[Paper Review] Analysis of a Canonical Labeling Algorithm for the Alignment of Correlated Erd\H{o}s-R\'enyi Graphs

Osman Emre Dai, Daniel Cullina+2 more|arXiv (Cornell University)|Apr 25, 2018

Advanced Graph Neural Networks27 references4 citations

TL;DR

This paper presents a seedless canonical labeling algorithm for aligning correlated Erdős-Rényi random graphs, achieving perfect alignment in O(n^{11/5} \log n) time. The method first matches high-degree vertices as anchors, then uses binary signatures of adjacencies to anchors for minimum-distance bipartite matching, proving success under specific edge probability regimes with high probability.

ABSTRACT

Graph alignment in two correlated random graphs refers to the task of identifying the correspondence between vertex sets of the graphs. Recent results have characterized the exact information-theoretic threshold for graph alignment in correlated Erd\H{o}s-R\'enyi graphs. However, very little is known about the existence of efficient algorithms to achieve graph alignment without seeds. In this work we identify a region in which a straightforward $O(n^{11/5} \log n )$-time canonical labeling algorithm, initially introduced in the context of graph isomorphism, succeeds in aligning correlated Erd\H{o}s-R\'enyi graphs. The algorithm has two steps. In the first step, all vertices are labeled by their degrees and a trivial minimum distance alignment (i.e., sorting vertices according to their degrees) matches a fixed number of highest degree vertices in the two graphs. Having identified this subset of vertices, the remaining vertices are matched using a alignment algorithm for bipartite graphs.

Motivation & Objective

• To develop a seedless algorithm for graph alignment in correlated random graphs without relying on pre-aligned vertex pairs.
• To characterize the information-theoretic regime where perfect alignment is achievable using an efficient algorithm.
• To establish theoretical performance bounds for a canonical labeling approach in terms of computational complexity and success probability.
• To bridge the gap between information-theoretic thresholds and practical, efficient algorithms for graph alignment.

Proposed method

• The algorithm uses a two-phase approach: first, it identifies the h highest-degree vertices in both graphs and matches them via degree ordering (the 'anchor' phase).
• In the second phase, each remaining vertex is labeled with a binary signature vector of length h indicating adjacency to the anchor vertices.
• The final alignment is computed via minimum-distance matching between signature vectors in the two graphs, equivalent to bipartite graph matching.
• The method relies on a probabilistic analysis of degree separation and signature distance using Chernoff bounds and generating functions.
• Theoretical guarantees are derived using union bounds over vertex pairs and concentration inequalities to bound failure probabilities.
• The analysis leverages intermediary results on bipartite matching performance under correlated edge distributions.

Experimental results

Research questions

• RQ1 Under what conditions does a canonical labeling algorithm achieve perfect graph alignment in correlated Erdős-Rényi graphs without seeds?
• RQ2 What is the computational complexity of achieving such alignment, and can it be bounded within a subquadratic time regime?
• RQ3 How does the performance of the algorithm depend on the edge probability p and the number of anchor vertices h?
• RQ4 Can the algorithm's success be rigorously characterized using probabilistic methods and concentration inequalities?
• RQ5 What is the role of degree-based anchoring in enabling efficient alignment in the absence of side information?

Key findings

• The algorithm achieves perfect graph alignment with high probability when the edge probability p satisfies p ≥ c \cdot \frac{\log n}{n} for a sufficiently large constant c.
• The algorithm runs in O(n^{11/5} \log n) time, representing a significant improvement over naive approaches.
• The success of the algorithm is guaranteed under a regime where the degree separation between non-anchor vertices exceeds a threshold k, with high probability.
• The probability of failure in the anchor phase is bounded by (2h + 1)e^{-\eta} / (1 - e^{-\eta}) for appropriately chosen η.
• The bipartite matching phase succeeds with probability at least 1 - 2e^{-h\rho^2} where ρ = \sqrt{q_0} - \sqrt{q_1} and q_0, q_1 are derived from edge co-occurrence probabilities.
• The theoretical analysis confirms that the algorithm operates successfully in a regime where prior efficient algorithms were not known to succeed.