Dirichlet Subgraphs: Theory & Applications

• Dirichlet subgraphs are defined by selecting interior and boundary vertices on a graph, where zero Dirichlet conditions enable precise spectral and potential analysis.
• They underpin spectral gap studies by leveraging the Dirichlet Laplacian and Cheeger inequalities to improve graph partitioning and clustering strategies.
• Extensions to fractional and higher-order Laplacians broaden applications in discrete potential theory, random walks, and nonlocal network analysis.

A Dirichlet subgraph is a subgraph of a larger (often infinite) graph, together with a distinguished set of boundary vertices, on which Dirichlet boundary conditions are imposed. Spectral, partitioning, and potential-theoretic problems on these subgraphs play a central role in discrete analysis, geometric graph theory, network science, and numerical methods. The Dirichlet subgraph framework underlies the analysis of Dirichlet Laplacians, Dirichlet-to-Neumann operators, minimal Dirichlet energy partitions, and generalizations to higher-order and fractional Laplacians.

1. Formal Definition and Boundary Conditions

Let $G = (V, E, w)$ be a (possibly infinite) weighted graph, with vertex set $V$, edge set $E$, and symmetric weight function $w$. A Dirichlet subgraph is typically described by a finite or infinite subset $S \subset V$ (the "interior") and a collection of "boundary" vertices $\partial S \subset V \setminus S$. The Dirichlet Laplacian $L^D$ is the restriction of the combinatorial Laplacian $L$ to $S$, with the constraint that any function $x \in \mathbb{R}^V$ in the relevant eigenproblem satisfies $x_v = 0$ for all $v \in \partial S$.

For finite graphs, $\partial S$ may simply be $V \setminus S$, or any distinguished set of vertices specified by the problem context. In infinite graphs, subtle distinctions arise among boundary points at infinity, ends, and various types of "Kiselman" boundary vertices (Perkins, 2014).

For an unweighted, finite, connected graph, the Dirichlet Laplacian $\mathcal{L}_D$ is often constructed by deleting from the Laplacian matrix all rows and columns corresponding to the boundary $\partial S$ (Tsiatas et al., 2011). For a function $u$ defined on $S$ (with $u|_{\partial S} = 0$), $(L^D u)(v) = \sum_{w \sim v} w_{vw} (u(v) - u(w))$, $v \in S$.

2. Spectral Theory of Dirichlet Subgraphs

The spectral properties of the Dirichlet Laplacian are central to the analysis of Dirichlet subgraphs. The Dirichlet spectrum, i.e., the collection of Dirichlet eigenvalues,

$$0 < \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_{|S|}$$

characterizes internal connectivity, expansion, heat dissipation, and mixing within $S$, with the Dirichlet boundary enforcing vanishing potential at $\partial S$. The smallest eigenvalue $\lambda_1$ is the Dirichlet spectral gap.

Dirichlet spectral gaps may behave radically differently from standard (Neumann) spectral gaps. In large finite graphs extracted from infinite graphs (e.g., finite balls in a tree), the standard spectral gap approaches zero, reflecting the impact of large isoperimetric "whiskers". The Dirichlet spectral gap, however, remains bounded away from zero in expanding domains, and in the infinite-volume limit recovers the spectral gap of the infinite graph (Tsiatas et al., 2011).

Table: Standard vs Dirichlet Spectral Gap in Real-World Networks (Tsiatas et al., 2011) | Dataset | Standard gap $\lambda$ | Dirichlet gap $\lambda_D$ | |---------|-----------------------|---------------------------| | 1221 | 0.00386 | 0.07616 | | 7018 | 0.00029 | 0.09531 | | 2D grid | 0.00025 | 0.00050 |

Cheeger inequalities relate the Dirichlet spectral gap to isoperimetric constants of $S$, namely,

$2 h_S \geq \lambda_D \geq \tfrac{1}{2} h_S^2$

where $h_S$ is the local Cheeger constant of $S$.

3. Dirichlet-to-Neumann Operators and Steklov Spectrum

For a Dirichlet subgraph $\Omega$, the Dirichlet-to-Neumann (DtN) operator assigns to boundary data $f$ on $\partial \Omega$ the normal derivative of the unique harmonic extension $u_f$ into the interior, evaluated at the boundary: $$\Lambda f(x) = \partial_n u_f(x) = \sum_{y \in \Omega, y \sim x} w_{xy}\,(u_f(x) - u_f(y)), \quad x \in \partial \Omega.$$ $\Lambda$ is a symmetric operator acting on $\ell^2(\partial \Omega)$, and its eigenvalues are called the Steklov spectrum. The first nontrivial Steklov eigenvalue $\lambda_2(\Omega)$ provides refined control on boundary-bottleneck phenomena.

In lattice graphs $\mathbb{Z}^n$, scaling estimates relate the first $n+1$ Steklov eigenvalues to $|\Omega|$: $$\lambda_k(\Omega) \leq C(n)\,|\Omega|^{-1/n}, \quad 2 \leq k \leq n+1,$$ as $|\Omega| \to \infty$, with the exponent $1/n$ sharp for cubic boxes (Han et al., 2019). In regular trees, $\lambda_2(\Omega)$ remains bounded below as $|\Omega| \to \infty$, illustrating the crucial difference for non-amenable underlying graphs.

For infinite graphs, Cheeger-type estimates for the DtN operator involve both interior and boundary isoperimetric constants, and higher-order Cheeger–Steklov constants $h_k(W)$, yielding two-sided polynomial bounds for higher Steklov eigenvalues (Hua et al., 2018).

