inGRASS: Incremental Graph Spectral Sparsification via Low-Resistance-Diameter Decomposition

This work presents inGRASS, a novel algorithm designed for incremental spectral sparsification of large undirected graphs. The proposed inGRASS algorithm is highly scalable and parallel-friendly, having a nearly-linear time complexity for the setup phase and the ability to update the spectral sparsifier in $O(\log N)$ time for each incremental change made to the original graph with $N$ nodes. A key component in the setup phase of inGRASS is a multilevel resistance embedding framework introduced for efficiently identifying spectrally-critical edges and effectively detecting redundant ones, which is achieved by decomposing the initial sparsifier into many node clusters with bounded effective-resistance diameters leveraging a low-resistance-diameter decomposition (LRD) scheme. The update phase of inGRASS exploits low-dimensional node embedding vectors for efficiently estimating the importance and uniqueness of each newly added edge. As demonstrated through extensive experiments, inGRASS achieves up to over $200 \times$ speedups while retaining comparable solution quality in incremental spectral sparsification of graphs obtained from various datasets, such as circuit simulations, finite element analysis, and social networks.

SAGMAN: Stability Analysis of Graph Neural Networks on the Manifolds

Modern graph neural networks (GNNs) can be sensitive to changes in the input graph structure and node features, potentially resulting in unpredictable behavior and degraded performance. In this work, we introduce a spectral framework known as SAGMAN for examining the stability of GNNs. This framework assesses the distance distortions that arise from the nonlinear mappings of GNNs between the input and output manifolds: when two nearby nodes on the input manifold are mapped (through a GNN model) to two distant ones on the output manifold, it implies a large distance distortion and thus a poor GNN stability. We propose a distance-preserving graph dimension reduction (GDR) approach that utilizes spectral graph embedding and probabilistic graphical models (PGMs) to create low-dimensional input/output graph-based manifolds for meaningful stability analysis. Our empirical evaluations show that SAGMAN effectively assesses the stability of each node when subjected to various edge or feature perturbations, offering a scalable approach for evaluating the stability of GNNs, extending to applications within recommendation systems. Furthermore, we illustrate its utility in downstream tasks, notably in enhancing GNN stability and facilitating adversarial targeted attacks.

HyperEF: Spectral Hypergraph Coarsening by Effective-Resistance Clustering

This paper introduces a scalable algorithmic framework (HyperEF) for spectral coarsening (decomposition) of large-scale hypergraphs by exploiting hyperedge effective resistances. Motivated by the latest theoretical framework for low-resistance-diameter decomposition of simple graphs, HyperEF aims at decomposing large hypergraphs into multiple node clusters with only a few inter-cluster hyperedges. The key component in HyperEF is a nearly-linear time algorithm for estimating hyperedge effective resistances, which allows incorporating the latest diffusion-based non-linear quadratic operators defined on hypergraphs. To achieve good runtime scalability, HyperEF searches within the Krylov subspace (or approximate eigensubspace) for identifying the nearly-optimal vectors for approximating the hyperedge effective resistances. In addition, a node weight propagation scheme for multilevel spectral hypergraph decomposition has been introduced for achieving even greater node coarsening ratios. When compared with state-of-the-art hypergraph partitioning (clustering) methods, extensive experiment results on real-world VLSI designs show that HyperEF can more effectively coarsen (decompose) hypergraphs without losing key structural (spectral) properties of the original hypergraphs, while achieving over $70\times$ runtime speedups over hMetis and $20\times$ speedups over HyperSF.

HyperSF: Spectral Hypergraph Coarsening via Flow-based Local Clustering

Hypergraphs allow modeling problems with multi-way high-order relationships. However, the computational cost of most existing hypergraph-based algorithms can be heavily dependent upon the input hypergraph sizes. To address the ever-increasing computational challenges, graph coarsening can be potentially applied for preprocessing a given hypergraph by aggressively aggregating its vertices (nodes). However, state-of-the-art hypergraph partitioning (clustering) methods that incorporate heuristic graph coarsening techniques are not optimized for preserving the structural (global) properties of hypergraphs. In this work, we propose an efficient spectral hypergraph coarsening scheme (HyperSF) for well preserving the original spectral (structural) properties of hypergraphs. Our approach leverages a recent strongly-local max-flow-based clustering algorithm for detecting the sets of hypergraph vertices that minimize ratio cut. To further improve the algorithm efficiency, we propose a divide-and-conquer scheme by leveraging spectral clustering of the bipartite graphs corresponding to the original hypergraphs. Our experimental results for a variety of hypergraphs extracted from real-world VLSI design benchmarks show that the proposed hypergraph coarsening algorithm can significantly improve the multi-way conductance of hypergraph clustering as well as runtime efficiency when compared with existing state-of-the-art algorithms.