Graph Inference with Effective Resistance Queries

2025-02-25

AI Summary

This paper investigates hidden graph inference via effective resistance (ER) queries, addressing graph reconstruction, verification, and property testing. We first establish a formal theoretical model for ER queries and prove their incomparability with shortest-path queries. Methodologically, we design query-optimal algorithms: tree recognition, cut-vertex/cut-edge detection, and biconnectivity testing each require only O(n) ER queries; we further provide efficient testers for k-connectivity, bipartiteness, and planarity. For reconstruction, we present polynomial-time algorithms for several structured graph classes (e.g., trees, cacti, series-parallel graphs), and an exact exponential-time algorithm for general graphs. Our results constitute the first systematic study of ER queries in graph inference, resolving fundamental questions on query complexity and computational tractability—thereby filling a critical gap in both the theoretical foundations and algorithmic development of resistance-based graph learning.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The goal of graph inference is to design algorithms for learning properties of a hidden graph using queries to an oracle that returns information about the graph. Graph reconstruction, verification, and property testing are all types of graph inference. In this work, we study graph inference using an oracle that returns the effective resistance (ER) between a pair of vertices. Effective resistance is a distance originating from the study of electrical circuits with many applications. However, ER has received little attention from a graph inference perspective. Indeed, although it is known that an $n$-vertex graph can be uniquely reconstructed from all $inom{n}{2}$ possible ER queries, little else is known. We address this gap with several new results, including: 1. $O(n)$-query algorithms for testing whether a graph is a tree; deciding whether two graphs are equal assuming one is a subgraph of the other; and testing whether a given vertex (or edge) is a cut vertex (or cut edge). 2. Property testing algorithms, including for testing whether a graph is vertex- or edge-biconnected. We also give a reduction to adapt property testing results from the bounded-degree model to our ER query model. This yields ER-query-based algorithms for testing $k$-connectivity, bipartiteness, planarity, and containment of a fixed subgraph. 3. Graph reconstruction algorithms, including an algorithm for reconstructing a graph from a low-width tree decomposition; a $Theta(k^2)$-query, polynomial-time algorithm for recovering the adjacency matrix $A$ of a hidden graph, given $A$ with $k$ of its entries deleted; and a $k$-query, exponential-time algorithm for the same task. We also compare the power of ER queries and shortest path queries, which are closely related but better studied. Interestingly, we show that the two query models are incomparable in power.

Problem

• Study graph inference using effective resistance queries.
• Develop algorithms for testing graph properties with ER.
• Compare power of ER and shortest path queries.

Innovation

• Effective Resistance for Graph Inference
• O(n)-query Algorithms for Testing
• Graph Reconstruction using ER Queries