11.2 Harmonic functions on graphs

Harmonic functions on graphs are discrete versions of continuous harmonic functions, satisfying a mean value property on graph vertices. They're crucial in discrete potential theory, with applications in conformal mappings and spectral graph theory.

These functions are defined using the graph Laplacian operator, which measures differences between a vertex's function value and its neighbors' average. Key properties include maximum and minimum principles, Harnack inequality, and Liouville's theorem, paralleling continuous harmonic functions.

Definition of harmonic functions on graphs

• Harmonic functions on graphs are a discrete analog of harmonic functions in continuous settings, capturing the notion of functions that satisfy a mean value property on the vertices of a graph
• They play a crucial role in discrete potential theory and have applications in various areas of mathematics and computer science, such as discrete conformal mappings and spectral graph theory

Laplacian operator on graphs

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• The Laplacian operator on a graph is a discrete analog of the continuous Laplacian operator, which measures the difference between a function's value at a vertex and the average of its values at neighboring vertices
• For a graph $G = (V, E)$ and a function $f: V \to \mathbb{R}$, the Laplacian of $f$ at a vertex $v$ is defined as: $\Delta f(v) = \sum_{u \sim v} (f(u) - f(v))$ where $u \sim v$ denotes that $u$ and $v$ are adjacent vertices
• The Laplacian matrix $L$ of a graph is a square matrix representing the Laplacian operator, with entries: \text{deg}(v_i) & \text{if } i = j \\ -1 & \text{if } i \neq j \text{ and } v_i \sim v_j \\ 0 & \text{otherwise} \end{cases}$$ where $\text{deg}(v_i)$ is the degree of vertex $v_i$

Local mean value property

• A function $f: V \to \mathbb{R}$ on a graph $G = (V, E)$ is said to be harmonic if it satisfies the local mean value property at every vertex $v \in V$: $f(v) = \frac{1}{\text{deg}(v)} \sum_{u \sim v} f(u)$
• Equivalently, a function is harmonic if $\Delta f(v) = 0$ for all $v \in V$, i.e., it is in the kernel of the Laplacian operator
• The local mean value property states that the value of a harmonic function at a vertex is the average of its values at the neighboring vertices, analogous to the mean value property for continuous harmonic functions

Existence and uniqueness of harmonic functions

• The existence and uniqueness of harmonic functions on graphs are closely related to the solvability of the Dirichlet problem and the properties of the Laplacian operator
• Understanding the conditions under which harmonic functions exist and are unique is crucial for the study of discrete potential theory and its applications

Dirichlet problem on graphs

• The Dirichlet problem on a graph $G = (V, E)$ with boundary $\partial V \subset V$ asks for a function $f: V \to \mathbb{R}$ that is harmonic on the interior vertices $V \setminus \partial V$ and takes prescribed values on the boundary vertices $\partial V$
• The existence and uniqueness of solutions to the Dirichlet problem depend on the connectivity of the graph and the prescribed boundary values
• For connected graphs, the Dirichlet problem has a unique solution if and only if the prescribed boundary values are feasible, i.e., they can be extended to a function on the entire graph

Comparison with continuous case

• The existence and uniqueness results for harmonic functions on graphs have similarities with the continuous case, but there are also some notable differences
• In the continuous setting, the Dirichlet problem for harmonic functions is well-posed for bounded domains with smooth boundaries, and the maximum principle ensures uniqueness
• On graphs, the discrete maximum principle holds under certain conditions (e.g., connected graphs), but the boundary conditions and the graph structure play a more significant role in determining the existence and uniqueness of harmonic functions

Properties of harmonic functions on graphs

• Harmonic functions on graphs satisfy various important properties that are analogous to those of continuous harmonic functions, such as the maximum and minimum principles, Harnack inequality, and Liouville's theorem
• These properties provide insights into the behavior of harmonic functions and are useful in the study of discrete potential theory and its applications

Maximum and minimum principles

• The maximum and minimum principles state that a harmonic function on a connected graph attains its maximum and minimum values on the boundary vertices
• Specifically, if $f: V \to \mathbb{R}$ is harmonic on a connected graph $G = (V, E)$ with boundary $\partial V$, then: $\min_{v \in \partial V} f(v) \leq f(v) \leq \max_{v \in \partial V} f(v) \quad \forall v \in V$
• The maximum and minimum principles are useful for proving uniqueness results and for understanding the behavior of harmonic functions on graphs

