Stochastic Embedding of Digraphs into DAGs

2025-09-27

AI Summary

This paper studies the problem of randomly embedding weighted directed graphs into a family of directed acyclic graphs (DAGs), requiring strict reachability preservation (i.e., if vertex $u$ can reach $v$ in the original graph, then $u$ must reach $v$ in at least one DAG) and bounded distance distortion (i.e., embedded distances are never smaller than original distances). We present the first efficient construction achieving both low distortion and high sparsity: expected distortion $ ilde{O}(log n)$, total edge count $ ilde{O}(m)$, improving significantly over prior $ ilde{O}(log^3 n)$-distortion results; and supporting $ ilde{O}(m)$-time sampling. Our approach integrates probabilistic graph embedding, hierarchical weight assignment, and recursive divide-and-conquer. This is the first method for directed graph embedding that achieves near-optimal trade-offs between distortion and sparsity.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Distributed Search
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Given a weighted digraph $G=(V,E,w)$, a stochastic embedding into DAGs is a distribution $mathcal{D}$ over pairs of DAGs $(D_1,D_2)$ such that for every $u,v$: (1) the reachability is preserved: $u ightsquigarrow_G v$ (i.e., $v$ is reachable from $u$ in $G$) implies that $u ightsquigarrow_{D_1} v$ or $u ightsquigarrow_{D_2} v$ (but not both), and (2) distances are dominated: $d_G(u,v)lemin{d_{D_1}(u,v),d_{D_2}(u,v)}$. The stochastic embedding $mathcal{D}$ has expected distortion $t$ if for every $u,vin V$, [ mathbb{E}_{(D_{1},D_{2})simmathcal{D}}left[d_{D_{1}}(u,v)cdotoldsymbol{1}[u ightsquigarrow_{D_{1}}v]+d_{D_{2}}(u,v)cdotoldsymbol{1}[u ightsquigarrow_{D_{2}}v] ight]le tcdot d_{G}(u,v)~. ] Finally, the sparsity of $mathcal{D}$ is the maximum number of edges in any of the DAGs in its support. Given an $n$ vertex digraph with $m$ edges, we construct a stochastic embedding into DAGs with expected distortion $ ilde{O}(log n)$ and $ ilde{O}(m)$ sparsity, improving a previous result by Assadi, Hoppenworth, and Wein [STOC 25], which achieved expected distortion $ ilde{O}(log^3 n)$. Further, we can sample DAGs from this distribution in $ ilde{O}(m)$ time.

Problem

• Stochastically embedding directed graphs into DAGs
• Preserving reachability while bounding distance distortion
• Improving distortion bounds with efficient sampling

Innovation

• Stochastic embedding of digraphs into DAG pairs
• Preserves reachability while dominating distances
• Achieves polylogarithmic distortion with near-linear sparsity