3.2.6 Equilibrium Solution of the Urban Value Field

The governing equations presented in the previous sections describe the mechanisms through which urban value is generated and propagated within an urban system. Under equilibrium conditions, the Urban Value Field can be represented by a closed-form expression that integrates intrinsic structural characteristics, background value propagation, and local interaction effects.

The normalized urban land value at location xxx is expressed as

Vgt(x)=(∏Si(x))⋅ϕ(dx)⋅μ(x)V_{gt}(x) = \left( \prod S_i(x) \right) \cdot \phi(d_x) \cdot \mu(x)Vgt​(x)=(∏Si​(x))⋅ϕ(dx​)⋅μ(x)

where

• ∏Si(x)\prod S_i(x)∏Si​(x) represents the intrinsic structural component of location xxx;

• ϕ(dx)\phi(d_x)ϕ(dx​) denotes the background propagation field originating from the urban core;

• μ(x)\mu(x)μ(x) represents local interaction effects generated by secondary value sources.

This equation describes the equilibrium distribution of urban value resulting from the combined influence of structural characteristics and spatial interactions.

Relationship with the Propagation Equation

The equilibrium expression above represents the final state of the Urban Value Field.

To explain how this equilibrium is formed, a propagation equation must be introduced.

Discrete Propagation Equation

Considering an urban network consisting of interconnected nodes linked by transportation infrastructure, the value at node iii can be represented as

Vi=Ai+∑j∈N(i)wijVjV_i = A_i + \sum_{j\in N(i)} w_{ij}V_jVi​=Ai​+j∈N(i)∑​wij​Vj​

where

• ViV_iVi​ is the urban value at node iii;

• N(i)N(i)N(i) denotes the neighboring nodes;

• wijw_{ij}wij​ is the propagation coefficient from node jjj to node iii;

• AiA_iAi​ is the intrinsic value source.

The intrinsic source is defined as

Ai=(∏Si)ϕ(di)μ(i)A_i = \left( \prod S_i \right) \phi(d_i) \mu(i)Ai​=(∏Si​)ϕ(di​)μ(i)

Thus, the equilibrium solution presented previously is interpreted as the steady-state solution of the discrete propagation process.

Matrix Representation

For the entire urban system, the propagation equation can be written in matrix form as

V=A+WV\mathbf{V} = \mathbf{A} + \mathbf{W}\mathbf{V}V=A+WV

which leads to

V=(I−W)−1A\mathbf{V} = (\mathbf{I}-\mathbf{W})^{-1}\mathbf{A}V=(I−W)−1A

where

• W\mathbf{W}W is the urban propagation matrix;

• A\mathbf{A}A is the source vector composed of structural, background, and local interaction components;

• V\mathbf{V}V is the equilibrium Urban Value Field.

This formulation provides an efficient framework for numerical computation, network simulation, and artificial intelligence applications.

Continuous Field Approximation

When the urban system is approximated as a continuous spatial domain, the Urban Value Field may be described by a diffusion–attenuation equation

−∇⋅(k∇V)+λV=A(x)-\nabla\cdot(k\nabla V)+\lambda V=A(x)−∇⋅(k∇V)+λV=A(x)

where

• V(x)V(x)V(x) is the Urban Value Field;

• kkk is the spatial propagation coefficient;

• λ\lambdaλ is the attenuation coefficient;

• A(x)A(x)A(x) is the distributed value source.

The Laplacian operator represents the spatial diffusion of urban value, while the attenuation term describes the gradual loss of influence with distance or spatial resistance.

Relationship Among Mathematical Representations

The four mathematical representations describe the same Urban Value Field at different levels of abstraction.

Representation

Scientific Meaning

Primary Application

Equilibrium solution

Closed-form value expression

Urban land valuation

Discrete propagation equation

Dynamic propagation on urban networks

Empirical modeling

Matrix formulation

System-wide computation

Simulation and AI

Continuous field equation

Continuous theoretical model

Mathematical analysis

These formulations are mathematically consistent and complementary. The equilibrium equation represents the steady-state solution of the propagation process, while the discrete, matrix, and continuous forms provide alternative representations for computation, simulation, and theoretical analysis.