Graph-Based Coarsening Approach

• Graph-based coarsening is a systematic process that reduces large graphs while preserving essential structural and spectral properties for scalable computing.
• The methodology groups nodes using matching-based and AMG-inspired schemes that leverage algebraic distance to guide effective edge contractions.
• Empirical findings show that integrating algebraic distance enhances partition quality and balance, optimizing performance for applications in scientific computing and machine learning.

A graph-based coarsening approach refers to the systematic reduction of a large graph into a smaller surrogate while preserving its essential structural and spectral properties. This transformation is typically applied within multilevel frameworks for graph partitioning, scientific computing, and machine learning, providing computational speedups and scalable downstream processing. The methodology consists of defining algorithmic strategies for grouping nodes, contracting edges, or aggregating subgraphs, all while employing measures that assess and guarantee the fidelity of the reduction in terms of cut, balance, and spectral similarity.

1. Multilevel Graph Coarsening Frameworks

Multilevel graph partitioning is an established paradigm in which an original graph $G_0$ is systematically reduced through a sequence of coarsened graphs $G_1, G_2, \ldots, G_k$. The purpose of the coarsening phase is to produce compressed graphs that are structurally similar to $G_0$ (Safro et al., 2012). The partitioning or optimization problem is solved on the coarsest graph, with the solution progressively projected (uncoarsened) back to finer levels. This design dramatically improves computational tractability for large-scale graphs, since both refinement and partitioning at coarser levels require fewer resources.

Coarsening identifies aggregates—groups of nodes to be merged—by exploiting either neighborhood proximities, matching algorithms, or spectral properties. Each aggregate forms a super-node in the coarsened graph, with inter-aggregate edges reflecting the sum or merging of connections from the original graph.

2. Matching-Based and AMG-Inspired Schemes

Two principal classes of coarsening algorithms are employed:

Matching-based coarsening constructs a matching $M \subset E$ (a set of edges with no shared endpoints), and contracts each matched edge. The node weights after contraction are

$c(x) = c(u) + c(v)$

(where $x$ is the contracted node from edge $\{u,v\}$). Edge weights incident to the same neighbor are merged by summing (Safro et al., 2012). High-quality matchings are found by maximizing edge rating functions, such as:

• $$\text{expansion}^{(*2)}(u,v) = \frac{\omega(\{u,v\})^2}{c(u) \cdot c(v)}$$
• $$\text{innerOuter}(u,v) = \frac{\omega(\{u,v\})}{\mathrm{Out}(u) + \mathrm{Out}(v) - 2\omega(\{u,v\})}$$, where $\mathrm{Out}(v)$ is the sum of weights of all edges incident to $v$.

The Global Paths Algorithm (GPA) and RandomGPA combine greedy edge ordering with dynamic path optimization for further matching quality improvements.

AMG (Algebraic Multigrid)-inspired coarsening relies on constructing a coarse graph via a coarse-fine mapping (projection matrix $P$) and projects the Laplacian:

$L_c = P L_f P^T,$

where $L_f$ is the fine-level Laplacian (Safro et al., 2012). The volume-normalized Laplacian $\tilde{L}$ is constructed to account for vertex balance. Coarse nodes are selected via a dominating set $C$, with node $i$ remaining fine if

$$\sum_{j \in C} \frac{1}{\rho_{ij}} \geq \Theta \sum_{j \in V_f} \frac{1}{\rho_{ij}},$$

where $\rho_{ij}$ is the algebraic distance reflecting the connectivity strength. The final interpolation matrix $P$ assigns fine nodes (possibly fractionally) to coarse aggregates using algebraic distance-based weights. Aggregates are limited in order (number of fine nodes assigned to each coarse node) to maintain both balance and compatibility with partitioning.

3. Role of Algebraic Distance

Algebraic distance, introduced as a key connectivity strength measure, is fundamental to both matching-based and AMG-type coarsening. It is computed through iterative relaxation (Jacobi overrelaxation) applied to random test vectors:

$H = (1-\alpha)I + \alpha \tilde{D}^{-1} \tilde{W}$

with test vectors relaxed via

$\chi^{k,r} = H^k \chi^{0,r}.$

The algebraic distance between nodes $i$ and $j$ is

$$\rho_{ij} = \left(\sum_{r=1}^R |\chi^{k,r}_i - \chi^{k,r}_j|^2 \right)^{1/2}$$

(Safro et al., 2012). This metric prevents early contractions of weakly coupled edges—those likely to span partitions—thus improving both the structural quality and balance of the partition.

In AMG-based schemes, algebraic distance both determines selection of seeds for coarsening and determines the interpolation weights in $P$. For matching-based coarsening, algebraic distance may be incorporated into the edge weighting/rating functions (e.g., by dividing edge expansion by $\rho_e$).

4. Spectral Considerations and Quality Metrics

Spectral properties, primarily derived from the graph Laplacian’s eigenvalues and eigenvectors, serve as proxies for the graph’s global and local structure. The aim is to maintain similarity between the spectrum of the original and the coarsened graph—ensuring preservation of community structure, cuts, and diffusion properties.

Spectral distances, such as the full and partial distances:

$$SD_{full}(G, G_c) = \sum_{i=1}^N |\lambda(i) - \lambda_l(i)|,\quad SD_{part}(G, G_c) = \sum_{i=1}^k |\lambda(i) - \lambda_c(i)| + \sum_{i=k+1}^n |\lambda_c(i) - \lambda(i+N-n)|,$$

provide theoretical guarantees for the similarity in structure between the original and coarsened graphs (Jin et al., 2018). These metrics guide algorithm design and post-hoc quality evaluation.

5. Computational Trade-Offs and Empirical Findings

Empirical evidence demonstrates substantial trade-offs in coarsening strategies (Safro et al., 2012):

• Incorporating algebraic distance (in AMG-ECO, ECO-ALG) yields consistently better partition quality, especially in irregular, scale-free, or “hard” instances.
• AMG-inspired schemes often lead to slightly longer coarsening times due to relaxation for algebraic distance computation; however, this overhead is insignificant compared to the total (including refinement), especially as refinement on denser coarse graphs dominates the runtime.
• For regular finite-element or VLSI graphs, matching-based and AMG methods perform comparably, though algebraic distance-enhanced configurations sometimes yield superior quality.

The integration of algebraic distance is particularly effective at avoiding premature merging of structurally remote regions—a critical property in preserving partition boundaries and ensuring scalable, high-quality solutions.

6. Extensions and Broader Impact

Graph-based coarsening is broadly applicable in:

• Partitioning and load balancing for scientific computations
• Preprocessing for faster graph embedding in machine learning
• Constructing hierarchical abstractions for scalable graph neural network (GNN) training

Contemporary research builds on these foundations via optimization-based coarsening (jointly learning projection and feature reduction) (Kumar et al., 2022), data-driven neural network parameterizations for edge weight assignment (Cai et al., 2021), and loss-aware approaches crafted for downstream tasks in ML and scientific computing.

Modern frameworks emphasize the necessity of preserving spectral, flow-based, and topological properties, with coarsening algorithms tailored to heterogeneous or dynamic graph scenarios and optimized for regularization in large-scale GNNs.

7. Conclusion

Graph-based coarsening approaches combine algorithmic rigor, balance constraints, spectral preservation, and computational efficiency to address the challenges of large graph processing. Matching-based and AMG-inspired coarsening schemes, particularly when guided by algebraic distance, provide quantifiable improvements in partitioning quality and runtime over traditional methods, with adaptability to graph heterogeneity and complexity. The continued development of coarsening algorithms remains central to scalable algorithms and systems in scientific computing, network analysis, and modern machine learning.