With the intensified trend of global economic integration and the integration of political economy, enterprises have entered the era of political competition. Some scholars have studied the issue of political connections from a global perspective. They found that corporate political ties are widespread in many countries. Political connections are especially prominent in countries and regions with weak property rights protection. Political success in many industries is as important as an economic success [1]. The business often has a political strategy as part of its overall strategy. Political connections have become the foundation of business success. Many enterprises pay attention to formulating and implementing political strategies while formulating market competition strategies to defeat their opponents. They access various government resources through the decision-making process that influences government policy. This can help improve business performance. The more a company has a sense of long-term development, the more emphasis will be placed on formulating and implementing political strategies. This has become an important part of strategic enterprise management.

In the context of China's transition and relationship-oriented social structure, the political connections of private enterprises are widespread. Political connections constitute an important part of the external environment of Chinese private enterprises [2]. A good relationship with the government is an important prerequisite for the success of some businesses. Through different channels, private enterprises have government resources to establish close ties with government departments at different levels. This can bring political and economic benefits. However, political ties can also increase costs for private firms and thus damage firm performance. It can be seen that political connection is a neutral concept. It can play a positive role in promoting the survival of private enterprises and have a negative impact. However, the research on political connections in the existing literature still focuses on qualitative analysis and quantitative research, failing to form a theoretical breakthrough. The current research lacks dynamic analysis on the inter-firm relationship formed under the influence of corporate political competitiveness and the flow of resources in the competitive space. And these are the problems that need to be solved urgently in the study of corporate political connections. This paper focuses on the concept of the internal mechanism, microstructure, channel characteristics, and competitive field of the politically connected competitiveness of private enterprises under the Chinese institutional environment [3]. A dynamic theoretical framework of political competitiveness is established for the first time. The evolution law of political competitiveness is analyzed. This provides a new perspective for the study of corporate political connections.

Politically connected competitiveness is an active process for private enterprises to gather effective political resources. It is also a process of organically combining subjective and objective conditions of private enterprises [4]. The degree of government control over resources, the degree of government intervention in business, and the degree of policy uncertainty directly affect the dependence of private enterprises on the government. This process takes the enterprise as the source of competition. It exerts a certain gravitational force on government agencies and their officials at different positions in social cyberspace. So we say that political competitiveness is defined in this social network space. If the competitiveness studied does not change over time, we call the field a static field. If it changes over time, we call it a dynamic field or a time-varying field. If the competitiveness understudy is scalar, we call the field a scalar field. If the competitiveness under study is a vector, we call the field a vector field.

A scalar politically connected arena can be denoted _{1}, _{2}, ⋯, _{n}_{i}_{1}, _{2}, ⋯, _{n}

A vector field of competition can be expressed as
_{i}_{i}

The annular areal density is defined as
_{n}F

We use the symbol q to represent the constituent elements of the politically connected competitiveness of private enterprises [8]. According to the specific situation of the political connection distribution of private enterprises, the political connection density can be used to describe their spatial distribution. We set the total amount of politically related elements distributed in the spatial volume element Δ

A very important concept must be raised in studying political competitiveness: the point element [9]. When the total amount of politically related elements q is concentrated on a geometric point whose spatial volume is zero, this geometric point becomes a point element q. The volume density

The total political connection factor Q in any volume V of the competition space can be given by:
_{i}

Suppose two business entities have the basic characteristics of point elements. The interaction force between two stationary enterprise entities (point elements) _{1} and _{2} in this space can be defined as
_{12} represents the force of business entity _{1} acting on a business entity _{2}. _{12} = _{2} − _{1} represents the distance vector of business entity _{1} to _{2}. _{12} = | _{12} |;

When a point element _{0} is placed at a certain point in space, the competitiveness of the point element is related to the amount of the elements it brings and the competition intensity of the point [10]. That is, there is a relation _{0}_{12} = | _{i}_{i}

This is the differential form of Gauss's theorem, and it is also an important basic equation of the politically connected field of competition in the competitive space. It reflects the relationship between the divergence of the spatial competitiveness of political associations and the distribution of spatial elements [11]. We integrate Gauss's theorem over the volume V of the competing space and use the Gaussian divergence theorem to have ∇ × _{C}

Suppose there is an element system consisting of two-point elements ±

The consistency of the orientation of elements with extreme political relevance reflects the degree of medium polarization. We describe it by the polarization vector _{i}_{1}. After the medium is polarized, there will be a macroscopic distribution of elements related to competition on its surface and inside [12]. The distribution of polarized elements is represented by the volumetric density

Take an arbitrary volume V in space. It has n element dipoles inside it. Its boundary is S. The positive element that passes through _{C} _{2} − _{1}). This formula is the boundary condition satisfied by the polarization vector

When the elements in the medium are polarized, the political competitiveness effect produced by the polarized elements in the medium is the same as the competition field produced by the main economic object of the competition field. Polarization factors also act as sources of divergence to stimulate the playing field [13]. The agglomeration effect of politically connected media can expand the intensity of the politically connected competitive field. Thus there is an ∇·

Assuming
_{S}

The static playing field is the active non-rotating playing field ∇ × ^{2}^{2}

Politically Connected Competitive Fields Field energies exist in any area of the competitive space. The ability of any business entity to utilize and integrate political resources requires the maintenance of energy. The energy density of a static competitive field in a medium can be expressed as
_{i}_{i}_{i}_{1} + _{2}, then the total energy of the competition field is
_{S}_{i}_{0} according to the energy change.

Assuming that there are N private enterprises in the competitive field, the total energy is
_{i}_{i}

Convexity theorem of information entropy in the political connection competition. The set R corresponding to the density distribution function of the competition field energy of political connection is convex. The information entropy

The principle of maximum entropy. The information entropy of the political-related competition field is an upward convex function with a maximum value.

The information entropy

China's private enterprises are not only operating in a complex and changing political and economic environment but also in the process of transitioning from a traditional planned economy to a socialist market economy with Chinese characteristics. There is still a certain distance from the mature market economy environment regarding market development and regulation. The government is still interfering with private companies at many levels. So private enterprises face more dynamic political changes. In a relationship-dominated environment, private enterprises have a strong incentive to establish political connections to enhance their competitiveness in utilizing and integrating various resources. They use political power and government resources to promote business development. We have established a dynamic analysis framework for the political competitiveness of private enterprises.

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