Graph-Skeleton: ~1% Nodes are Sufficient to Represent Billion-Scale Graph

TL;DR

This paper proposes a novel Graph-Skeleton model, which properly fetches the background nodes, and further condenses the semantic and topological information of background nodes within similar target-background local structures, and achieves highly comparable performance while only containing 1.8% nodes of the original graph.

Abstract

Due to the ubiquity of graph data on the web, web graph mining has become a hot research spot. Nonetheless, the prevalence of large-scale web graphs in real applications poses significant challenges to storage, computational capacity and graph model design. Despite numerous studies to enhance the scalability of graph models, a noticeable gap remains between academic research and practical web graph mining applications. One major cause is that in most industrial scenarios, only a small part of nodes in a web graph are actually required to be analyzed, where we term these nodes as target nodes, while others as background nodes. In this paper, we argue that properly fetching and condensing the background nodes from massive web graph data might be a more economical shortcut to tackle the obstacles fundamentally. To this end, we make the first attempt to study the problem of massive background nodes compression for target nodes classification. Through extensive experiments, we reveal two critical roles played by the background nodes in target node classification: enhancing structural connectivity between target nodes, and feature correlation with target nodes. Followingthis, we propose a novel Graph-Skeleton1 model, which properly fetches the background nodes, and further condenses the semantic and topological information of background nodes within similar target-background local structures. Extensive experiments on various web graph datasets demonstrate the effectiveness and efficiency of the proposed method. In particular, for MAG240M dataset with 0.24 billion nodes, our generated skeleton graph achieves highly comparable performance while only containing 1.8% nodes of the original graph.

Figures (11)

• Figure 1: (a) Explorations of background nodes influences. Upper: results on DGraph dgraph. Lower: results on ogbn-arxiv ogb_dataset. (****$p$ < 1e-4, **$p$ < 1e-2, paired t-tests; errorbars represent the standard deviation). (b) Feature correlation between target nodes and their neighboring background nodes. (c) Illustration of target nodes and the corresponding essential background nodes.
• Figure 2: Graph-Skeleton framework: it generates a synthetic skeleton subgraph from original graph with rich information for target prediction while enjoying the benefits of small scale. It first fetches the essential background nodes under the guidance of structural connectivity and feature correlation (Left), then condenses the information of background nodes (Middle). The generated skeleton graph is highly informative and friendly for storage and graph model deployment (Right).
• Figure 3: Illustration of Strategy-$\alpha$, where the background nodes sharing a identical structural multiple-set ($MSS$) $\{\left\langle T, d\rangle\right\}$ (within a same shadow envelope) will be condensed into one synthetic node.
• Figure 4: Illustration of Strategy-$\beta$. The background nodes sharing the same structural multiple-set $MSS^{'}$$\{\left\langle T\rangle\right\}$ (within the same shadow envelope) will be condensed into one node. To maintain the relative distance information between different nodes, we encode the distance information of target nodes by weighting the edges of skeleton graph.
• Figure 5: Illustration of Strategy-$\gamma$. Based on condensation-$\beta$, we further condense the affiliation nodes to the corresponding target node.

Theorems & Definitions (6)

• definition 1: Node Pair Equivalence Class
• definition 2: Linear message passing operation
• definition 3: Linear message path passing
• definition 4: LMPP equivalence class
• proposition 1
• definition 5: Linear message path passing