Harnack inequality on graphs

• The Harnack inequality is a quantitative version of the maximum principle, providing an upper bound for the ratio of the values of a non-negative harmonic function at two vertices
• For a connected graph $G = (V, E)$ and a non-negative harmonic function $f: V \to \mathbb{R}$, the Harnack inequality states that: $f(v) \leq C f(u) \quad \forall v, u \in V$ where $C$ is a constant depending on the graph structure and the choice of vertices $v$ and $u$
• The Harnack inequality is a powerful tool for studying the regularity and growth properties of harmonic functions on graphs

Liouville's theorem for graphs

• Liouville's theorem states that bounded harmonic functions on infinite graphs must be constant, under certain conditions on the graph structure
• Specifically, if $G = (V, E)$ is an infinite, connected graph with bounded degree, and $f: V \to \mathbb{R}$ is a bounded harmonic function, then $f$ is constant
• Liouville's theorem for graphs is analogous to the classical Liouville's theorem for continuous harmonic functions on Euclidean spaces
• The theorem has important implications for the study of harmonic functions on infinite graphs and their asymptotic behavior

Discrete Green's functions

• Discrete Green's functions are fundamental objects in discrete potential theory, serving as the discrete analog of the continuous Green's functions
• They play a crucial role in solving the discrete Poisson equation, representing the response of the Laplacian operator to a unit impulse, and have deep connections to harmonic functions on graphs

Definition and properties

• For a connected graph $G = (V, E)$ with boundary $\partial V$, the discrete Green's function $G: V \times V \to \mathbb{R}$ is defined as the solution to the following boundary value problem: \Delta_v G(v, u) = \delta_{vu} & \forall v \in V \setminus \partial V \\ G(v, u) = 0 & \forall v \in \partial V \end{cases}$$ where $\Delta_v$ denotes the Laplacian operator with respect to the variable $v$, and $\delta_{vu}$ is the Kronecker delta (1 if $v = u$, 0 otherwise)
• The discrete Green's function satisfies properties such as symmetry ($G(v, u) = G(u, v)$), positivity (for $v, u \in V \setminus \partial V$), and the reproducing property ($\Delta_v G(v, u) = \delta_{vu}$)

Relationship to harmonic functions

• Discrete Green's functions are closely related to harmonic functions on graphs
• Given a harmonic function $f: V \to \mathbb{R}$ on a connected graph $G = (V, E)$ with boundary $\partial V$, the Green's representation formula states that: $f(v) = \sum_{u \in \partial V} f(u) \cdot \frac{\partial G}{\partial n_u}(v, u) \quad \forall v \in V \setminus \partial V$ where $\frac{\partial G}{\partial n_u}(v, u)$ is the discrete normal derivative of the Green's function at the boundary vertex $u$
• This formula expresses a harmonic function in terms of its boundary values and the discrete Green's function, highlighting the deep connection between these two concepts

Computation of discrete Green's functions

• Computing discrete Green's functions is an important task in discrete potential theory and its applications
• For finite graphs, the discrete Green's function can be computed by solving a linear system of equations involving the Laplacian matrix and the boundary conditions
• Efficient numerical methods, such as the Cholesky decomposition or the conjugate gradient method, can be employed to solve this linear system and obtain the discrete Green's function
• For infinite graphs, the computation of discrete Green's functions is more challenging and often requires techniques from spectral graph theory or the study of random walks on graphs

Discrete Poisson equation

• The discrete Poisson equation is a fundamental problem in discrete potential theory, serving as the discrete analog of the continuous Poisson equation
• It is closely related to harmonic functions on graphs and has numerous applications in mathematics, physics, and computer science

Formulation and solvability

• For a connected graph $G = (V, E)$ with boundary $\partial V$ and a function $f: V \to \mathbb{R}$, the discrete Poisson equation asks for a function $u: V \to \mathbb{R}$ that satisfies: \Delta u(v) = f(v) & \forall v \in V \setminus \partial V \\ u(v) = g(v) & \forall v \in \partial V \end{cases}$$ where $g: \partial V \to \mathbb{R}$ is a prescribed boundary function
• The solvability of the discrete Poisson equation depends on the compatibility of the boundary conditions and the right-hand side function $f$
• For connected graphs, the discrete Poisson equation has a unique solution if and only if the prescribed boundary values and the right-hand side function satisfy a discrete version of the Fredholm alternative

Connection to harmonic functions

• The discrete Poisson equation is intimately connected to harmonic functions on graphs
• If $f \equiv 0$ in the discrete Poisson equation, the problem reduces to finding a harmonic function with prescribed boundary values, i.e., the Dirichlet problem for harmonic functions
• The solution to the discrete Poisson equation can be expressed using the discrete Green's function and the Green's representation formula, highlighting the connection between these concepts

