networks]] <ref>Klein B, Hoel E. The emergence of informative higher scales in complex networks[J]. Complexity, 2020, 20201-12.</ref>. The authors defined the [[Markov chain]] in the network with the help of [[random walkers]]. Placing random walkers on nodes is equivalent to intervening on nodes. They then defined the transition probability matrix between nodes based on the random walk probability. At the same time, the authors establish a connection between EI and the connectivity of the network. Connectivity is characterized by the uncertainty in the weights of outgoing and incoming edges of the nodes. Based on this, the EI in complex networks is defined. For detailed methods, refer to [[causal emergence in complex networks]].

The authors conducted experimental comparisons in artificial networks such as [[random network]] (ER), the [[preferential attachment network model]] (PA), and four types of real networks. They found that, for ER networks, the degree of EI only depends on the connection probability <math>p</math>, and as the network size increases, it will converge to the value <math>-\log_2p</math>. At the same time, a key finding shows that there is a phase transition point in the EI value, which approximately appears at the position where the [[average degree]] (<math><k></math>) of the network is equal to <math>\log_2N</math>. This also corresponds to the phase transition point where the random network structure does not contain more information as its scale increases with the increase of the connection probability. For preferential attachment model networks, the EI depends on the power-law exponent α of the network's degree distribution. When <math>\alpha<1.0</math>, the degree of EI increases as the network size grows. Conversely, when <math>\alpha>1.0</math>, the degree of EI decreases with network size. The case of <math>\alpha = 1.0</math> corresponds to the scale-free network, making the critical boundary for network growth. For real networks, the authors found that biological networks have the lowest EI because they exhibit significant noice. However, we can remove this noise through effective coarse-graining, which makes biological networks show more significant causal emergence phenomena than other types of networks. On the other hand, technical type networks are sparser and non-degenerate, resulting in higher average efficiency, more specific node relationships, and the highest EI. However, it is challenging to increase the degree of causal emergence in these networks through coarse-graining.

The authors conducted experimental comparisons in artificial networks such as [[random network]] (ER), the [[preferential attachment network model]] (PA), and four types of real networks. They found that, for ER networks, the degree of EI only depends on the connection probability <math>p</math>, and as the network size increases, it will converge to the value <math>-\log_2p</math>. At the same time, a key finding shows that there is a phase transition point in the EI value, which approximately appears at the position where the [[average degree]] (<math><k></math>) of the network is equal to <math>\log_2N</math>. This also corresponds to the phase transition point where the random network structure does not contain more information as its scale increases with the increase of the connection probability. For preferential attachment model networks, the EI depends on the power-law exponent α of the network's degree distribution. When <math>\alpha<1.0</math>, the degree of EI increases as the network size grows. Conversely, when <math>\alpha>1.0</math>, the degree of EI decreases with network size. The case of <math>\alpha = 1.0</math> corresponds to the scale-free network, making the critical boundary for network growth. For real networks, the authors found that biological networks have the lowest EI because they exhibit significant noise. However, we can remove this noise through effective coarse-graining, which makes biological networks show more significant causal emergence phenomena than other types of networks. On the other hand, technical type networks are sparser and non-degenerate, resulting in higher average efficiency, more specific node relationships, and the highest EI. However, it is challenging to increase the degree of causal emergence in these networks through coarse-graining.

In this article, the authors use the [[greedy algorithm]] to coarse-grain the network. However, for large-scale networks, this algorithm is very inefficient. Subsequently, Griebenow et al. <ref>Griebenow R, Klein B, Hoel E. Finding the right scale of a network: efficient identification of causal emergence through spectral clustering[J]. arXiv preprint arXiv:190807565, 2019.</ref> proposed a method based on [[spectral clustering]] to identify causal emergence in [[preferential attachment networks]]. Compared with the [[greedy algorithm]] and the [[gradient descent algorithm]], the [[spectral clustering]] algorithm has less computation time and the causal emergence of the found macroscopic network is also more significant.

In this article, the authors use the [[greedy algorithm]] to coarse-grain the network. However, for large-scale networks, this algorithm is very inefficient. Subsequently, Griebenow et al. <ref>Griebenow R, Klein B, Hoel E. Finding the right scale of a network: efficient identification of causal emergence through spectral clustering[J]. arXiv preprint arXiv:190807565, 2019.</ref> proposed a method based on [[spectral clustering]] to identify causal emergence in [[preferential attachment networks]]. Compared with the [[greedy algorithm]] and the [[gradient descent algorithm]], the [[spectral clustering]] algorithm has less computation time and the causal emergence of the found macroscopic network is also more significant.