The network illustrates the interrelationships among diverse entities and has attracted considerable attention. A wide range of applications makes the network a popular modeling tool, including but not limited to social networks [1], business administration [2], and city construction [3]. In these fields, quantifying network value is essential. By examining network value, companies gain insights into consumer behavior, allowing them to refine marketing strategies accordingly [4]. Similarly, investigating network value can assist urban planners in better understanding the utilization and demands for public facilities, ultimately optimizing city construction [5]. To better understand network value, researchers have established some influential laws that describe network value in terms of the number of neighbors, edges, and subgraphs.Sarnoff 's Law. In the 1940s, David Sarnoff proposed that the value of a broadcasting network is directly proportional to the number of nodes (audience), which is also known as Sarnoff's law [6]. For example, in the case of TV programs, the network value increases linearly with the number of the audience, because the growing number of audiences allows advertisers to access more potential customers. This results in more advertisers, increased revenues, and higher broadcast media demand. Formally, for a network with n nodes, the communication value is Θ(n). This law was initially applied to the film industry, later extended to television, and usually represents one-way communication. Broadcasting can only transmit messages unidirectionally to users, but cannot spread information within users.Metcalfe's Law. With the conferral of the Turing Award on Robert Metcalfe in 2022, Metcalfe's law[7] has regained attention. Metcalfe's law was proposed on the background of the increasing number of Ethernet users and growing attention to the interconnection value of networks. Metcalfe argues that if the value of each node (terminal) is equal, the value of a network is proportional to the number of edges. One of the classic illustrations of Metcalfe's law lies in communication networks, where a network with n users can provide interconnection value proportional to approximately n 2 , i.e. Θ(n 2 ), as each user can communicate with the other n − 1 users in the network. Despite the many challenges associated with the development of the network, such as network scale, connection quality, and network design, Metcalfe's law remains applicable in some scenarios, especially in the field of cloud computing [8]. The increasing number of cloud computing users leads to the availability of more resources, which in turn attracts additional users, creating a virtuous cycle of growth. As a theory for describing the interconnection value of networks, Metcalfe's law is held in high regard and possesses significant implications for communication networks, the Internet, and social networks.Reed's Law. In 2000, David Reed proposed Reed's law [9] for group-forming networks (GFNs). He argued that network value depends not ...