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

Ali Aghdaei, Zhuo Feng

TL;DR

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.

Abstract

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.

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

TL;DR

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.

Abstract

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 time for each incremental change made to the original graph with 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 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.
Paper Structure (22 sections, 3 theorems, 6 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 22 sections, 3 theorems, 6 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Spectral Graph Theory
  4. Spectral Graph Decomposition
  5. Effective Resistance
  6. inGRASS: Incremental Graph Sparsification
  7. Overview of inGRASS Algorithm
  8. Setup Phase
  9. Update Phase
  10. The Setup Phase of inGRASS
  11. Scalable Estimation of Effective Resistances
  12. Multilevel Resistance Embedding
  13. Multilevel Sparse Data Structure
  14. The Update Phase of inGRASS
  15. Spectral Distortion Estimation
  16. ...and 7 more sections

Key Result

Lemma 2.1

Spectral sparsification of an undirected graph $G = (V, E, w)$, with its Laplacian denoted by $L_G$, can be achieved by leveraging a short-cycle decomposition algorithm. This algorithm produces a sparsified graph $H = (V, E',w')$, where $E' << E$, with its Laplacian denoted by $L_H$, such that for a

Figures (4)

  • Figure 1: The proposed inGRASS algorithm for incremental spectral sparsification. Given the initial input graph $G^{(0)}$ and its sparsifier $H^{(0)}$, inGRASS constructs the updated sparsifiers $H^{(1)}, H^{(2)}, \cdots$ with newly added edges.
  • Figure 2: A 4-level resistance embedding of the initial graph sparsifier achieved through the proposed LRD decomposition. Since the embedding vectors for nodes $5$ and $9$ are $[3,3,3,2]^\top$ and $[6,2,2,2]^\top$, the effective-resistance distance between them is bounded by the resistance diameter of cluster $2$ shown in (d).
  • Figure 3: (a) The original graph featuring three newly introduced edges highlighted in red. (b) The edge included into the graph sparsifier, marked in red, alongside the edges with adjusted weights, denoted in blue.
  • Figure 4: Runtime scalability comparison between GRASS and inGRASS.

Theorems & Definitions (3)

  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2