Numerical methods for solving discrete Poisson equation

• Solving the discrete Poisson equation efficiently is crucial for many applications in discrete potential theory and related fields
• Numerical methods for solving the discrete Poisson equation include direct methods, such as Gaussian elimination or Cholesky decomposition, and iterative methods, such as Jacobi, Gauss-Seidel, or multigrid methods
• The choice of the numerical method depends on the size and structure of the graph, the sparsity of the Laplacian matrix, and the desired accuracy and efficiency of the solution
• Exploiting the graph structure and using techniques from graph algorithms and linear algebra can significantly improve the performance of numerical solvers for the discrete Poisson equation

Applications of harmonic functions on graphs

• Harmonic functions on graphs have numerous applications in various fields, including discrete potential theory, discrete conformal mappings, and discrete minimal surfaces
• These applications demonstrate the power and versatility of the concept of harmonic functions in the discrete setting

Discrete potential theory

• Discrete potential theory is the study of harmonic functions, Green's functions, and related concepts on graphs, serving as a discrete analog of classical potential theory
• Applications of discrete potential theory include:
  • Solving boundary value problems on graphs, such as the Dirichlet and Neumann problems
  • Studying random walks and electrical networks on graphs
  • Investigating the spectra of graph Laplacians and their connections to graph properties
• Discrete potential theory provides a framework for understanding the behavior of harmonic functions on graphs and their role in various mathematical and physical problems

Discrete conformal mappings

• Discrete conformal mappings are a discrete analog of continuous conformal mappings, aiming to preserve angles and local structure when mapping between graphs
• Harmonic functions play a crucial role in the construction and study of discrete conformal mappings
• Applications of discrete conformal mappings include:
  • Surface parameterization and mesh processing in computer graphics
  • Discrete complex analysis and the study of discrete analytic functions
  • Discrete Riemann surfaces and their applications in geometry and topology
• The theory of harmonic functions on graphs provides a foundation for the development and analysis of discrete conformal mappings and their properties

Discrete minimal surfaces

• Discrete minimal surfaces are a discrete analog of continuous minimal surfaces, representing surfaces with minimal area subject to certain boundary conditions
• Harmonic functions are closely related to discrete minimal surfaces, as they arise as the coordinate functions of such surfaces
• Applications of discrete minimal surfaces include:
  • Modeling and visualization of soap films and bubbles
  • Geometry processing and mesh optimization in computer graphics
  • Study of discrete curvature and Gaussian curvature on polyhedral surfaces
• The connection between harmonic functions and discrete minimal surfaces allows for the use of techniques from discrete potential theory in the study and computation of these surfaces

Generalizations and related topics

• The theory of harmonic functions on graphs can be generalized and extended in various directions, leading to new research areas and connections with other fields
• Some notable generalizations and related topics include harmonic functions on weighted graphs, infinite graphs, and connections to spectral graph theory

Harmonic functions on weighted graphs

• Weighted graphs are graphs with edge weights representing the strength or importance of the connections between vertices
• The concept of harmonic functions can be extended to weighted graphs by modifying the Laplacian operator and the mean value property to account for the edge weights
• Harmonic functions on weighted graphs have applications in:
  • Modeling diffusion processes and heat transfer on networks
  • Studying electrical networks with varying resistances
  • Investigating the spectra of weighted graph Laplacians and their relation to graph properties
• The theory of harmonic functions on weighted graphs provides a more general framework for discrete potential theory and its applications

Harmonic functions on infinite graphs

• Infinite graphs are graphs with an infinite number of vertices, edges, or both
• The study of harmonic functions on infinite graphs requires additional tools and techniques from functional analysis and spectral theory
• Some important topics in the study of harmonic functions on infinite graphs include:
  • Existence and uniqueness of bounded harmonic functions
  • Behavior of harmonic functions at infinity and their relation to the graph structure
  • Connections to random walks and Brownian motion on infinite graphs
• The theory of harmonic functions on infinite graphs has applications in statistical physics, percolation theory, and the study of infinite electrical networks

Connections to spectral graph theory

• Spectral graph theory studies the properties and behavior of graphs using the eigenvalues and eigenvectors of graph matrices, such as the adjacency matrix or the Laplacian matrix
• Harmonic functions on graphs are closely related to the eigenfunctions of the graph Laplacian, as they are in the kernel of the Laplacian operator
• Some important connections between harmonic functions and spectral graph theory include:
  • Characterization of harmonic functions using the eigenvalues and eigenvectors of the Laplacian
  • Relation between the spectra of the Laplacian and the existence and uniqueness of harmonic functions
  • Applications of spectral techniques in the computation and approximation of harmonic functions
• The interplay between harmonic functions and spectral graph theory provides a rich source of ideas and techniques for the study of graphs and their properties