4. Higher-Order, Fractional, and Poly-Laplacians on Dirichlet Subgraphs

The Dirichlet Laplacian extends to fractional and poly-Laplacian powers. The fractional Dirichlet Laplacian $(-\Delta)_D^s$ on a Dirichlet subgraph $\Omega \subset \mathbb{Z}^n$ ($0 < s < 1$) is defined either semigroup-theoretically or via the integral kernel

$$(-\Delta)_D^s u(x) = C_{n,s} \sum_{y \in \mathbb{Z}^n \setminus \{x\}} (u^*(x) - u^*(y)) |x-y|^{-n-2s}, \quad x \in \Omega,$$

where $u^*$ is the zero-extension of $u$ outside $\Omega$ (Wang, 2023).

For poly-Laplacians, $( -\Delta )^{\ell}$, the Dirichlet poly-Laplacian is defined as $(\Delta^\ell u^*)|_{V(G)}$ for $u$ vanishing on the boundary. On such operators, sharp Li–Yau–Kröger-type estimates bound the sum of the first $k$ Dirichlet eigenvalues: $$\frac{1}{k}\sum_{j=1}^k \lambda_j^{(\ell)} \leq (2\pi)^{2\ell} \frac{d}{d+2\ell}(V_d |G|)^{-2\ell/d} k^{2\ell/d} + O(|\partial^\ell G|/|G|),$$ with corresponding lower bounds, and monotonicity $\lambda_k^{(2\ell)} \geq (\lambda_k^{(\ell)})^2$ (Hua et al., 2024). For the fractional case, similar upper and lower eigenvalue sum bounds extend the classical Li–Yau and Kröger framework to the discrete, nonlocal setting (Wang, 2023).

5. Dirichlet Subgraph Partitioning and Clustering

Graph partitioning via Dirichlet energy minimization replaces cut-size or conductance criteria with objectives based on the sum of first Dirichlet eigenvalues of the components: $$\Lambda_k^* = \min_{S_1 \cup \cdots \cup S_k = V,\, S_i \cap S_j = \emptyset} \sum_{i=1}^k \lambda_1(S_i).$$ This combinatorial partition problem is NP-hard in $k$. Osting–White–Oudet introduce a relaxation in which each cluster is represented by soft indicator functions $\phi_i: V \to [0,1]$ with $\sum_{i=1}^k \phi_i(v) = 1$. The penalized eigenvalue,

$$\lambda^\alpha(\phi) = \lambda_{\min}(L + \alpha \operatorname{diag}(1-\phi)),$$

approximates the Dirichlet eigenvalue for hard partitions as $\alpha \to \infty$ (Osting et al., 2013). The rearrangement algorithm iteratively updates cluster assignments by maximizing the corresponding eigenvectors' components. The energy strictly decreases in each non-fixed iteration and converges in finitely many steps to a local minimum at a hard partition, as confirmed by a "bang–bang" theorem for local minimizers.

The same framework supports semi-supervised extensions by pinning known labels, and provides cluster representatives via maximizing eigenvector components. Empirical evaluation on synthetic, manifold, and UCI datasets shows geometric partitioning superior to perimeter-based methods, and convergence in a finite number of steps (Osting et al., 2013).

Alternative approaches employ Dirichlet spectral clustering. For the normalized Laplacian $\mathcal{L}$, removing boundary rows/columns yields $\mathcal{L}_D$; its lowest eigenvectors embed the interior, and clustering suppresses spurious "bags of whiskers" characteristic of standard spectral clustering (Tsiatas et al., 2011).

6. Dirichlet Problems, Infinite Graphs, and Potential Theory

Discrete potential theory on Dirichlet subgraphs with mixed boundary (finite plus ends) generalizes classical Dirichlet problems to infinite graphs with ends (Perkins, 2014). For a quasi-reversible infinite weighted graph with finitely many ends and suitable boundary data $f$, there exists a unique bounded harmonic function $h$ with $h|_{\partial X} = f$ and $h$ harmonic on the interior.

Constructive analytic approaches employ iterative application of the walk operator. Probabilistic representations interpret $h(x)$ as the expected value of $f$ at the exit (boundary hitting or escape to an end) for random walks. Approximation by finite Dirichlet subgraphs via exhaustion provides convergence guarantees for practical computation.

Explicit formulas for the harmonic solution are available in models such as the biased integer lattice (gambler's-ruin formula) and regular trees (Martin kernel). The approximation error is controlled by tails of the Green's function as the finite subgraph exhausts the infinite domain.

7. Applications and Further Developments

Dirichlet subgraphs find applications across spectrum-driven network analysis, geometric partitioning, communication networks, percolation, potential theory, and discrete random processes. The Dirichlet spectral gap quantifies expansion in real-world networks more robustly than standard gaps, and Dirichlet-based spectral clustering produces single-component clusters with accurate identification of network bottlenecks (Tsiatas et al., 2011). In amenable graphs, Steklov eigenvalues vanish as subgraph size increases, while in non-amenable graphs (e.g., trees), they remain strictly positive (Han et al., 2019, Hua et al., 2018).

Ongoing research extends these ideas to nonlocal fractional Laplacians, higher-order Laplacians, random walks, manifold discretizations, and graphs with complex boundary at infinity (Wang, 2023, Hua et al., 2024, Perkins, 2014). Connections with isoperimetry, harmonic analysis, and combinatorics continue to drive advances in understanding the analytic and geometric structure of general networks through the lens of Dirichlet subgraphs.