Canonization of a random graph by two matrix-vector multiplications

TL;DR

The paper shows that for a random graph G ∼ G(n,1/2), a canonical labeling is obtained by mapping each vertex x to the 3-way walk-vector w_3^G(x), which can be computed with two matrix-vector multiplications in time essentially linear in the input size. It provides a detailed probabilistic analysis showing w_3^G(x) distinguishes all vertices whp, with a complementary bound establishing that two-step walk counts already collide with probability vanishing as n grows. Beyond the random-graph result, the work compares the WM and CR algorithmic paradigms, proving that CR subsumes WM in terms of the information it uses for canonization and isomorphism testing, and it even constructs a Shrikhande-graph-based example where CR succeeds while WM fails, illustrating fundamental separations. Overall, the paper bridges linear-algebraic and combinatorial approaches to graph canonization, showing that simple walk-based invariants can yield near-linear-time canonical labeling for almost all graphs and clarifying the relative strengths of WM and CR.

Abstract

We show that a canonical labeling of a random $n$-vertex graph can be obtained by assigning to each vertex $x$ the triple $(w_1(x),w_2(x),w_3(x))$, where $w_k(x)$ is the number of walks of length $k$ starting from $x$. This takes time $O(n^2)$, where $n^2$ is the input size, by using just two matrix-vector multiplications. The linear-time canonization of a random graph is the classical result of Babai, Erdős, and Selkow. For this purpose they use the well-known combinatorial color refinement procedure, and we make a comparative analysis of the two algorithmic approaches.

Figures (3)

• Figure 1: Two colored versions of the Shrikhande graph. Unexposed edges are obtainable by identification of the arrows according to the standard square representation of a torus.
• Figure 2: The colorings of $A$ and $B$ after the first color refinement round. For each $i=1,2,3$, the vertices $a_i$ and $b_i$ have the same unique color. The color of each non-individualized vertex is determined by its adjacency to the individualized vertices. For example, the color of a vertex in $A$ means that this vertex is adjacent to $a_1$ and $a_2$ but not to $a_3$.
• Figure 3: Construction of $G$.

Theorems & Definitions (11)

• Theorem 1
• Corollary 2
• Theorem 3
• Lemma 4: Chernoff's bound; see, e.g., AlonS16
• Lemma 5
• proof
• Lemma 6: The local de Moivre--Laplace theorem; see, e.g., Feller
• Remark 7
• Proposition 8: An ILT-to-LLT conversion (Mukhin Mukhin)
• Lemma 9